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Class 11 Chemistry Revision Notes for Structure of Atom of Chapter 2

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• Discovery of Electron—Discharge Tube Experiment
In 1879, William Crooks studied the conduction of electricity through gases at low pressure. He performed the experiment in a discharge tube which is a cylindrical hard glass tube about 60 cm in length. It is sealed at both the ends and fitted with two metal electrodes as shown in Fig. 2.1.

The electrical discharge through the gases could be observed only at very low pressures and at very high voltages.
The pressure of different gases could be adjusted by evacuation. When sufficiently high voltage is applied across the electrodes, current starts flowing through a stream of particles moving in the tube from the negative electrode (cathode) to the positive electrode (anode). These were called cathode rays or cathode ray particles.
• Properties of Cathode Rays
(i) Cathode rays travel in straight line.
(ii) Cathode rays start from cathode and move towards the anode.
(iii) These rays themselves are not visible but their behaviour can be observed with the help of certain kind of materials (fluorescent or phosphorescent) which glow when hit by them.
(iv) Cathode rays consist of negatively charged particles. When electric field is applied on the cathode rays with the help of a pair of metal plates, these are found to be deflected towards the positive plate indicating the presence of negative charge.
(v) The characteristics of cathode rays do not depend upon the material of electrodes and the nature of gas present in the cathode ray’tube.
• Determination of Charge/Mass (elm) Ratio for Electrons
J. J. Thomson for the first time experimentally determined charge/mass ratio called elm ratio for the electrons. For this, he subjected the beam of electrons released in the discharge tube as cathode rays to influence the electric and magnetic fields. These were acting perpendicular to one another as well as to the path followed by electrons.
According to Thomson, the amount of deviation of the particles from their path in presence of electrical and magnetic field depends upon following factors:
(i) Greater the magnitude of the charge on the particle, greater is the interaction with the electric or magnetic field and thus greater is the deflection.
(ii) The mass of the particle — lighter the particle, greater the deflection.
(iii) The deflection of electrons from their original path increases with the increase in the voltage across the electrodes or strength of the magnetic field.
By carrying out accurate measurements on the amount of deflections observed by the electrons on the electric field strength or magnetic field strength, Thomson was able to determine the value of
e/me = 1.758820 x 10¹¹ C kg^-1 where me = Mass of the electron in kg
e = magnitude of charge on the electron in coulomb (C).
• Charge on the Electron
R.A. Millikan devised a method known as oil drop experiment to determine the charge on the electrons.

• Discovery of Proton—Anode Rays
In 1886, Goldstein modified the discharge tube by using a perforated cathode. On reducing the pressure, he observed a new type of luminous rays passing through the holes or perforations of the cathode and moving in a direction opposite to the cathode rays. These rays were named as positive rays or anode rays or as canal rays. Anode rays are not emitted from the anode but from a space between anode and cathode.
• Properties of Anode Rays
(i) The value of positive charge (e) on the particles constituting anode rays depends upon the nature of the gas in the discharge tube.
(ii) The charge to mass ratio of the particles is found to depend on the gas from which these originate.
(iii) Some of the positively charged particles carry a multiple of the fundamental unit of electrical charge.
(iv) The behaviour of these particles in the magnetic or electric field is opposite to that observed for electron or cathode rays.
• Proton
The smallest and lightest positive ion was obtained from hydrogen and was called proton. Mass of proton = 1.676 x 10^-27 kg
Charge on a proton = (+) 1.602 x 10^-19 C
• Neutron
It is a neutral particle. It was discovered by Chadwick (1932).
By the bombardment of thin sheets of beryllium with fast moving a-particles he observed • that highly penetrating rays consist of neutral particles which were named neutrons.
• Thomson Model of Atom

(i) J. J. Thomson proposed that an atom may be regarded as a sphere of approximate radius 1CT8 cm carrying positive charge due to protons and in which negatively charged electrons are embedded.
(ii) In this model, the atom is visualized as a pudding or cake of positive charge with electrons embedded into it.
(iii) The mass of atom is considered to be evenly spread over the atom according to this model.
Drawback of Thomson Model of Atom
This model was able to explain the overall neutrality of the atom, it could not satisfactorily, explain the results of scattering experiments carried out by Rutherford in 1911.
• Rutherford’s a-particle Scattering Experiment
Rutherford in 1911, performed some scattering experiments in which he bombarded thin foils of metals like gold, silver, platinum or copper with a beam of fast moving a-particles. The thin gold foil had a circular fluorescent zinc sulphide screen around it. Whenever a-particles struck the screen, a tiny flash of light was produced at that point.
From these experiments, he made the following observations:
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(i) Most of the a-particles passed through the foil without undergoing any deflection,
(ii) A few a-particles underwent deflection through small angles.
(iii) Very few mere deflected back i.e., through an angle of nearly 180°.
From these observations, Rutherford drew the following conclusions:
(i) Since most of the a-particles passed through the foil without undergoing any deflection, there must be sufficient empty space within the atom.
(ii) A small fraction of a-particles was deflected by small angles. The positive charge has to be concentrated in a very small volume that repelled and deflected a few positively charged a-particles. This very small portion of the atom was called nucleus.
(iii) The volume of nucleus is very small as compared to total volume of atom.
• Rutherford’s Nuclear Model of an Atom
(i) The positive charge and most of the mass of the atom was densely concentrated in an extremely small region. This very small portion of the atom was called nucleus by Rutherford.
(ii) The nucleus is surrounded by electrons that move around the nucleus with a very high speed in circular paths called orbits.
(iii) Electrons and nucleus are held together by electrostatic forces of attraction.
• Atomic Number
The number of protons present in the nucleus is equal to the atomic number (z). For example, the number of protons in the hydrogen nucleus is 1, in sodium atom it is 11, therefore, their atomic numbers are 1 and 11. In order to keep the electrical neutrality, the number of electrons in an atom is equal to the number of protons (atomic number, z). For example, number of electrons in hydrogen atom and sodium atom are 1 and 11 respectively.
Atomic Number (z) = Number of protons in the nucleus of an atom.
= Number of electrons in a neutral atom.
• Mass Number
Number of protons and neutrons present in the nucleus are collectively known as nucleons. The total number of nucleons is termed as mass number (A) of the atom.
Mass Number (A) = Number of protons (p) + Number of neutrons (n).
• Isotopes
Atoms with identical atomic number but different atomic mass number are known as Isotopes.
Isotopes of Hydrogen:

These three isotopes are shown in the figure below:


Characteristics of Isotopes
(i) Since the isotopes of an element have the same atomic number, but different mass number, the nuclei of isotopes contain the same number of protons, but different number of neutrons.
(ii) Since, the isotopes differ in their atomic masses, all the properties of the isotopes depending upon the mass are different.
(iii) Since, the chemical properties are mainly determined by the number of protons in the nucleus, and the number of electrons in the atom, the different isotopes of an element exhibit similar chemical properties. For example, all the isotopes of carbon on burning give carbon dioxide.
• Isobars

• Drawbacks of Rutherford Model
(i) When a body is moving in an orbit, it achieves acceleration. Thus, an electron moving around nucleus in an orbit is under acceleration.
According to Maxwell’s electromagnetic theory, charged particles when accelerated must emit electromagnetic radiations. Therefore, an electron in an orbit will emit radiations, the energy carried by radiation comes from electronic motion. Its path will become closer to nucleus and ultimately should spiral into nucleus within . 10-8 s. But actually this does not happen.
Thus, Rutherford’s model cannot explain the stability of atom if the motion of electrons is described on the basis of classical mechanics and electromagnetic theory.
(ii) Rutherford’s model does not give any idea about distribution of electrons around the nucleus and about their energies.
• Developments Leading to the Bohr’s Model of Atom
Two developments played a major role in the formulation of Bohr’s model of atom. These were:
(i) Dual character of the electromagnetic radiation which means that radiations possess both wave like and particle like properties.
(ii) Experimental results regarding atomic spectra which can be explained only by assuming quantized electronic energy levels in atoms.
• Nature of Electromagnetic Radiation (Electromagnetic Wave Theory)
This theory was put forward by James Clark Maxwell in 1864. The main points of this theory are as follows:
(i) The energy is emitted from any source (like the heated rod or the filament of a bulb through which electric current is passed) continuously in the form of radiations and is called the radiant energy.
(ii) The radiations consist of electric and magnetic fields oscillating perpendicular to each other and both perpendicular to the direction of propagation of the radiation.
(iii) The radiations possess wave character and travel with the velocity of light 3 x 10⁸ m/sec.
(iv) These waves do not require any material medium for propagation. For example, rays from the sun reach us through space which is a non-material medium.
• Characteristics of a Wave
Wavelength: It is defined as the distance between any two consecutive crests or troughs. It is represented by X and its S.I. unit is metre.

Frequency: Frequency of a wave is defined as the number of waves passing through a point in one second. It is represented by v (nu) and is expressed in Hertz (Hz).
1 Hz = 1 cycle/sec.
Velocity: Velocity of a wave is defined as the linear distance travelled by the wave in one second.
It is represented by c and is expressed in cm/sec or m/sec.
Amplitude: Amplitude of a wave is the height of the crest or the depth of the through. It is represented by V and is expressed in the units of length.
Wave Number: It is defined as the number of waves present in 1 metre length. Evidently it will be equal to the reciprocal of the wavelength. It is represented by bar v (read as nu bar).

Electromagnetic Spectrum: When electromagnetic radiations are arranged in order of their increasing wavelengths or decreasing frequencies, the complete spectrum obtained is called electromagnetic spectrum.

• Limitations of Electromagnetic Wave Theory
Electromagnetic wave theory was successful in explaining properties of light such as interference, diffraction etc; but it could not explain the following:
(i) The phenomenon of black body radiation.
(ii) The photoelectric effect.
(iii) The variation of heat capacity of solids as a function of temperature.
(iv) The line spectra of atoms with reference to hydrogen.
• Black Body Radiation
The ideal body, which emits and absorbs all frequencies is called a black body and the radiation emitted by such a body is called black body radiation. The. exact frequency distribution of the emitted radiation from a black body depends only on its temperature.

At a given temperature, intensity of radiation emitted increases with decrease of wavelength, reaches a maximum value at a given wavelength and then starts decreasing with further decrease of wavelength as shown in Fig 2.6.
• Planck’s Quantum Theory
To explain the phenomenon of ‘Black body radiation’ and photoelectric effect, Max Planck in 1900, put forward a theory known as Planck’s Quantum Theory.
This theory was further extended by Einstein in 1905. The main points of this theory was as follows: ,
(i) The radiant energy emitted or absorbed in the form of small packets of energy. Each such packets of energy is called a quantum.
(ii) The energy of each quantum is directly proportional to the frequency of the radiation

where h is a proportionality constant, called Planck’s constant. Its value is equal to 6.626 x 10^-34 Jsec.
• Photoelectric Effect
Hertz, in 1887, discovered that when a beam of light of certain frequency strikes the surface of some metals, electrons are emitted or ejected from the metal surface. The phenomenon is called photoelectric effect.

Observations in Photoelectric Effect
(i) Only photons of light of certain minimum frequency called threshold frequency (v₀) can cause the photoelectric effect. The value of v₀ is different for different metals.
(ii) The kinetic energy of the electrons which are emitted is directly proportional to the frequency of the striking photons and is quite independent of their intensity.
(iii) The number of electrons that are ejected per second from the metal surface depends upon the intensity of the striking photons or radiations and not upon their frequency.
Explanation of Photoelectric Effect
Einstein in (1905) was able to give an explanation of the different points of the photoelectric effect using Planck’s quantum theory as under:
(i) Photoelectrons are ejected only when the incident light has a certain minimum frequency (threshold frequency v₀)
(ii) If the frequency of the incident light (v) is more than the threshold frequency (v₀), the excess energy (hv – hv₀) is imparted to the electron as kinetic energy.
K.E. of the ejected electron

energy of the emitted electron.
(iii) On increasing the intensity of light, more electrons are ejected but the energies of the electrons are not altered.
• Dual Behaviour of Electromagnetic Radiation
From the study of behaviour of light, scientists came to the conclusion that light and other electromagnetic radiations have dual nature. These are wave nature as well as particle nature. Whenever radiation interacts with matter, it displays particle like properties in contrast to the wavelike properties (interference and diffraction) which it exhibits when it propagates. Some microscopic particles, like electrons, also exhibit this wave-particle duality.
• Spectrum
When a ray of white light is passed through a prism the wave with shorter wavelength bends more than the one with a longer wavelength. Since ordinary white light consists of waves with all the wavelengths in the visible range, array of white light is spread out into a series of coloured bands called spectrum. The light of red colour which has longest wavelength is deviated the least while the violet light, which has shortest wavelength is deviated the most.
Continuous Spectrum
When a ray of white light is analysed by passing through a prism it is observed that it splits up into seven different wide bands of colours from violet to red (like rainbow). These colours are so continuous that each of them merges into the next. Hence, the spectrum is called continuous spectrum.
Emission Spectra
Emission Spectra is noticed when the radiations emitted from a source are passed through a prism and then received on the photographic plate. Radiations can be emitted in a number of ways such as:
(i) from sun or glowing electric bulb.
(ii) by passing electric discharge through a gas at low pressure.
(iii) by heating a substance to high temperature.
Line Spectra
When the vapours of some volatile substance are allowed to fall on the flame of a Bunsen burner and then analysed with the help of a spectroscope. Some specific coloured lines appear on the photographic plate which are different for different substances. For example, sodium or its salts emit yellow light while potassium or its salts give out violet light.
Absorption Spectra
When white light is passed through the vapours of a substance and the transmitted light is then allowed to strike a prism, dark lines appear in the otherwise continuous spectrum. The dark lines indicate that the radiations corresponding to them were absorbed by the substance from the white light. This spectrum is called absorption spectrum.
Dark lines appear exactly at the same positions where the lines in the emission spectra appear.
• Line Spectrum of Hydrogen
When electric discharge is passed through hydrogen gas enclosed in discharge tube under low pressure and the emitted light is analysed by a spectroscope, the spectrum consists of a large number of lines which are grouped into different series. The complete spectrum is known as hydrogen spectrum.
On the basis of experimental observations, Johannes Rydberg noted that all series of lines in the hydrogen spectrum could be described by the following expression:

Rydberg in 1890, and has given a simple theoretical equation for the calculation of wavelengths and wave numbers of the spectral lines in different series of hydrogen spectrum. The equation is known as Rydberg formula (or equation).

This relation is valid for hydrogen atom only. For other species,

where Z is the atomic number of the species.
Here RH = constant, called Rydberg constant for hydrogen and n1 , n2 are integers (n2 > n1)
For any particular series, the value of n1 is constant while that of n2 changes. For example,
For Lyman series, n₁= 1, n₂= 2, 3, 4, 5………..
For Balmer series, n₁ = 2, n₂ = 3, 4, 5, 6………..
For Paschen series, n₁= 3, n₂ = 4, 5, 6, 7………..
For Brackett series,n₁ = 4, n₂ = 5, 6, 7, 8………..
For Pjund series, n₁ =5, n₂ = 6, 7, 8, 9………..
Thus, by substituting the values of n₁ and n₂ in the above equation, wavelengths and wave number of different spectral lines can be calculated. When n₁ = 2, the expression given above is called Balmer’s formula.
• Bohr’s Model of Atom
Niels Bohr in 1913, proposed a new model of atom on the basis of Planck’s Quantum Theory. The main points of this model are as follows:
(i) In an atom, the electrons revolve around the nucleus in certain definite circular paths called orbits.
(ii) Each orbit is associated with definite energy and therefore these are known as energy
levels or energy shells. These are numbered as 1, 2, 3, 4……….. or K, L, M, N………..
(iii) Only those energy orbits are permitted for the electron in which angular momentum of the electron is a whole number multiple of h/2π
Angular momentum of electron (mvr) = nh/2π (n = 1, 2, 3, 4 etc).
m = mass of the electron.
v = tangential velocity of the revolving electron.
r = radius of the orbit.
h = Planck’s constant.
n is an integer.
(iv) As long as electron is present in a particular orbit, it neither absorbs nor loses energy and its energy, therefore, remains constant.
(v) When energy is supplied to an electron, it absorbs energy only in fixed amounts as quanta and jumps to higher energy state away from the nucleus known as excited state. The excited state is unstable, the electron may jump back to the lower energy state and in doing so, it emits the same amount of energy. (∆E = E₂ – E₁).
• Achievements of Bohr’s Theory
1. Bohr’s theory has explained the stability of an atom.
2. Bohr’s theory has helped in calculating the energy of electron in hydrogen atom and one electron species. The mathematical expression for the energy in the nth orbit is,


3. Bohr’s theory has explained the atomic spectrum of hydrogen atom.
• Limitations of Bohr’s Model
(i) The theory could not explain the atomic spectra of the atoms containing more than one electron or multielectron atoms.
(ii) Bohr7s theory failed to explain the fine structure of the spectral lines.
(iii) Bohr’s theory could not offer any satisfactory explanation of Zeeman effect and Stark effect.
(iv) Bohr’s theory failed to explain the ability of atoms to form molecule formed by chemical bonds.
(v) It was not in accordance with the Heisenberg’s uncertainty principle.
• Dual Behaviour of Matter (de Broglie Equation)
de Broglie in 1924, proposed that matter, like radiation, should also exhibit dual behaviour i.e., both particle like and wave like properties. This means that like photons, electrons also have momentum as well as wavelength.
From this analogy, de Broglie gave the following relation between wavelength (λ) and momentum (p) of a material particle.

• Heisenberg’s Uncertainty Principle
It states that, “It is impossible to determine simultaneously, the exact position and exact momentum (or velocity) of an electron”.

• Significance of Uncertainty Principle
(i) It rules out existence of definite paths or trajectories of electrons and other similar particles.
(ii) The effect of Heisenberg’s uncertainty principle is significant only for microscopic objects and is negligible for macroscopic objects.
• Reasons for the Failure of Bohr Model
(i) The wave character of the electron is not considered in Bohr Model.
(ii) According to Bohr Model an orbit is a clearly defined path and this path can completely be defined only if both the position and the velocity of the electron are known exactly at the same time. This is not possible according to the Heisenberg’s uncertainty principle.
• Quantum Mechanical Model of Atom
Quantum mechanics: Quantum mechanics is a theoretical science that deals with the study of the motions of the microscopic objects that have both observable wave like and particle like properties.
Important Features of Quantum Mechanical Model of Atom
(i) The energy of electrons in atom is quantized i.e., can only have certain values.
(ii) The existence of quantized electronic energy level is a direct result of the wave like properties of electrons.
(iii) Both the exact position and exact velocity of an electron in an atom cannot be determined simultaneously.
(iv) An atomic orbital has wave function φ. There are many orbitals in an atom. Electron occupy an atomic orbital which has definite energy. An orbital cannot have more than two electrons. The orbitals are filled in increasing order of energy. All the information about the electron in an atom is stored in orbital wave function φ.
(v) The probability of finding electron at a point within an atom is proportional to square of orbital wave function i.e., |φ²|at that point. It is known as probability density and is always positive.
From the value of φ² at different points within atom, it is possible to predict the region around the nucleus where electron most probably will be found.
• Quantum Numbers
Atomic orbitals can be specified by giving their corresponding energies and angular momentums which are quantized (i.e., they have specific values). The quantized values can be expressed in terms of quantum number. These are used to get complete information about electron i.e., its location, energy, spin etc.
Principal Quantum Number (n)
It is the most important quantum number since it tells the principal energy level or shell to which the electron belongs. It is denoted by the letter V and can have any integral value except zero, i.e., n = 1, 2, 3, 4……….. etc.
The various principal energy shells are also designated by the letters, K, L, M, N, O, P ….. etc. Starting from the nucleus.
The principal quantum number gives us the following information:
(i) It gives the average distance of the electron from the nucleus.
(ii) It completely determines the energy of the electron in hydrogen atom and hydrogen like particles.
(iii) The maximum number of electrons present in any principal shell is given by 2n² where n is the number of the principal shell.
Azimuthal or Subsidiary or Orbital Angular Quantum Number (l)
It is found that the spectra of the elements contain not only the main lines but there are many fine lines also present. This number helps to explain the fine lines of the spectrum.
The azimuthal quantum number gives the following information:
(i) The number of subshells present in the main shell.
(ii) The angular momentum of the electron present in any subshell.
(in) The relative energies of various subshells.
(iv) The shapes of the various subshells present within the same principal shell.
This quantum number is denoted by the letter T. For a given value of n, it can have any value ranging from 0 to n – 1. For example,
For the 1st shell (k), n = 1, l can have only one value i.e., l = 0 For n = 2, the possible value of l can be 0 and 1.
Subshells corresponding to different values of l are represented by the following symbols:
value of l 0 1 2 3 4 5 ……………..
Notation for subshell s p d f g h ………………..
Magnetic Orbital Quantum Number (m or m1)
The magnetic orbital quantum number determines the number of preferred orientations of the electrons present in a subshell. Since each orientation corresponds to an orbital, therefore, the magnetic orbital quantum number determines the number of orbitals present in any subshell.
The magnetic quantum number is denoted by letter m or ml and for a given value of l, it can have all the values ranging from – l to + l including zero.
Thus, for energy value of l, m has 2l + 1 values.
For example,
For l = 0 (s-subshell), ml can have only one value i.e., m1 = 0.
This means that s-subshell has only one orientation in space. In other words, s-subshell has only one orbital called s-orbital.
Spin Quantum Number (S or ms)
This quantum number helps to explain the magnetic properties of the substances. A spinning electron behaves like a micromagnet with a definite magnetic moment. If an orbital contains two electrons, the two magnetic moments oppose and cancel each other.
• Shapes of s-orbitals
s-orbital is present in the s-subshell. For this subshell, l = 0 and ml = 0. Thus, s-orbital with only one orientation has a spherical shape with uniform electron density along all the three axes.
The probability of Is electron is found to be maximum near the nucleus and decreases with the increase in the distance from the nucleus. In 2s electron, the probability is also maximum near the nucleus and decreases to zero probability. The spherical empty shell for 2s electron is called nodal surface or simply node.

• Shapes of p-orbitals
p-orbitals are present in the p-subshell for which l = 1 and m₁ can have three possible orientations – 1, 0, + 1.
Thus, there are three orbitals in the p-subshell which are designated as p_x, p_y and p_z orbitals depending upon the axis along which they are directed. The general shape of a p-orbital is dumb-bell consisting of two portions known as lobes. Moreover, there is a plane passing through the nucleus along which finding of the electron density is almost nil. This is known as nodal plane as shown in the fig.

From the dumb-bell pictures, it is quite obvious that unlike s-orbital, a p-orbital is directional in nature and hence it influences the shapes of the molecules in the formation of which it participates.
• Shapes of d-orbitals
d-orbitals are present in d-subshell for which l = 2 and m[ = -2, -1, 0, +1 and +2. This means that there are five orientations leading to five different orbitals.

• Aufbau Principle
The principle states: In the ground state of the atoms, the orbitals are filled in order of their increasing energies.
In other words, electrons first occupy the lowest energy orbital available to them and enter into higher energy orbitals only after the lower energy orbitals are filled.
The order in which the energies of the orbitals increase and hence the order in which the orbitals are filled is as follows:
Is, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, id, 5p, 6s, if, 3d, 6p, 7s, 5f 6d, 7p
The order may be remembered by using the method given in fig. 2.11.

• Pauli Exclusion Principle
According to this principle, no two electrons in an atom can have the same set of four quantum numbers.
Pauli exclusion principle can also be stated as: Only two electrons may exist in the same orbital and these electrons must have opposite spins.
• Hund’s Rule of Maximum Multiplicity
It states that: pairing of electrons in the orbitals belonging to the same subshell (p, d or f) does not take place until each orbital belonging to that subshell has got one electron each i.e., it is singly occupied.
• Electronic Configuration of Atoms
The distribution of electrons into orbitals of an atom is called its electronic configuration. The electronic configuration of different atoms can be represented in two ways.
For example:

• Causes of Stability of Completely Filled and Half Filled Subshells
The completely filled and half filled subshells are stable due to the following reasons:

1. Symmetrical distribution of electrons: The completely filled or half filled subshells have symmetrical distribution of electrons in them and are therefore more stable.
2. The stabilizing effect arises whenever two or more electrons with same spin are present in the degenrate orbitals of a subshell. These electrons tend to exchange their positions
and the energy released due to their exchange is called exchange energy. The number of exchanges that can takes place is maximum when the subshell is either half filled or completely filled.
-As a result the exchange energy is maximum and so is the stability.

Class 11 Chemistry Revision Notes for Some Basic Concepts of Chemistry of Chapter 1

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• Importance of Chemistry
Chemistry has a direct impact on our life and has wide range of applications in different fields. These are given below:
(A) In Agriculture and Food:
(i) It has provided chemical fertilizers such as urea, calcium phosphate, sodium nitrate, ammonium phosphate etc.
(ii) It has helped to protect the crops from insects and harmful bacteria, by the use ‘ of certain effective insecticides, fungicides and pesticides.
(iii) The use of preservatives has helped to preserve food products like jam, butter, squashes etc. for longer periods.
(B) In Health and Sanitation:
(i) It has provided mankind with a large number of life-saving drugs. Today, dysentery and pneumonia are curable due to discovery of sulpha drugs and penicillin life-saving drugs. Cisplatin and taxol have been found to be very effective for cancer therapy and AZT (Azidothymidine) is used for AIDS victims.
(ii) Disinfectants such as phenol are used to kill the micro-organisms present in drains, toilet, floors etc.
(iii) A low concentration of chlorine i.e., 0.2 to 0.4 parts per million (ppm) is used ’ for sterilization of water to make it fit for drinking purposes.
(C) Saving the Environment:
The rapid industrialisation all over the world has resulted in lot of pollution.
Poisonous gases and chemicals are being constantly released in the atmosphere. They are polluting environment at an alarming rate. Scientists are working day and night to develop substitutes which may cause lower pollution. For example, CNG (Compressed Natural Gas), a substitute of petrol, is very effective in checking pollution caused by automobiles.
(D) Application in Industry:
Chemistry has played an important role in developing many industrially ^ manufactured fertilizers, alkalis, acids, salts, dyes, polymers, drugs, soaps,
detergents, metal alloys and other inorganic and organic chemicals including new materials contribute in a big way to the national economy.
• Matter
Anything which has mass and occupies space is called matter.
For example, book, pencil, water, air are composed of matter as we know that they have
mass and they occupy space.
• Classification of Matter
There are two ways of classifying the matter:
(A) Physical classification
(B) Chemical classification
(A) Physical Classification:
Matter can exist in three physical states:
1. Solids 2. Liquids 3. Gases
1. Solids: The particles are held very close to each other in an orderly fashion and there is not much freedom of movement.
Characteristics of solids: Solids have definite volume and definite shape.
2. Liquids: In liquids, the particles are close to each other but can move around. Characteristics of liquids: Liquids have definite volume but not definite shape.
3. Gases: In gases, the particles are far apart as compared to those present in solid or liquid states. Their movement is easy and fast.
Characteristics of Gases: Gases have neither definite volume nor definite shape. They completely occupy the container in which they are placed.
(B) Chemical Classification:
Based upon the composition, matter can be divided into two main types:
1. Pure Substances 2. Mixtures.
1. Pure substances: A pure substance may be defined as a single substance (or matter) which cannot be separated by simple physical methods.
Pure substances can be further classified as (i) Elements (ii) Compounds
(i) Elements: An element consists of only one type of particles. These particles may be atoms or molecules.
For example, sodium, copper, silver, hydrogen, oxygen etc. are some examples of elements. They all contain atoms of one type. However, atoms of different elements are different in nature. Some elements such as sodium . or copper contain single atoms held together as their constituent particles whereas in some others two or more atoms combine to give molecules of the element. Thus, hydrogen, nitrogen and oxygen gases consist of molecules in which two atoms combine to give the respective molecules of the element.
(ii) Compounds: It may be defined as a pure substance containing two or more elements combined together in a fixed proportion by weight and can be decomposed into these elements by suitable chemical methods. Moreover, the properties of a compound are altogether different from the constituting elements.
The compounds have been classified into two types. These are:
(i) Inorganic Compounds: These are compounds which are obtained from non-living sources such as rocks and minerals. A few
examples are: Common salt, marble, gypsum, washing soda etc.
(ii) Organic Compounds are the compounds which are present in plants and animals. All the organic compounds have been found to contain carbon as their essential constituent. For example, carbohydrates, proteins, oils, fats etc.
2. Mixtures: The combination of two or more elements or compounds which are not chemically combined together and may also be present in any proportion, is called mixture. A few examples of mixtures are: milk, sea water, petrol, lime water, paint glass, cement, wood etc.
Types of mixtures: Mixtures are of two types:
(i) Homogeneous mixtures: A mixture is said to be homogeneous if it has a uniform composition throughout and there are no visible boundaries of separation between the constituents.
For example: A mixture of sugar solution in water has the same sugar water composition throughout and all portions have the same sweetness.
(ii) Heterogeneous mixtures: A mixture is said to be heterogeneous if it does not have uniform composition throughout and has visible boundaries of separation between the various constituents. The different constituents of a heterogeneous mixture can be seen even with naked eye.
For example: When iron filings and sulphur powder are mixed together, the mixture formed is heterogeneous. It has greyish-yellow appearance and the two constituents, iron and sulphur, can be easily identified with naked eye.
• Differences between Compounds and Mixtures
Compounds
1. In a compound, two or more elements are combined chemically.
2. In a compound, the elements are present in the fixed ratio by mass. This ratio cannot change.
3. CompoUnds are always homogeneous i.e., they havethe same composition throughout.
4 In a compound, constituents cannot be separated by physical methods
5. In a compound, the constituents lose their identities i.e., i compound does not show the characteristics of the constituting elements.
Mixtures
1. In a mixture, or more elements or compounds are simply mixed and not combined chemically.
2. In a mixture the constituents are not present in fixed ratio. It can vary
3. Mixtures may be either homogeneous or heterogeneous in nature.
4. Constituents of mixtures can be separated by physical methods.
5, In a mixture, the constituents do not lose their identities i.e., a mixture shows the characteristics of all the constituents .
We have discussed the physical and chemical classification of matter. A flow sheet representation of the same is given below.

• Properties of Matter and Their Measurements
Physical Properties: Those properties which can be measured or observed without changing the identity or the composition of the substance.
Some examples of physical properties are colour, odour, melting point, boiling point etc. Chemical Properties: It requires a chemical change to occur. The examples of chemical properties are characteristic reactions of different substances. These include acidity, basicity, combustibility etc.
• Units of Measurement
Fundamental Units: The quantities mass, length and time are called fundamental quantities and their units are known as fundamental units.
There are seven basic units of measurement for the quantities: length, mass, time, temperature, amount of substance, electric current and luminous intensity.
Si-System: This system of measurement is the most common system employed throughout the world.
It has given units of all the seven basic quantities listed above.

• Definitions of Basic SI Units
1. Metre: It is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.
2. Kilogram: It is the unit of mass. It is equal to the mass of the international prototype
of the kilogram. ,
3. Second: It is the duration of 9192631, 770 periods of radiation which correspond to the transition between the two hyper fine levels of the ground state of caesium- 133 atom.
4. Kelvin: It is the unit of thermodynamic temperature and is equal to 1/273.16 of the thermodynamic temperature of the triple point of water.
5. Ampere: The ampere is that constant current which if maintained in two straight parallel conductors of infinite length, of negligible circular cross section and placed, 1 metre apart in vacuum, would produce between these conductors a force equal to 2 x 10-7 N per metre of length.
6. Candela: It may be defined as the luminous intensity in a given direction, from a source which emits monochromatic radiation of frequency 540 x 1012 Hz and that has a radiant intensity in that direction of 1/ 683 watt per steradian.
7. Mole: It is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon -12. Its symbol is ‘mol’.
• Mass and Weight
Mass: Mass of a substance is the amount of matter present in it.
The mass of a substance is constant.
The mass of a substance can be determined accurately in the laboratory by using an analytical
balance. SI unit of mass is kilogram.

Weight: It is the force exerted by gravity on an object. Weight of substance may vary from one place to another due to change in gravity.
Volume: Volume means the space occupied by matter. It has the units of (length)3. In SI units, volume is expressed in metre3 (m3). However, a popular unit of measuring volume, particularly in liquids is litre (L) but it is not in SI units or an S.I. unit.
Mathematically,
1L = 1000 mL = 1000 cm3 = 1dm3.
Volume of liquids can be measured by different devices like burette, pipette, cylinder, measuring flask etc. All of them have been calibrated.

Temperature: There are three scales in which temperature can be measured. These are known as Celsius scale (°C), Fahrenheit scale (°F) and Kelvin scale (K).

-> Thermometres with Celsius scale are calibrated from 0°C to 100°C.
-> Thermometres with Fahrenheit scale are calibrated from 32°F to 212°F.
-> Kelvin’scale of temperature is S.I. scale and is very common these days.Temperature on this scale is shown by the sign K.
The temperature on two scales are related to each other by the relationship

Density: Density of a substance is its amount of mass per unit volume. So, SI unit of density can be obtained as follows:

This unit is quite large and a chemist often expresses density in g cm³ where mass is expressed in gram and volume is expressed in cm³.
• Uncertainty in Measurements
All scientific measurements involve certain degree of error or uncertainty. The errors which arise depend upon two factors.
(i) Skill and accuracy of the worker (ii) Limitations of measuring instruments.
• Scientific Notation
It is an exponential notation in which any number can be represented in the form N x 10ⁿ where n is an exponent having positive or negative values and N can vary between 1 to 10. Thus, 232.508 can be written as 2.32508 x 10² in scientific notation.
Now let us see how calculations are carried out with numbers expressed in scientific notation.
(i) Calculation involving multiplication and division

(ii) Calculation involving addition and subtraction: For these two operations, the first numbers are written in such a way that they have the same exponent. After that, the coefficients are added or subtracted as the case may be. For example,

• Significant Figures
Significant figures are meaningful digits which are known with certainty. There are certain rules for determining the number of significant figures. These are stated below:
1. All non-zero digits are significant. For example, in 285 cm, there are three significant figures and in 0.25 mL, there are two significant figures.
2. Zeros preceding to first non-zero digit are not significant. Such zeros indicates the position of decimal point.
For example, 0.03 has one significant figure and 0.0052 has two significant figures.
3. Zeros between two non-zero digits are significant. Thus, 2.005 has four significant figures.
4. Zeros at the end or right of a number are significant provided they are on the right side of the decimal point. For example, 0.200 g has three significant figures.
5. Counting numbers of objects. For example, 2 balls or 20 eggs have infinite significant figures as these are exact numbers and can be represented by writing infinite number of zeros after placing a decimal.
i.e., 2 = 2.000000
or 20 = 20.000000
• Addition and Subtraction of Significant Figures
In addition or subtraction of the numbers having different precisions, the final result should be reported to the same number of decimal places as in the term having the least number of decimal places.
For example, let us carry out the addition of three numbers 3.52, 2.3 and 6.24, having different precisions or different number of decimal places.

The final result has two decimal places but the answer has to be reported only up to one decimal place, i.e., the answer would be 12.0.
Subtraction of numbers can be done in the same way as the addition.

The final result has four decimal places. But it has to be reported only up to two decimal places, i.e., the answer would be 11.36.
• Multiplication and Division of Significant Figures
In the multiplication or division, the final result should be reported upto the same number of significant figures as present in the least precise number.
Multiplication of Numbers: 2.2120 x 0.011 = 0.024332
According to the rule the final result = 0.024
Division of Numbers: 4.2211÷3.76 = 1.12263
The correct answer = 1.12
• Dimensional Analysis
Often while calculating, there is a need to convert units from one system to other. The method used to accomplish this is called factor label method or unit factor method or dimensional analysis.
• Laws of Chemical Combinations
The combination of elements to form compounds is governed by the following five basic laws.
(i) Law of Conservation of Mass
(ii) Law of Definite Proportions
(iii) Law of Multiple Proportions
(iv) Law of Gaseous Volume (Gay Lussac’s Law)
(v) Avogadro’s Law
(i) Law of Conservation of Mass
The law was established by a French chemist, A. Lavoisier. The law states:
In all physical and chemical changes, the total mass of the reactants is equal to that of the products.
In other words, matter can neither be created nor destroyed.
The following experiments illustrate the truth of this law.
(a) When matter undergoes a physical change.

It is found that there is no change in weight though a physical change has taken place.
(b) When matter undergoes a chemical change.
For example, decomposition of mercuric oxide.

During the above decomposition reaction, matter is neither gained nor lost.
(ii) Law of Definite Proportions
According to this law:
A pure chemical compound always consists of the same elements combined together in a fixed proportion by weight.
For example, Carbon dioxide may be formed in a number of ways i.e.,

(iii) Law of Multiple Proportions
If two elements combine to form two or more compounds, the weight of one of the elements which combines with a fixed weight of the other in these compounds, bears simple whole number ratio by weight.
For example,

(iv) Gay Lussac’s Law of Gaseous Volumes
The law states that, under similar conditions of temperature and pressure, whenever gases combine, they do so in volumes which bear simple whole number ratio with each other and also with the gaseous products. The law may be illustrated by the following examples.
(a) Combination between hydrogen and chlorine:

(b) Combination between nitrogen and hydrogen: The two gases lead to the formation of ammonia gas under suitable conditions. The chemical equation is

(v) Avogadro’s Law: Avogadro proposed that, equal volumes of gases at the same temperature and pressure should contain equal number of molecules.
For example,
If we consider the reaction of hydrogen and oxygen to produce water, we see that two volumes of hydrogen combine with one volume of oxygen to give two volumes of water without leaving any unreacted oxygen.

• Dalton’s Atomic Theory
In 1808, Dalton published ‘A New System of Chemical Philosophy’ in which he proposed the following:
1. Matter consists of indivisible atoms.
2. All the atoms of a given element have identical properties including identical mass. Atoms of different elements differ in mass.
3. Compounds are formed when atoms of different elements combine in a fixed ratio.
4. Chemical reactions involve reorganisation of atoms. These are neither created nor destroyed in a chemical reaction.
• Atomic Mass
The atomic mass of an element is the number of times an atom of that element is heavier than an atom of carbon taken as 12. It may be noted that the atomic masses as obtained above are the relative atomic masses and not the actual masses of the atoms.
One atomic mass unit (amu) is equal to l/12th of the mass of an atom of carbon-12 isotope. It is also known as unified mass.
Average Atomic Mass
Most of the elements exist as isotopes which are different atoms of the same element with different mass numbers and the same atomic number. Therefore, the atomic mass of an element must be its average atomic mass and it may be defined as the average relative mass of an atom of an element as compared to the mass of carbon atoms (C-12) taken as 12w.
Molecular Mass
Molecular mass is the sum of atomic masses of the elements present in a molecule. It is obtained by multiplying the atomic mass of each element by number of its atoms and adding them together.
For example,
Molecular mass of methane (CH4)
= 12.011 u + 4 (1.008 u)
= 16.043 u
Formula Mass
Ionic compounds such as NaCl, KNO₃, Na₂C0₃ etc. do not consist of molecules i.e., single entities but exist “as ions closely packed together in a three dimensional space as shown in -Fig. 1.5.

In such cases, the formula is used to calculate the formula mass instead of molecular mass. Thus, formula mass of NaCl = Atomic mass of sodium + atomic mass of chlorine
= 23.0 u + 35.5 u = 58.5 u.
• Mole Concept
It is found that one gram atom of any element contains the same number of atoms and one gram molecule of any substance contains the same number of molecules. This number has been experimentally determined and found to be equal to 6.022137 x 10²³ The value is generally called Avogadro’s number or Avogadro’s constant.
It is usually represented by NA:
Avogadro’s Number, NA = 6.022 × 10²³
• Percentage Composition
One can check the purity of a given sample by analysing this data. Let us understand by taking the example of water (H₂0). Since water contains hydrogen and oxygen, the percentage composition of both these elements can be calculated as follows:

• Empirical Formula
The formula of the compound which gives the simplest whole number ratio of the atoms of yarious elements present in one molecule of the compound.
For example, the formula of hydrogen peroxide is H₂0₂. In order to express its empirical formula, we have to take out a common factor 2. The simplest whole number ratio of the atoms is 1:1 and the empirical formula is HO. Similarly, the formula of glucose is C₆H₁₂0₆. In order to get the simplest whole number of the atoms,
Common factor = 6
The ratio is = 1 : 2 : 1 The empirical formula of glucose = CH₂0
• Molecular Formula
The formula of a compound which gives the actual ratio of the atoms of various elements present in one molecule of the compound.
For example, molecular formula of hydrogen peroxide = H₂0₂and Glucose = C₆H₁₂0₆
Molecular formula = n x Empirical formula
Where n is the common factor and also called multiplying factor. The value of n may be 1, 2, 3, 4, 5, 6 etc.
In case n is 1, Molecular formula of a compound = Empirical formula of the compound.
• Stoichiometry and Stoichiometric Calculations
The word ‘stoichiometry’ is derived from two Greek words—Stoicheion (meaning element) and metron (meaning measure). Stoichiometry, thus deals with the calculation of masses (sometimes volume also) of the reactants and the products involved in a chemical reaction. Let us consider the combustion of methane. A balanced equation for this reaction is as given below:

Limiting Reactant/Reagent
Sometimes, in alchemical equation, the reactants present are not the amount as required according to the balanced equation. The amount of products formed then depends upon the reactant which has reacted completely. This reactant which reacts completely in the reaction is called the limiting reactant or limiting reagent. The reactant which is not consumed completely in the reaction is called excess reactant.
Reactions in Solutions
When the reactions are carried out in solutions, the amount of substance present in its given volume can be expressed in any of the following ways:
1. Mass percent or weight percent (w/w%)
2. Mole fraction
3. Molarity
4. Molality
1. Mass percent: It is obtained by using the following relation:

2. Mole fraction: It is the ratio of number of moles of a particular component to the total number of moles of the solution. For a solution containing n2 moles of the solute dissolved in n1 moles of the solvent,

3. Molarity: It is defined as the number of moles of solute in 1 litre of the solution.

4. Molality: It is defined as the number of moles of solute present in 1 kg of solvent. It is denoted by m.

• All substances contain matter which can exist in three states — solid, liquid or gas.
• Matter can also be classified into elements, compounds and mixtures.
• Element: An element contains particles of only one type which may be atoms or molecules.
• Compounds are formed when atoms of two or more elements combine in a fixed ratio to each other.
• Mixtures: Many of the substances present around us are mixtures.
• Scientific notation: The measurement of quantities in chemistry are spread over a wide rhnge of 10^-31to 10²³. Hence, a convenient system of expressing the number in scientific notation is used.
• Scientific figures: The uncertainty is taken care of by specifying the number of significant figures in which the observations are reported.
• Dimensional analysis: It helps to express the measured quantities in different systems of units.
• Laws of Chemical Combinations are:
(i) Law of Conservation of Mass
(ii) Law of Definite Proportions
(iii) Law of Multiple Proportions
(iv) Gay Lussac’s Law of Gaseous Volumes
(v) Avogadro’s Law.
• Atomic mass: The atomic mass of an element is expressed relative to 12C isotope of carbon which has an exact value of 12u.
• Average atomic mass: Obtained by taking into account the natural aboundance of different isotopes of that element.
• Molecular mass: The molecular mass of a molecule is obtained by taking sum of atomic masses of different atoms present in a molecule.
• Avogadro number: The number of atoms, molecules or any other particles present in a given system are expressed in terms of Avogadro constant.
= 6.022 x 10²³
• Balanced chemical equation: A balanced equation has the same number of atoms of each element on both sides of the equation.
• Stoichiometry: The quantitative study of the reactants required or the products formed is called stoichiometry. Using stoichiometric calculations, the amounts of one or more reactants required to produce a particular amount of product can be determined and vice-versa.

Waves Notes Class 11 Physics

Class XI: Physics

Waves

Key Learning:

1. Waves carry energy from one place to another.

2. The amplitude is the magnitude of the maximum displacement of the elements from their equilibrium positions as the wave passes through them.

3. The wavelength λ of a wave is the distance between repetitions of the shape of the wave. In a stationary wave, it is twice the distance between two consecutive nodes or anti nodes.

4. The period T of oscillation of a wave is the time any string element takes to move through one full oscillation.

5. A mechanical wave travels in some material called the medium. Mechanical waves are governed by Newton’s Laws.

6. The speed of the wave depends on the type of wave and the properties of the medium.

7. The product of wavelength and frequency equals the wave speed.

9. In transverse waves the particles of the medium oscillate perpendicular to the direction of wave propagation.

10. In longitudinal waves the particles of the medium oscillate along the direction of wave propagation.

11. Progressive wave is a wave that moves from one point of medium to another.

12. The speed of a transverse wave on a stretched string is set by the properties of the string. The speed on a string with tension T and linear mass density μ is v=√T/μ

13. Sound waves are longitudinal mechanical waves that can travel through solids, liquids, or gases. The speed v of sound wave in a fluid having bulk modulus B and density ρ is  v=√B/ρ

The speed of longitudinal waves in a metallic bar of Young’s modulus Y and density ρ isY  v=√ Y/ρ

For gases, since B =g P , the speed of sound is v=√gP/ρ

12. When two or more waves traverse the same medium, the displacement of any element of the medium is the algebraic sum of the displacements due to each wave. This is known as the principle of superposition of waves

13. Two sinusoidal waves on the same string exhibit interference, adding or canceling according to the principle of superposition.

14. A traveling wave, at a rigid boundary or a closed end, is reflected with a phase reversal but the reflection at an open boundary takes place without any phase change.

For an incident wave
yi (x, t) = a sin (kx – ωt )
The reflected wave at a rigid boundary is
yr (x, t) = – a sin (kx + ωt )
For reflection at an open boundary
yr (x,t ) = a sin (kx + ωt)

15. The interference of two identical waves moving in opposite directions produces standing waves. For a string with fixed ends, the standing wave is given by
y (x, t) = [2a sin kx ] cos ωt

16. Standing waves are characterized by fixed locations of zero displacement called nodes and fixed locations of maximum displacements called antinodes. The separation between two
consecutive nodes or antinodes is λ/2.

17. A stretched string of length L fixed at both the ends vibrates with frequencies given by
ν = 1/2 V/2L , 1,2,3…..

The set of frequencies given by the above relation are called the normal modes of oscillation of the system. The oscillation mode with lowest frequency is called the fundamental mode or the first harmonic. The second harmonic is the oscillation mode with n = 2 and so on.

16. A string of length L fixed at both ends or an air column closed at one end and open at the other end, vibrates with frequencies called its normal modes. Each of these frequencies is a resonant frequency of the system.

17. Beats arise when two waves having slightly different frequencies, ν1 and ν2 and comparable amplitudes, are superposed. The beat frequency, νbeat = ν1 ~ ν2

18. The Doppler’s effect is a change in the observed frequency of a wave when the source and the observer move relative to the medium.
5. The velocity of sound changes with change in pressure, provided temperature remains constant.

16. The plus/minus sign is decided by loading/filling any of the prongs of either tuning fork.

17. on loading a fork, its frequency decrease and on filling its frequency increases.

Oscillations And Waves Class 11 Physics Revision Notes

Class 11 Physics students should refer to the following concepts and notes for Oscillations And Waves in standard 11. These exam notes for Grade 11 Physics will be very useful for upcoming class tests and examinations and help you to score good marks

Oscillations And Waves Notes Class 11 Physics

Oscillations and Waves

•Periodic Motion: A motion which repeats itself over and over again after a regular interval of time.

• Oscillatory Motion: A motion in which a body moves back and forth repeatedly about a fixed point.

• Periodic function: A function that repeats its value at regular intervals of its argument is called periodic function. The following sine and cosine functions are periodic with period T.

f(t) =[ sin 2π /T]   and      g(t) = [cos  2π /T]

These are called Harmonic Functions.

Note :- All Harmonic functions are periodic but all periodic functions are not harmonic.

One of the simplest periodic functions is given by

f(t) = A cos ωt       [ω = 2π/T]

If the argument of this function ωt is increased by an integral multiple of 2π radians, the value of the function remains the same. The function f(t) is then periodic and its period, T is given by

T = 2π /ω

Thus the function f(t) is periodic with period T

f(t) = f(t +T)

Linear combination of sine and cosine functions

f(t) = A sin ωt + B cos ωt

A periodic function with same period T is given as

A = D cos ø and B = D sin ø

f(t) = D sin (ωt + ø)

D = √ A² + B² and ø = tan^-1 x /α

• Simple Harmonic Motion (SHM): A particle is said to execute SHM if it moves to and fro about a mean position under the action of a restoring force which is directly proportional to its displacement from mean position and is always directed towards mean position. Restoring Force Displacement

F α x

F= -kx

Where ‘k’ is force constant.

• Amplitude: Maximum displacement of oscillating particle from its mean position.

xMax = + A

•Time Period: Time taken to complete one oscillation.

• Frequency: 1 /r . Unit of frequency is Hertz (Hz).

1 Hz = 1 S^-1

•  Angular Frequency:

= 2π /T = 2πν

S.I unit ω = rad S^-1

• Phase:

1. The Phase of Vibrating particle at any instant gives the state of the particle as regards its position and the direction of motion at that instant.

It is denoted by ø.

2. Initial phase or epoch: The phase of particle corresponding to time t = 0.

It is denoted by ø.

• Displacement in SHM :

  X=A cos( ωt+ ø0)

Where,X= Displacement,

A = Amplitude

ωt = Angular Frequency

ø0 = Initial Phase.

Chapter 13 Kinetic Theory Of Gases Hand written Notes Class 11 Physics

Chapter 13 Kinetic Theory Of Gases Notes Class 11 Physics

Kinetic Theory of Matter:-

(a) Solids:- It is the type of matter which has got fixed shape and volume. The force of attraction between any two molecules of a solid is very large.

(b) Liquids:- It is the type of matter which has got fixed volume but no fixed shape. Force of attraction between any two molecules is not that large as in case od solids.

(c) Gases:- It is the type of matter does not have any fixed shape or any fixed volume.

• Ideal Gas:- A ideal gas is one which has a zero size of molecule and zero force of interaction between its molecules.
• Ideal Gas Equation:- A relation between the pressure, volume and temperature of an ideal gas is called ideal gas equation.

PV/T = Constant  or PV = nRT

Here, n is the number of moles and R is the universal gas constant.

Gas Constant:-

(a) Universal gas constant (R):-

R= P₀ V₀/T₀

=8.311 J mol^-1K^-1

(b) Specific gas constant (r):- 

PV= (R/M) T = rT,              

Here,  r = R/M

• Real Gas:-The gases which show deviation from the ideal gas behavior are called real gas.
• Vander wall’s equation of state for a real gas:-

[P+(na/V)²?][V-nb] = nRT

Here n is the number of moles of gas.

Avogadro’s number (N):- Avogadro’s number (N), is the number of carbon atoms contained in 12 gram of carbon-12.

N = 6.023×10²³

(a) To calculate the mass of an atom/molecule:-

Mass of one atom = atomic weight (in gram)/N

Mass of one molecule = molecular weight (in gram)/N

(b) To calculate the number of atoms/molecules in a certain amount of substance:-

Number of atoms in m gram = (N/atomic weight)×m

Number of molecules in m gram = (N/molecular weight)×m

(c) Size of an atom:-

Volume of the atom, V = (4/3)πr³

Mass of the atom, m = A/N

Here, A is the atomic weight and N is the Avogadro’s number.

Radius, r =[3A/4πNρ]^1/3\

Here ρ is the density.

Gas laws:-

(a) Boyle’s law:- It states that the volume of a given amount of gas varies inversely as its pressure, provided its temperature is kept constant.

PV = Constant

(b) Charlers law or Gey Lussac’s law:- It states that volume of a given mass of a gas varies directly as its absolute temperature, provided its pressure is kept constant.

V/T= Constant

V–V₀/V₀t = 1/273 = γ_p

Here γ_p (=1/273) is called volume coefficient of gas at constant pressure.

Volume coefficient of a gas, at constant pressure, is defined as the change in volume per unit volume per degree centigrade rise of temperature.

(c) Gay Lussac’s law of pressure:- It states that pressure of a given mass of a gas varies directly as its absolute temperature provided the volume of the gas is kept constant.

P/T = P₀/T₀ or P – P₀/P₀t = 1/273 = γ_p

Here γ_p (=1/273) is called pressure coefficient of the gas at constant volume.

Pressure coefficient of a gas, at constant volume, is defined as the change in pressure per unit pressure per degree centigrade rise of temperature.

(d) Dalton’s law of partial pressures:-

Partial pressure of a gas or of saturated vapors is the pressure which it would exert if contained alone in the entire confined given space.

P= p₁+p₂+p₃+……..

nRT/V = p₁+p₂+p₃+……..

(e) Grahm’s law of diffusion:- Grahm’s law of diffusion states that  the rate of diffusion of gases varies inversely as the square root of the density of gases.

R∝1/√ρ    or R₁/R₂ =√ρ₂/ ρ₁

So, a lighter gas gets diffused quickly.

(f) Avogadro’s law:- It states that under similar conditions of pressure and temperature, equal volume of all gases contain equal number of molecules.

For m gram of gas, PV/T = nR = (m/M) R

• Pressure of a gas (P):- P = 1/3 (M/V) C² = 1/3 (ρ) C²
• Root mean square (r.m.s) velocity of the gas:- Root mean square velocity of a gas is the square root of the mean of the squares of the velocities of individual molecules.

C= √[c₁²+ c₂²+ c₃²+…..+ c_n²]/n = √3P/ ρ

• Pressure in terms of kinetic energy per unit volume:- The pressure of a gas is equal to two-third of kinetic energy per unit volume of the gas.

P= 2/3 E

• Kinetic interpretation of temperature:- Root mean square velocity of the molecules of a gas is proportional to the square root of its absolute temperature.

C= √3RT/M

Root mean square velocity of the molecules of a gas is proportional to the square root of its absolute temperature.

At, T=0, C=0

Thus, absolute zero is the temperature at which all molecular motion ceases.

• Kinetic energy per mole of gas:-

K.E. per gram mol of gas = ½ MC² = 3/2 RT

• Kinetic energy per gram of gas:-

½ C² = 3/2 rt

Here, ½ C² = kinetic energy per gram of the gas and r = gas constant for one gram of gas.

• Kinetic energy per molecule of the gas:-

Kinetic energy per molecule = ½ mC² = 3/2 kT

Here, k (Boltzmann constant) = R/N

Thus, K.E per molecule is independent of the mass of molecule. It only depends upon the absolute temperature of the gas.

• Regnault’s law:- P∝T
• Graham’s law of diffusion:-

R₁/R₂ = C₁/C₂ = √ρ₂/ ρ₁

Distribution of molecular speeds:-

(a) Number of molecules of gas possessing velocities between v and v+dv :-

(b) Number of molecules of gas possessing energy between u and u+dv:-

(c) Number of molecules of gas possessing momentum between p and p+dp :-

(d) Most probable speed:- It is the speed, possessed by the maximum number of molecules of a gas contained in an enclosure.

V_m= √[2kT/m]

(e) Average speed (V_av):- Average speed of the molecules of a gas is the arithmetic mean so the speeds of all the molecules.

V_av= √[8kT/πm]

(f) Root mean square speed (Vrms):- It is the square root of the mean of the squares of the individual speeds of the molecules of a gas.

V_rms = √[3kT/m]

• V_rms > V_av > V_m
• Degree of Freedom (n):- Degree of freedom, of a mechanical system, is defined as the number of possible independent ways, in which the position and configuration of the system may change.

In general, if N is the number of particles, not connected to each other, the degrees of freedom n of such a system will be,

n = 3N

If K is the number of constraints (restrictions), degree of freedom n of the system will be,

n = 3N –K

Degree of freedom of a gas molecule:-

(a) Mono-atomic gas:- Degree of freedom of monoatomic molecule, n = 3

(b) Di-atomic gas:-

At very low temperature (0-250 K):- Degree of freedom, n = 3

At medium temperature (250 K – 750 K):- Degree of freedom, n = 5 (Translational = 3, Rotational = 2)

At high temperature (Beyond 750 K):- Degree of freedom, n = 6 (Translational = 3, Rotational = 2, Vibratory =1), For calculation purposes, n = 7

• Law of equipartition of energy:- In any dynamical system, in thermal equilibrium, the total energy is divided equally among all the degrees of freedom and energy per molecule per degree of freedom is ½ kT.

E = ½ kT

• Mean Energy:- K.E of one mole of gas is known as mean energy or internal energy of the gas and is denoted by U.

U = n/2 RT

Here n is the degree of freedom of the gas.

(a) Mono-atomic gas(n = 3):- U = 3/2 RT

(b) Diatomic gas:-

At low temperature (n=3):- U = 3/2 RT

At medium temperature (n=5):- U = 5/2 RT

At high temperature (n=7):- U = 7/2 RT

Relation between ratio of specific heat capacities (γ) and degree of freedom (n):-

γ = C_p/C_v = [1+(2/n)]

(a) For mono-atomic gas (n=3):- γ = [1+(2/n)] = 1+(2/3) = 5/3=1.67

(b) For diatomic gas (at medium temperatures (n=5)):- γ = [1+(2/5)] = 1+(2/5) = 7/5=1.4

(c) For diatomic gas (at high temperatures (n=7)):- γ = [1+(2/7)] = 9/7 = 1.29

Link-1 Chapter 12 Thermodynamics Class 11 Hand written notes Physics

Link-2 Chapter 12 Thermodynamics Class 11 Hand written notes Physics

Thermodynamics Class 11 notes Physics Chapter 12

Introduction

The foundation of thermodynamics is the conservation of energy and the fact that heat flows spontaneously from hot to cold body and not the other way around. The study of heat and its transformation to mechanical energy is called thermodynamics. It comes from a Greek word meaning “Movement of Heat”.

In this chapter, we shall study the laws of thermodynamics, various process, basic theory of heat engines, refrigerators and Carnot engine.

Thermal Equilibrium

When the temperature of the mixture becomes almost stable with the surrounding there is no further exchange of energy. This state in thermodynamics is called thermal equilibrium. So we may say in thermal equilibrium, the temperatures of the two systems are equal.

Zeroth Law of Thermodynamics

Zeroth law of thermodynamics states that “If two systems are in thermal equilibrium with a third system separately are in thermal equilibrium with each other.” Physical quantity whose value is equal for two systems in thermal equilibrium is called temperature (T).

Heat and Internal Energy

(i). Heat

Heat is that form of energy which gets transferred between a system and its surrounding because of temperature difference between them. Heat flows from the body at a higher temperature to the body at lower temperature. The flow stops when the temperature equalises. i.e., the two bodies are then in thermal equilibrium.

(ii). Internal Energy

It is sum of the kinetic energies and potential energies of all the constituent molecules of the system. It is denoted by ‘U’. U depends only on the state of the system. It is a state variable which is independent of the path taken to arrive at that state.

Work Done by a Gas

A container of cross sectional area A is fitted with a movable piston. Let the pressure of gas is P. Due to force applied by gas on piston, piston is displaced by Δx

Work done by gas,

W=F.△rW=F.△r

W=F△xcos0W=F△xcos0

W=F△xW=F△x

W=PA△xW=PA△x

W=P△VW=P△V

First Law of Thermodynamics

The first law of thermodynamics is a particular form of the general law of conservation of energy. Suppose the amount of heat Q is supplied to a system. It is normally spent in two ways.

1. Partially, it is spent in increasing internal energy of system.
2. The remaining part of it is spent in expanding the body against the external pressure, i.e. in doing external work W.

If ΔU the change in internal energy “since energy can neither be created nor destroyed but only convert from one form to another“, we have then

Q = ΔU + W…………(1)

If dQ, dU and dW are infinitesimal changes in heat, internal energy and work respectively, then equation (1) becomes

dQ = dU + dW

This equation represents the differential form of first law of thermodynamics.

Limitations of First Law of Thermodynamics

The first law of thermodynamics plays an important role in thermodynamics as it can be applied to know how much work will be obtained by transferring a certain amount of heat energy in a given thermodynamic process. However, first law of thermodynamics suffers from the following limitations :

• First law of thermodynamics does not indicate the direction of heat transfer.
• First law of thermodynamics does not tell anything about the conditions under which heat can be transformed into work.
• The first law does not indicate as to why the whole of the heat energy cannot be continuously converted into mechanical work.

Specific Heat Capacity

Specific heat capacity of a substance is defined as the heat required to raise the temperature of unit mass through 1°C (or 1 K).

Heat capacity of a substance is given by

S=△Q△TS=△Q△T

If we divide S by mass of the substance m in kg, we get

C=Sm=1m△Q△TC=Sm=1m△Q△T

here s is known as the specific heat capacity of the substance. It depends on the nature of the substance and its temperature. The unit of s is J kg^–1 K^–1.

The specific heat at constant volume C_v

It is defined as the amount of heat required to raise the temperature of a 1 mole of a gas through 1°C when its volume is kept constant. It is denoted by (C_v) and given by

CV=(△Q△T)VCV=(△Q△T)V

The specific heat at constant pressure C_p

It is defined as the amount of heat required to raise the temperature of 1 mole of the gas through 1°C when its pressure is kept constant. It is denoted by (C_p) and given by

CP=(△Q△T)PCP=(△Q△T)P

Derivation of Mayer’s Formula

From 1st law,

ΔQ = ΔU + ΔW = ΔU + PΔV

At constant volume ΔV = 0 so ΔQ = ΔU

Cv=(ΔQΔT)v=(ΔUΔT)vCv=(ΔQΔT)v=(ΔUΔT)v

Cv=ΔUΔTCv=ΔUΔT

On the other hand, at constant pressure,

ΔQ = ΔU + PΔV

Cp=(ΔQΔT)p=(ΔUΔT)p+P(ΔVΔT)pCp=(ΔQΔT)p=(ΔUΔT)p+P(ΔVΔT)p

Now, for a mole of an ideal gas

PV = RT

ΔVΔT=RPΔVΔT=RP

Cp=(ΔUΔT)p+P×RPCp=(ΔUΔT)p+P×RP

Cp=Cv+RCp=Cv+R

Cp−Cv=RCp-Cv=R

This formula is known as Mayer’s Formula. All the three quantities (C_p), (C_v) and R in this equation should be expressed in the same units either in joule/mole°C or in cal/mole°C.

Thermodynamic state variables and equation of state

The parameters or variables which describe equilibrium states of the system are called state variables.

(i). Intensive Variable

These are the variables which are independent of the size. e.g., pressure, density and temperature.

(ii). Extensive Variable

These are the variables which depend on the size of the system. e.g., volume, mass, internal energy.

(iii). Equation of State

The relation between the state variables is called the equation of state.

Thermodynamic processes

Any change in the thermodynamic coordinates of a system is called a process. The following are familiar processes in the thermodynamics.

(i). Isothermal Process

When a thermodynamic system undergoes a process under the condition that its temperature remains constant, then the process is said to be isothermal process. The essential condition for an isothermal process is that the system must be contained in a perfectly conducting chamber.

For isothermal process,

ΔU = 0

from the first lawof thermodynamics,

ΔU = Q-W

0 = Q-W

Q = W

Hence, for an ideal gas all heat is converted into work in isothermal process.

(ii). Adiabatic Process

When a thermodynamic system undergoes a process under the condition that no heat comes into or goes out of the system, then the process is said to be adiabatic process. Such a process can occur when a system is perfectly insulated from the surroundings.

For adiabatic process,

Q = 0

from the first lawof thermodynamics,

ΔU = Q-W

ΔU = 0-W

ΔU = -W

(iii). Isobaric Process

If the working substance is taken in expanding chamber in which the pressure is kept constant, the process is called isobaric process. In this process the gas either expands or shrinks to maintain a constant pressure and hence a net amount of work is done by the system or on the system.

(iv). Isochoric Process

If a substance undergoes a process in which the volume remains unchanged, the process is called an isochoric process. The increase of pressure and temperature produced by the heat supplied to a working substance contained in a non-expanding chamber is an example of isochoric process.

For isochoric process,

ΔV = 0,     W = PΔV,       W = 0

from the first law of thermodynamics,

ΔU = Q-W

ΔU = Q-0

ΔU = Q

(v). Quasi Static Process

A quasi-static process is defined as the process in which the deviation from thermodynamics equilibrium is infinitesimal and all the states through which the system passes during quasi-static process may be treated as aquarium states. Thus it may be defined as a succession of equilibrium states.

Heat engines

Any “cyclic” device by which heat is converted into mechanical work is called a heat engine. For a heat engine there are three essential requirements :

• Source:- A hot body, at a fixed high temperature T₁ from which the heat can be drawn heat, is called source or hot reservoir.
• Sink:- A cold body at a fixed lower temperature T₂ to which any amount of heat can be rejected, is called sink or cold reservoir.
• Working Substance:- The material, which on being supplied with heat, performs mechanical work is called the working substance.

Heat Engine

In a heat engine, the working substance takes in heat from the source, converts a part of it into external work, gives out the rest to the sink and returns to its initial state. This series of operations constitute a cycle. The work can be continuously obtained by performing the same cycle over and over again.

Suppose the working substance takes in an amount of heat Q₁ from the source, and gives out an amount Q₂ to the sink. Let W be the amount of work obtained. The net amount of heat absorbed by the substance is Q₁ – Q₂, which has been actually converted into work. Applying the  of thermodynamics to one complete cycle. We get

Q₁ – Q₂ = W

Thermal Efficiency

The thermal efficiency (e) of an engine is defined as the ratio of the work obtained to the heat taken in from the source, that is,

e=WQ1=Q1−Q2Q1e=WQ1=Q1-Q2Q1

e=1−Q2Q1e=1-Q2Q1

This equation indicates that the efficiency of the heat engine will be unity (efficiency 100%) when Q₂ = 0. This is, however, not possible in practice, This means that the engine cannot convert all the heat taken in from the source into work.

Refrigerators and heat pumps

(i). Reversible Process

A reversible process is one which can be retraced in opposite order by slightly changing the external conditions. The working substance in the reverse process passes through all the stages as in the direct process in such a way that all changes occurring in the direct process are exactly repeated in the opposite order and inverse sense and no changes are left in any of the bodies participating in the process or in the surroundings.

For reversible process,

ΔU = 0

from the first law of thermodynamics,

ΔU = Q-W

0 = Q-W

Q = W

(ii). Irreversible Process

Those process which can not be retraced in the opposite order by reversing the controlling factors are known as irreversible processes.

Second law of thermodynamics

This has two statements. First is Kelvin-Planck statement which is based upon the performance of heat engine and second is Clausius statement which is based upon the performance of refrigerator.

Kelvin-Planck statement

This may be stated as, “It is impossible to construct a device which operating in a cycle, has a sole effect of extracting heat from a reservoir at performing an equivalent amount of work“. Thus, a single reservoir at a single temperature can not continuously transfer heat into work.

Clausius statement

This may be stated as, “It is impossible for a self-acting machines working in a cycle process, unaided by any external agency to transfer heat from a body at a lower temperature to a body at a higher temperature.” In other words it may be stated as “Heat cannot flow itself from a colder to a hotter body”.

Reversible and Irreversible Processes

Reversible Process: A thermodynamic process is said to be reversible if the process can be turned back such that both the system and the surroundings return to their original states, with no other change anywhere else in the universe. Ex- extension of springs, slow adiabatic compression or expansion of gases.

Irreversible Process: An irreversible process can be defined as a process in which the system and the surroundings do not return to their original condition once the process is initiated. Ex- Relative motion with friction, Heat transfer.

Carnot Engine

A reversible heat engine operating between two temperatures is called a Carnot engine and the sequences of steps constituting one cycle is called the Carnot cycle.

Carnot Theorem

Carnot gave the most important results which are:

• No engine can have efficiency more than that of the Carnot engine.
• The efficiency of the Carnot engine is independent of the nature of the working substance.

Link-1 Chapter 11 Thermal Properties of Matter Handwritten notes Class 11 Physics

Link-2 Chapter 11 Thermal Properties of Matter Handwritten notes Class 11 Physics

Thermal Properties of Matter Class 11 notes Physics Chapter 11

Introduction

In this chapter, we shall examine some of the thermal properties of matter. We will first consider thermal expansion which plays an important role in everyday life and then discuss changes of phase and latent heat. At the end, we will discuss the phenomenon of heat transfer.

When a body is heated, various changes take place. Temperature is a measure of ‘hotness’ of a body. A kettle with boiling water is hotter than a box containing ice. When water boils or freezes, its temperature does not change during these processes even though a great deal of heat is flowing into or out of it.

Temperature and Heat

(i). Temperature

Temperature is a relative measure, or indication of hotness or coldness. A hot cooker is said to have a high temperature, and ice cube to have a lower temperature. An object at a higher temperature is said to be hotter than the one at a lower temperature. The SI unit of temperature is kelvin (K), whereas degree celsius (°C) is a commonly used unit of temperature.

(ii). Heat

When you put a cold spoon into a cup of hot coffee, the spoon warms up and the coffee cools down as they were trying to equalise the temperature. Energy transfer that takes place solely because of a temperature difference is called heat flow or heat transfer and energy transferred in this way is called heat. The SI unit of heat energy transferred is expressed in joule (J).

Measurement of Temperature

A physical property that changes with temperature is called a thermometric property. When a thermometer is put in contact with a hot body, the mercury expands, increasing the length of the mercury column.

(i). Celsius Scale

It defines ice-point temperature as 0°C and the steam point temperature as 100°C. The space between 0°C and 100°C marks is equally divided into 100 intervals.

(ii). Fahrenheit Scale

It defines the ice-point temperature as 32°F and the steam point temperature as 212°F. The space between 32°F and 212°F is divided into 180 equal intervals.

(iii). Kelvin Scale

Kelvin Scale is a scale of measuring of temperature, the melting point of ice is taken as 273 K and the boiling point of water as 373 K the space between these two points is divided into 100 equal intervals.

(iv). Relation between Different Scales of Temperatures

To convert a temperature from one scale to the other, we must take into account the fact that zero temperatures of the two scales are not the same.

C100=F−32180=K−273100=R80C100=F-32180=K-273100=R80

Note: The normal temperature of the human body measured on the Celsius scale is 37°C which is 98.6°F.

Ideal Gas Equation and Absolute Temperature

(i). Ideal Gas Equation

An equation which follows the law of Boyal, law of Charls and llaw of Avogadro is called ideal gas equation.

At constant temperature,

V∝1PV∝1P …( From Boyal’s law)

At constant pressure,

V∝TV∝T …( From charl’s law)

At constant T and P,

V∝nV∝n …( From Avogadro’s law)

By combinig all above equation, we get

V∝TnPV∝TnP

V=nRTPV=nRTP

PV=nRTPV=nRT

where, n = Number of moles of gas

R = Universal gas constant (R = 8.31 J mol–1 K–1)

P = Pressure of gas

V = Volume of gas

(ii). Absolute Temperature

The absolute minimum temperature is equal to –273.15ºC. This is also known as absolute zero. Absolute zero is the foundation of the kelvin temperature scale or absolute scale temperature.

Thermal Expansion

Increase in size of any matter on heating is called thermal expansion. There are three types of thermal expansion.

(i). Linear Expansion

The expansion in length is called linear expansion and the fractional change in length, ΔL/L is given by ΔL/L = αΔT where α is called coefficient of linear expansion.

(ii). Area Expansion

The expansion in area is called area expansion or superficial expansion and the fractional change in area, ΔA/A is given by ΔA/A = βΔT where β is called coefficient of area expansion.

(iii). Volume Expansion

The expansion in volume is called volume expansion and the fractional change in area, ΔV/V is given by ΔV/V = γΔT where γ is called coefficient of volume expansion.

(iv). Relation Between

α : β : γ = 1 : 2 : 3

αβ=12αβ=12 …..[ β = 2α ]

αγ=13αγ=13 …..[ γ = 3α ]

Specific Heat Capacity

If an amount of heat Q, when given to a body of mass m, increases its temperature by an amount ΔT, then

Q = mcΔT

where c is a constant and is called the specific heat capacity or simply specific heat of the material of the body.

If m = 1 kg and ΔT = 1C° then c = Q

Specific heat of the material of a substance is the amount of heat required to raise the temperature of unit mass of the substance through 1C°.

In SI, the unit of c is J/kg K.

Calorimetry

Calorimetry deals with the measurement of heat. The vessel which is largely used in such a measurement is called a calorimeter.

When two bodies at different temperatures are allowed to share heat, they attain a common temperature. If it is assumed that no heat is received from or given to any body outside the system and if there is no chemical action involved in the process of sharing, then

Heat gained = Heat lost

This simple statement based on the law of conservation of energy is called the principle of calorimetry.

Change of State

Depending on temperature and pressure, all matter can exist in a solid, liquid or gaseous state. These states or forms of matter are also called the phases of matter.

The change of state from solid to liquid is called melting and from liquid to solid is called fusion. It is observed that the temperature remains constant until the entire amount of the solid substance melts. That is, both the solid and the liquid states of the substance coexist in thermal equilibrium during the change of states from solid to liquid.

The temperature at which the solid and the liquid states of the substance is in thermal equilibrium with each other is called its melting point. The change of state from liquid to vapour (or gas) is called vaporisation. It is observed that the temperature remains constant until the entire amount of the liquid is converted into vapour.

The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point. The change from solid state to vapour state without passing through the liquid state is called sublimation, and the substance is said to sublime.

Latent Heat

Latent heat is defined as the heat or energy that is absorbed or released during a phase change of a substance. It could either be from a gas to a liquid or liquid to solid and vice versa. Latent heat is related to a heat property called enthalpy. It is denoted by L and its SI unit is J/kg.

L=QmL=Qm

There are two types of latent heat.

(i). Latent Heat of melting

It is a amount of heat which is required to change of phase from solid to liquid for unit mass at constant temperature. Ex- Latent heat of melting of ice is 3.33 x 10⁵ J/kg.

(ii). Latent Heat of Vaporization

It is a amount of heat which is required to change of phase from liquid to vapor for unit mass at constant temperature. Ex- Latent heat of vaporization of water is 22.6 x 10⁵ J/kg.

Heat Transfer

There are three mechanisms of heat transfer which name is given as- conduction, convection and radiation. Conduction occurs within a body or between two bodies in contact. Convection depends on motion of mass from one region of space to another. Radiation is heat transfer by electromagnetic radiation, such as sunshine, with no need for matter to be present in the space between bodies.

(i). Conduction

Conduction is the mechanism of transfer of heat between two adjacent parts of a body because of their temperature difference. Suppose, one end of a metallic rod is put in a flame, the other end of the rod will soon be so hot that you cannot hold it by your bare hands.

Here, heat transfer takes place by conduction from the hot end of the rod through its different parts to the other end. Gases are poor thermal conductors, while liquids have conductivities intermediate between solids and gases.

(ii). Convection

Convection is a mode of heat transfer by actual motion of matter. It is possible only in fluids. Convection can be natural or forced. In natural convection, gravity plays an important part. When a fluid is heated from below, the hot part expands and, therefore, becomes less dense. Because of buoyancy, it rises and the upper colder part replaces it. This again gets heated, rises up and is replaced by the relatively colder part of the fluid. The process goes on.

In forced convection, material is forced to move by a pump or by some other physical means. The common examples of forced convection systems are forced-air heating systems in home.

(iii). Radiation

Radiation is the transfer of heat by electromagnetic waves such as visible light, infrared, and ultraviolet rays. Everyone has felt the warmth of the sun’s radiation and intense heat from a charcoal grill or the glowing coals in a fireplace. Most of the heat from these bodies reaches you not by conduction or convection in the intervening air but by radiation. This heat transfer would occur even if there were nothing but vacuum between you and the source of heat.

Black Body Radiation

(i). Emissive Power

The amount of heat energy rediated per unit area of the surface of a body, per unit time and per unit wavelength range is constant which is called as the ’emissive power’ (e_λ) of the given surface, given temperature and wavelength. Its S.I. unit is Js^-1 m-^-2.

(ii). Absorptive Power

The ‘absorptive power’ of a surface at a given temperature and for a given wavelength is the ratio of the heat energy absorbed by a surface to the total energy incident on it at a certain time. It is represented by (a_λ). It has no unit as it is a ratio.

(iii) Perfect Black Body

A body is said to be a perfect black body, if its absorptivity is 1. It neither reflects nor transmits but absorbs all the thermal radiations incident on it irrespetive of their wavelengths.

(iv) Wein’s Displacement Law

This law states that as the temperature increases, the maximum value of the radiant energy emitted by the black body, move towards shorter wavelengths. Wein found that “The product of the peak wavelength (λ_m) and the Kelvin temperature (T) of the black body should remain constant.”

λm×T=bλm×T=b

Where b is constant known as Wein’s constant. Its value is 2.898 x 10^-3 mk.

(v) Stefan’s Law

This law states that the thermal radiations energy emitted per second from the surface of a black body is directly proportional to its surface area A and to the fourth power of its absolute temperature T.

Emission coefficient or degree of blackness of a body is represented by a dimensionless quantity ε, 0 < ε < 1. If ε = 1 then the body is perfectly black body. Hence

E∝AT4E∝AT4

E=σAT4E=σAT4

Where σ is a Stefan’s constant and its value is 5.67 x 10^-8 W m^-2 K^-4.

Newton’s Law of Cooling

According to Newton’s law of cooling, “The rate of loss of heat of a body is directly proportional to the excess of the temperature (T–T₀) of the body with respect to the surroundings”.

−dTdt∝(T−T0)-dTdt∝(T-T0)

Summary

• Temperature : The relative measure of hotness or coldness of a body is called its temperature.
• Heat : The energy that flows between two bodies by virtue of temperature difference between them is called heat. It flows from a hot body to cold body.
• Specific Heat Capacity : The amount of heat per unit mass absorbed or rejected by a substance to change its temperature by one unit is called its specific heat capacity (C).
• Molar Specific Heat Capacity : The amount of heat per unit mole absorbed or rejected by a substance to change its temperature by one unit is called its molar specific heat capacity (C).
• Calorimeter : A device in which heat measurement can be made is called a calorimeter.
• Melting Point : The temperature at which the solid and the liquid states of a substance exist in thermal equilibrium with each other is called its melting point.
• Boiling Point : The temperature at which the liquid and the vapour states of the substance coexist is called its boiling point.
• Triple Point : The temperature and pressure at which all the three phases of a substance co exist is called its triple point.
• Latent Heat : The heat per unit mass required (absorbed or evolved) to change the state of a substance at the same temperature and pressure is called its latent heat.
• The temperature of an object is measured with a device called thermometer.
• Heat transfer can take place by three modes namely, conduction, convection and radiation. Radiation is fastest of them all and does not require a material medium.

Chapter 10 Mechanical Properties of Fluids Handwritten notes Class 11 Physics

Mechanical Properties of Fluids Class 11 notes Physics Chapter 10

Introduction

In this chapter, we shall study some common physical properties of liquids and gases. How are fluids different from solids? What is common in liquids and gases? Unlike a solid, a fluid has no definite shape of its own. Solids and liquids have a fixed volume, whereas a gas fills the entire volume of its container.

Liquids and gases can flow and are therefore, called fluids. Earth has an envelop of air and two-thirds of its surface is covered with water. All the processes occurring in living beings including plants are mediated by fluids.

Pressure

We can define pressure as the normal force acting per unit area of a surface. It is denoted by P and SI unit of it is pascal (Pa). It is a scalar quantity.

P=FAP=FA

We observe that the same force (weight) exerts different pressures for different areas in contact. Lesser the area, more is the pressure exerted for a given force.

Density

The density of any material is defined as its mass per unit its volume. If a fluid of mass m occupies a volume V, then its density is given as

Density=mVDensity=mV

Density is usually denoted by the symbol ρ. It is a positive scalar quantity. Its SI unit is kg m^–3 and its dimensions are [ML^–3].

Pascal’s Law

This law states that the pressure in a fluid at rest is same at all points which are at the same height.

Imagine a small element of fluid in the shape of a right angled prism. All its points lie at the same depth inside the liquid. Therefore the effect of gravity is same at all these points. The forces exerted by rest of the fluid on different surfaces of this fluid element are as follows:

F1=F2=F3F1=F2=F3

If A₁, A₂ and A₃ are the surface areas of faces respectively, then by dividing.

F1A1=F2A2=F3A3F1A1=F2A2=F3A3

p1=p2=p3p1=p2=p3

So, the pressure exerted is same in all directions in a fluid at rest, at the points at equal height.

Variation of Pressure with Depth

Let’s find the difference in pressures at two points, whose levels differ by a height h in a fluid at rest. Let P₁ and P₂ be the pressures at two points 1 and 2 inside a fluid. Point 1 is at a height h above the point 2.

Imagine a fluid element in the shape of a cylinder as shown. If A be the area of the top and the bottom of this cylinder, then

F1=P1AF1=P1A

F2=P2AF2=P2A

Since, the fluid remains at rest, therefore the force F₂, which acts upwards should balance the two downward forces. These are, the force F₁ exerted at the top of the cylinder, and the weight W of the fluid confined within the cylinder.

F2=F1+mgF2=F1+mg

If ρ is the density of the fluid, then

F2=F1F2=F1+ρ.(volume of the cylinder).g

P2A=P1A+ρ.(A.h).gP2A=P1A+ρ.(A.h).g

P2=P1+ρghP2=P1+ρgh

P2−P1=ρghP2-P1=ρgh

This result tells us that as we go deep down a liquid the pressure goes on increasing. This pressure depends only on the height of the liquid column above the point.

Atmospheric Pressure

The atmospheric pressure at a point is equal to the weight of a column of air of unit cross-sectional area extending from that point to the top of the atmosphere. Its value is 1.013 × 10⁵ Pa at sea level. Atmospheric pressure is measured using an instrument called barometer.

Units os Atmospheric Pressure

• SI unit of pressure is N m^–2 or Pascal (Pa)
• Atmosphere, 1 atm = 1.013 × 10⁵ Pa = 760 mm of Hg
• 1 torr = 133 Pa
• 1 mm of Hg = 1 torr
• 1 bar = 10⁵ Pa
• 1 millibar = 100 Pa

Gauge Pressure

When we remove atmospheric pressure from total pressure of any system then this remaining pressure is called Gauge Pressure. The excess pressure P–P_a, at depth h is called a gauge pressure at that point.

Archimedes’ Principle

When a body is partially or completely immersed in a liquid, it loses some of its weight. The loss in weight of the body in the liquid is equal to the weight of the liquid displaced by the immersed part of the body. The upward force excerted by the liquid displaced when a body is immersed is called buoyancy. Due to this, there is apparent loss in the weight experienced by the body.

Law of Floatation

A body floats in a liquid if weight of the liquid displaced by the immersed portion of the body is equal to the weight of the body. When a body is immersed partially or wholly in a liquid, then the various forces acting on the body are

1. Upward thrust (T) acting at the centre of buoyancy and whose magnitude is equal to the weight of the liquid displaced
2. The weight of the body (W) which acts vertically downward through its centre of gravity.(a) When W > T, the body will sink in the liquid;(b) When W = T, then the body will remain in equilibrium inside the liquid;(c) When W < T, then the body will come upto the surface of the liquid.

Streamline Flow

When a liquid flows such that each particle of the liquid passing a given point moves along the same path and has the same velocity as its predecessor had at that point, the flow is called streamlined or steady flow. The path followed by a fluid particle in steady flow is called streamline.

Equation of Continuity

According to this theorem, “For the streamline flow of an incompressible fluid through a pipe of varying cross-section, product of cross-section area and velocity of streamline flow (Av) remains constant throughout the flow”.

Av = constant

Bernoulli’s Principle

It may be stated as follows: As we move along a streamline, the sum of the pressure (P), the kinetic energy per unit volume (ρv22ρv22) and the potential energy per unit volume (ρgh) remains a constant.

P+ρgh+12ρv2=P+ρgh+12ρv2=constant

Limitations of Bernoulli’s Equation

1. The equation is valid only for incompressible fluids having streamline flow. It is because it does not take into account the elastic energy of the fluids.
2. It is assumed that no energy is dissipated due to frictional force exerted by different layers of fluid on each other.
3. It does not hold for non-steady flow. In such situation velocity and pressure constantly fluctuate with time.

Surface Tension

Surface tension is defined as the surface energy per unit area or the force per unit length acting in the plane of the interface between the plane of the liquid and any other substance. The surface tension of a liquid usually decreases with increase in temperature.

S=FlS=Fl

Capillary Rise

A tube of very fine bore is called a capillary. ‘Capilla’ is a Latin word which means hair. Thus, capillary is a very thin tube. When such a tube, open at both ends, is dipped in a beaker containing water, water rises in it against gravity.

Let us find an expression for the height h, upto which a liquid rises in a capillary tube. Let a capillary tube of radius r be dipped in a liquid of surface tension S and density ρ.

Thus, capillary rise

h=2Scosθrρgh=2Scosθrρg

If the angle of contact for this liquid and the capillary tube is acute, the liquid forms a concave meniscus.

Viscosity

When a fluid moves, it flows in the form of parallel layers. These layers exert a force on each other which tends to oppose their relative motion. This is similar to what a frictional force does when two solids in contact move or tend to move over each other. The property of fluid which gives rise to such frictional force in them, is called viscosity. It is denoted by ‘η’. Its SI unit is N s m^–2 or Pa s which is also called poiseuille (Pl). The dimensions of viscosity are [ML^–1T^–1].

η=FAdvdxη=FAdvdx

where, dv/dx = velocity gardient

F = frictional force between layer of water

A = area of layer

Angle of Contact

The angle of contact is defined as the angle that the tangent to the liquid surface at the point of contact makes with the solid surface inside the liquid. The angle of contact depends on the nature of the solid and the liquid in contact. At the point of contact, the surface forces between the three media must be in equilibrium.

Stokes’ Law

When a spherical ball is dropped in a liquid, he observed that the viscous force F experienced by the ball is proportional to the

– velocity of the object through the fluid, v

– viscosity of the fluid, η

– radius of the sphere, r

Thus    F ∝ ηrv

Here, the constant of proportionality is found to be 6π.

Thus    F = –6πηrv   ← Stokes’ law.

The negative sign in the above expression just indicates that the retarding force is opposite to the direction of motion of the object.

Terminal Velocity

The maximum constant velocity acquired by a body while falling through a viscous medium is called its terminal velocity. It is usually denoted by V_T.

When the body acquires terminal velocity,

the upward viscous force + the upward buoyant force = weight of the ball

Reynolds Number

Whether a flow will be turbulent or not, is decided by a dimensionless parameter called Reynolds number R_e. This parameter is given by the relation

Re=ρvdηRe=ρvdη

where, ρ = density of the fluid

v = velocity of the fluid

η = viscosity of the fluid

d = diameter of the pipe through which the fluid flows

• If R_e < 1000, the flow is streamline or laminar
• If Re > 2000, the flow becomes turbulent
• If 1000 < Re < 2000, flow is unsteady i.e., it may change from laminar to turbulent and vice versa.

Poiseuille’s Formula

The Poiseuille’s formula gives an expression for volume flow rate through a capillary tube of inner radius r and length l due to a pressure difference between its ends, P. The volume flow rate is represented by Q such that

Q=dVdtQ=dVdt

According to Poiseuille’s,

dVdt=(π8)×(Pl)×r4ηdVdt=(π8)×(Pl)×r4η

Here, η is the coefficient of viscosity and all symbols have standard meaning.

Chapter 9 Mechanical Properties of Solids Hand Written Notes Class 11 Physics

Chapter 9 Mechanical Properties of Solids Class 11 notes Physics

Introduction

Can we design an aeroplane which is very light but sufficiently strong? Can we design an artificial limb which is lighter but stronger? Why does a railway track have a particular shape like I? Why is glass brittle while brass is not?

In this chapter, we will introduce the concepts of stress, strain and elastic modulus and a simple principle called Hooke’s law that help us predict what deformation will occur when forces are applied to a real kind of body.

Elastic Behaviour of Solids

In a solid, each atom or molecule is surrounded by neighbouring atoms or molecules. These are bonded together by interatomic or intramolecular forces and stay in a stable equilibrium position. When a solid is deformed, the atoms or molecules are displaced from their equilibrium positions causing a change in the interatomic distance.

When the deforming force is removed, the interatomic forces tend to drive them back to their original position. Thus the body regains its original shape and size.

(i) Deforming Force

If a force applied on a body produces a change in the normal positions of the molecules of the body, it is called deforming force.

(ii) Elasticity

The property of the body due to which, it tries to regain its original configuration when the deforming forces are removed is called elasticity.

(iii) Perfectly Elastic body

A body which completely regains its original configuration after the removal of deforming force, is called perfectly elastic body. Quartz and phosphor bronze are closest to perfectly elastic body known.

(iv) Perfectly Plastic Body

A body which does not regain its original configuration at all on the removal of deforming force, how so ever small the deforming force may be is called perfectly plastic body. For example, clay behaves like a perfectly plastic body.

(v) Restoring Force

When a deforming force is applied to a body to change its shape, the body develops an opposing force due to its elasticity. This opposing force tries to restore the original shape of the body, it is called restoring force.

Stress and Strain

(A) Strain

The strain is the relative change in dimensions of a body resulting from the external forces.

Strain = change in length / original length

It is a fractional quantity so, it has no unit.

(i) Tensile Strain

The tensile strain of the object is equal to the fractional change in length, which is the ratio of the elongation Δl to the original length l.

Tensile Strain=ΔlL=ΔlL

(ii) Shear Strain

We define shear strain as the ratio of the displacement x to the transverse dimension L.

Shear Strain=xL=xL

(iii) Bulk Strain

The fractional change in volume that is, the ratio of the volume change ΔV to the original volume V is called Bulk Strain.

Bulk Strain=ΔVV=ΔVV

(B) Stress

The restoring force developed per unit area in a body is called stress.

Stress = Restoring Force / area

In SI system, stress is measured in N / m² (pascal) and in CGS system in dyne/cm². The dimensional formula for stress is [M L^–1T^–2]

(i) Tensile Stress

We define the tensile stress at the cross-section as the ratio of the force F_⊥ to the cross-sectional area A.

Tensile Stress=F⊥A=F⊥A

(ii) Shear Stress

We define the shear stress as the force F_|| acting tangent to the surface, divided by the area A on which it acts.

Shear Stress=F∣∣A=F∣∣A

(iii) Bulk Stress

If an object is immersed in a fluid (liquid or gas) at rest, the fluid exerts a force on any part of the surface of the object. This force is perpendicular to the surface. The force F_⊥ per unit area that the fluid exerts on the surface of an immersed object is called the pressure p in the fluid (Bulk Stress).

Hooke’s Law

For small deformations the stress and strain are proportional to each other. This is known as Hooke’s law.

Thus,

stress ∝ strain

stress = k × strain

where k is the proportionality constant and is known as modulus of elasticity.

Elastic Moduli

The ratio of stress and strain, called modulus of elasticity, is found to be a characteristic of the material.

(i) Young’s Modulus

For a sufficiently small tensile stress, stress and strain are proportional. The corresponding elastic modulus is called Young’s modulus, denoted by Y.

Y = Tensile-stress / Tensile-strain=FAΔlL=FAΔlL

Y=FLAΔlY=FLAΔl

(ii) Shear Modulus

If the forces are small enough that Hooke’s law is obeyed, the shear strain is proportional to the shear stress. The corresponding elastic modulus is called the shear modulus, denoted by G. It is also called the modulus of rigidity.

G = Shear-stress / Shear-strain=FAΔxL=FAΔxL

G=F×LA×ΔxG=F×LA×Δx

G=FAθG=FAθ

SI unit of shear modulus is Nm^–2 or Pa.

(iii) Bulk Modulus

When Hooke’s law is obeyed, an increase in Bulk stress produces a proportional Bulk strain. The corresponding elastic modulus (ratio of stress to strain) is called the Bulk modulus, denoted by B.

When the pressure on a body changes by a small amount Δp, from p to (p+Δp), and the resulting Bulk strain is ΔV/V, Hooke’s law takes the form

B = Normal-stress / Volume-strain=−ΔpΔVV=-ΔpΔVV

We include a minus sign in this equation because an increase of pressure always causes a decrease in volume. The Bulk modulus B itself is a positive quantity.

SI unit of bulk modulus is Nm^–2 or Pa.

Compressibility

The reciprocal of the Bulk modulus is called the compressibility and is denoted by K. From equation

K=1BK=1B

The units of compressibility are those of reciprocal pressure, Pa^–1 or atm^–1.

Elastic Potential Energy

The excess of the energy of interaction between all atoms/molecules of a deformed object is elastic energy. When we remove the external force the body becomes undeformed and the elastic energy, will be retrieved back and converted into vibrational energy followed by heat, light, sound etc.

The elastic potential energy

ΔU=12=12stress × strain × volume

Also, the elastic potential energy per unit volume, i.e.,

ΔUvolume=12ΔUvolume=12stress × strain

Poisson’s Ratio

When a body is linearly extended, it contracts in the direction at right angles. Poisson’s ratio, σ is the ratio of the lateral strain to the longitudinal strain.

Longitudinal strain = Δl/L

Lateral strain = – ΔR/R

The Poisson’s ratio is given as,

σ = lateral strain / longitudinal strain

σ=–ΔRRΔlLσ=–ΔRRΔlL

–ve sign shows that if the length increases, then the radius of wire decreases. Poisson’s ratio is a unit less and dimensionless quantity.

Relation between Y, K, η and σ

• Y = 3K (1 – 2σ)
• Y = 2η (1 + σ)
• σ = (3K – 2η) /(2η + 6K)
• 9/Y = 1/K + 3/η

Applications of Elastic Behaviour of Materials

In our daily life, most of the materials which we use, undergo some kind of stress. That is why, while designing a structure of the material we give due consideration to the possible stresses, the material might suffer at one stage or the other. The following examples illustrate this concept.

1. The metallic parts of the machinery are never subjected to a stress beyond elastic limit, otherwise they will get permanently deformed.
2. The crane which is used to lift and move the heavy load is provided with thick and strong metallic ropes to which the load to be lifted is attached. The rope is pulled by using pulleys and motor.
3. The bridges are designed in such a way that they do not bend much or break under the load of heavy traffic, force of strongly blowing wind and its own weight.
4. Maximum height of a mountain on earth can be estimated from the elastic behaviour of earth. At the base of mountain, the pressure is given by p = ρgh, where h is the height of mountain, ρ is the density of material of mountain and g is the acceleration due to gravity.

Summary

• Elasticity : Elasticity is that property of the material of a body due to which the body opposes any change in its shape and size when deforming forces are applied on it and recovers its original configuration partially or wholly as soon as the deforming forces are removed.
• Stress : It is defined as the internal restoring force per unit area of cross-section of object.
• Strain : The change in dimensions of an object per unit original dimensions is called strain.
• Hooke’s law : For small deformation, the stress is proportional to strain.
• Young’s modulus : The ratio of tensile (or compressive) stress to the corresponding longitudinal strain is called Young’s modulus.
• Bulk modulus : The ratio of volumetric stress to volumetric strain is called Bulk modulus.
• Shear modulus : It is the ratio of shear stress to shearing strain.
• Poisson’s ratio : The lateral strain is proportional to longitudinal strain within the elastic limit and the ratio of two strains is called Poisson’s ratio.
• Elastic after effect : The slow process of recovering the original state by an object after the removal of the deforming force is called elastic after effect.

Chapter 8 Gravitation Handwritten Notes Class 11 notes Physics

Chapter 8 Gravitation Class 11 notes Physics

Summary

• Gravitational force: It is a force of attraction between the two bodies by the virtue of their masses.
• Acceleration due to gravity: The acceleration produced in the motion of a body freely falling towards earth under the force of gravity is known as acceleration due to gravity.
• Gravitational potential energy: The amount of work done in displacing the particle from infinity to a point under consideration.
• Gravitational potential: The gravitational potential due to the gravitational force of the earth is defined as the potential energy of a particle of unit mass at that point.
• Escape speed: The minimum speed with which the body has to be projected vertically upwards from the surface of the earth is called escape speed.
• Orbital speed: The minimum speed required to put the satellite into the given orbit around earth is called orbital speed.
• Satellite: It is a body which revolves continuously in an orbit around a comparatively much larger body.
• Polar satellite: It is the satellite which revolves in polar orbit around the earth.
• Geostationary satellite: It is the satellite which appears at a fixed position and at a definite height to an observer on earth.
• Kepler’s Ist law: All planets move in elliptical orbits, with the sun at one of foci of the ellipse.
• Kepler’s IInd law: The line that joins any planet to the sun sweeps out equal areas in equal intervals of time.
• Kepler’s IIIrd law: The square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse traced out by the planet.
• Newton’s universal law of gravitation: Every particle in the universe attracts every other particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
• Gravitational force is a conservative force.
• The value of acceleration due ot gravity is maximum at the surface of the earth while zero at the centre of earth.
• Henry Cavendish was the first person who found the value of G experimentally.
• Gravitational force on a particle inside a spherical shell is zero.
• Gravitational shielding is not possible.
• An astronaut experiences weightlessness in a space satellite. It is because both the astronaut and the satellite are in “free fall” towards the earth.
• The value of g increases from equator to poles.
• The escape speed from a point on the surface of the earth may depend on its location on the earth e.g., escape speed is more on poles and less on equator.
• The orbital speed of satellite is independent of mass of the satellites.
• Kepler’s laws hold equally well for satellites.