Journey Inside the Atom
Chapter 8 — full, step-by-step solutions for every worked calculation, In-Text “Pause & Ponder” question, and end-of-chapter “Revise, Reflect, Refine” exercise, with clean diagrams and maths.
Worked Examples
The two quantitative calculations worked through in the chapter text.
- Convert the thickness to metres$0.1\ \text{mm} = 0.1\times10^{-3}\ \text{m} = 10^{-4}\ \text{m}$
- Divide sheet thickness by atomic diameter$\text{Number of atoms} \approx \dfrac{10^{-4}\ \text{m}}{10^{-10}\ \text{m}} = 10^{6}$
So about one million atoms stacked together make up the thickness of a single sheet of paper — a vivid sense of how unimaginably small a single atom is.
A simple average ignores how common each isotope is:
$$\text{Simple average}=\frac{35+37}{2}=36\ \text{u}$$
But the two isotopes are not equally abundant, so we use a weighted average — each mass is multiplied by its fractional abundance:
- Multiply each mass by its abundance$\left(35\times\dfrac{75}{100}\right)+\left(37\times\dfrac{25}{100}\right)$
- Simplify$=\dfrac{105}{4}+\dfrac{37}{4}=\dfrac{142}{4}$
- Result$=35.5\ \text{u}$
In-Text — Pause & Ponder
Every “Pause and Ponder” question that appears within the chapter.
- (i) The negative charge now outweighs the positive charge, so the whole model carries a net negative charge. It no longer represents a neutral atom — it behaves like a negative ion (anion).
- (ii) No. If the clay is also slightly negative, then both parts are negative and nothing supplies positive charge to balance the beads. The total charge is negative, so the model cannot represent a neutral atom — neutrality needs the positive and negative charges to be exactly equal.
Where it matches: the soft pulp plays the role of the spread-out positive charge, and the seeds embedded within it act like electrons scattered through that positive matter — exactly the ‘plum-pudding’ picture.
Where it falls short:
- Seeds in a fruit are clustered in segments, not evenly spread, while Thomson imagined electrons distributed throughout.
- The pulp is not actually electrically charged, and the number of seeds has nothing to do with balancing charge, so the analogy can’t show neutrality.
- Seeds are huge compared to the fruit, whereas electrons are vanishingly small compared with the atom.
In his cathode-ray experiments, the rays were always streams of the same negatively charged particles (electrons) — and crucially, their nature did not depend on the metal used for the cathode or on the gas filling the tube.
Since the same electrons appeared no matter which material was used, they could not belong to one special substance. Thomson therefore concluded that electrons are a fundamental constituent of every atom.
A positive nucleus repels positive α-particles, which is why some bounced back. Negative particles would instead be attracted towards the positive nucleus.
- Most would still travel through the empty space almost undeflected.
- Those passing close to a nucleus would be pulled toward it (their paths curving inward), rather than being pushed back.
- We would therefore not see the sharp backward bounce that repulsion produced; the deflection pattern would reflect attraction instead.
In the plum-pudding model the positive charge is spread thinly and evenly across the whole atom. Such a diffuse charge could only ever nudge a fast α-particle slightly — it could never stop one and throw it backward.
A bounce of more than 90° needs a near head-on encounter with something tiny, very dense and strongly positive. That can only exist if the positive charge and mass are concentrated in a minute central nucleus — the opposite of a spread-out sphere.
This is an open, reflective question — there is no single ‘correct’ answer. A thoughtful example:
A good question shows curiosity about either the reasoning behind the nuclear idea or the open puzzles he left (such as what keeps the electrons from collapsing in).
Reason (R): According to Thomson’s model, electrons are embedded in a uniformly distributed positive-charge sphere.
- Both A and R true, R is the correct explanation of A.
- Both A and R true, R is not the correct explanation of A.
- A true, R false.
- A false, R true.
A is true: the large-angle scattering of α-particles led Rutherford to the dense central nucleus.
R is true: that is a correct description of Thomson’s model.
But R describes a different model and says nothing about why Rutherford concluded the mass is central — that came from his own scattering experiment. So R does not explain A.
Sample: name the element ‘Aaravium’ after Aarav. A valid symbol would be Av.
Why it follows IUPAC norms:
- The first letter is a capital (A) and the second is lowercase (v) — e.g. Av, never AV.
- It is built from letters of the element’s name.
- It is not already taken by an existing element, so there is no clash.
- Constant confusion and miscommunication — the same symbol could mean different elements to different people.
- Errors in writing chemical formulae and equations, which could be dangerous in medicine or industry.
- Scientists across countries and languages could no longer share results or reproduce each other’s work reliably.
A single, internationally agreed set of symbols (now maintained by IUPAC) removes all this ambiguity.
- Protons = atomic number$p = Z = 26$
- Electrons = protons (neutral atom)$e = 26$
- Neutrons = mass number − protons$n = A – Z = 56 – 26 = 30$
- Neutrons = mass number − protons$n = A – p = 41 – 20 = 21$
- Mass number = protons + neutrons$A = Z + n = 17 + 18 = 35$
- Protons = electrons (neutral atom)$p = e = 11$, so $Z = 11$
- Neutrons = mass number − protons$n = A – Z = 23 – 11 = 12$
- (i) Carbon ($Z=6$): configuration 2, 4 → outermost shell has 4 electrons.
- (ii) Fluorine ($Z=9$): configuration 2, 7 → outermost shell has 7 electrons.
- (iii) Silicon ($Z=14$): configuration 2, 8, 4 → outermost shell has 4 electrons.
Fill shells in order K (max 2), L (max 8), M (max 8 for these):
- $Z=12$ (Magnesium): 2, 8, 2
- $Z=16$ (Sulfur): 2, 8, 6
- $Z=18$ (Argon): 2, 8, 8
11 protons means atomic number 11 — the element is sodium (Na), a soft metal that reacts vigorously with water.
- Neutrons = mass number − protons$n = A – p = 23 – 11 = 12$
- Atomic numbersBoth have 11 protons, so both have $Z = 11$ — same atomic number.
- Mass numbers$A_1 = 11+12 = 23$ and $A_2 = 11+13 = 24$ — different mass numbers.
Same atomic number means they are the same element (sodium). Same element but different mass numbers → they are isotopes.
- Weight each mass by its abundance$\left(79\times\dfrac{49.7}{100}\right)+\left(81\times\dfrac{50.3}{100}\right)$
- Compute each term$=\dfrac{3926.3}{100}+\dfrac{4074.3}{100}$
- Add$=\dfrac{8000.6}{100}=80.006\ \text{u}$
Exercise — Revise, Reflect, Refine
Full solutions to the end-of-chapter questions.
- It clearly showed the existence of neutrons in the nucleus.
- The results disproved the plum-pudding model and led to the idea of a nucleus.
- The large deflection of a few α-particles showed most of the mass and positive charge are packed into a tiny centre.
- The deflections showed how electrons move around the nucleus.
- (i) Incorrect. Neutrons were discovered much later (Chadwick, 1932); this experiment said nothing about them.
- (ii) Correct. Back-scattering could not fit a spread-out positive sphere, so it overturned plum-pudding and pointed to a central nucleus.
- (iii) Correct. Only a tiny, dense, highly positive centre could deflect heavy α-particles through large angles.
- (iv) Incorrect. The experiment revealed the nucleus, not the motion of electrons.
- Electrons lose energy in fixed orbits and slowly fall into the nucleus.
- Electrons can exist anywhere around the nucleus with no fixed energy.
- Electrons revolve in orbits of fixed energy without losing energy.
- Electrons can be found between energy levels.
- (i) Incorrect. The whole point of Bohr’s model is that in a fixed orbit (stationary state) an electron does not lose energy.
- (ii) Incorrect. Electrons may only occupy fixed orbits, each with a definite energy — not just anywhere.
- (iii) Correct. This is exactly Bohr’s key postulate, which explains atomic stability.
- (iv) Incorrect. Electrons cannot exist between allowed shells.
First find each mass number $A = p + n$:
- X: $A = 18+19 = 37$, $Z = 18$
- Y: $A = 17+18 = 35$, $Z = 17$
- Z: $A = 17+20 = 37$, $Z = 17$
(i) Y and Z: same number of protons (17) but different neutrons → same element with different mass numbers (35 and 37) → they are isotopes.
(ii) Z and X: different atomic numbers (17 vs 18) but the same mass number (37) → they are isobars.
Because only a very few α-particles were strongly deflected while most passed straight through, Rutherford concluded that the positive charge is:
- Concentrated in an extremely small central region — the nucleus — not spread out across the atom.
- Dense and massive, carrying most of the atom’s mass, so it could throw back heavy α-particles.
- Positively charged, since it repelled the positive α-particles.
- Surrounded by mostly empty space, which is why most particles went through undeflected.
The models developed in this order:
- (iv) DaltonAtoms are tiny indivisible particles — the first scientific atomic theory (1808).
- (ii) ThomsonElectrons embedded in a positive sphere — the plum-pudding model (1897–1904).
- (iii) RutherfordGold-foil scattering reveals a dense central nucleus (1911).
- (i) BohrElectrons occupy fixed energy levels, explaining stability (1913).
The nucleus is positively charged (protons) and electrons are negatively charged. The electrostatic force of attraction between them pulls the electrons toward the nucleus.
For an electron moving in a circular path, this inward pull provides exactly the centripetal force needed to keep it in orbit — balancing its tendency to fly off in a straight line. In Bohr’s picture the electron stays in a fixed energy level without losing energy, so it neither escapes nor falls in.
Reason (R): The number of electrons equals the number of protons in an atom.
- Both A and R true, R is the correct explanation of A.
- Both A and R true, R is not the correct explanation of A.
- A true, R false.
- A false, R true.
A is true: finding electrons, protons and neutrons is precisely what let scientists build models of the atom.
R is true: in a neutral atom electrons do equal protons.
But R is just one fact about neutrality; it does not explain why discovering the particles advanced our understanding. So R is not the correct explanation of A.
- (i) Protons = atomic number$p = Z = 12$
- (ii) Neutrons = mass number − protons$n = A – Z = 24 – 12 = 12$
- (iii) Electrons = protons (neutral atom)$e = 12$
- Electron arrangementFill K, L, M → configuration 2, 8, 2
Reading the shells of each atom:
- (a) 2, 1 → Lithium (Li): total e = 3, valence e = 1, valency = 1, protons = 3, $Z$ = 3.
- (b) 2, 5 → Nitrogen (N): total e = 7, valence e = 5, valency = 3 (gains/shares $8-5=3$), protons = 7, $Z$ = 7.
- (c) 2, 8, 3 → Aluminium (Al): total e = 13, valence e = 3, valency = 3 (loses 3), protons = 13, $Z$ = 13.
- (d) 2, 7 → Fluorine (F): total e = 9, valence e = 7, valency = 1 (gains $8-7=1$), protons = 9, $Z$ = 9.
Rutherford’s problem: an electron moving in a circle is constantly accelerating. By classical physics an accelerating charge must continuously radiate energy; losing energy, it would spiral inward and crash into the nucleus in a fraction of a second. His model therefore predicted that atoms could not be stable.
Bohr’s fix: Bohr postulated special stationary states (fixed orbits) in which an electron revolves without radiating any energy. As long as the electron stays in such a level its energy is constant, so it never spirals in — which explains why atoms are stable.
- Protons = electrons (neutral atom)$p = e = 31$
- Neutrons = mass number − protons$n = A – p = 70 – 31 = 39$
- (i) Neutrons = mass number − protons$n = 197 – 79 = 118$
- (ii) Electrons = protons (neutral atom)$e = 79$
Use $A = p + n$, $Z = p = e$ (neutral atom):
| Atomic no. (Z) | Mass no. (A) | Neutrons | Protons | Electrons | Element |
|---|---|---|---|---|---|
| 5 | 11 | 6 | 5 | 5 | Boron |
| 7 | 14 | 7 | 7 | 7 | Nitrogen |
| 12 | 24 | 12 | 12 | 12 | Magnesium |
| 15 | 31 | 16 | 15 | 15 | Phosphorus |
| 1 | 1 | 0 | 1 | 1 | Hydrogen |
- Row 1: $Z=5,\ n=6 \Rightarrow p=5,\ e=5,\ A=11$ → Boron.
- Row 2: $A=14,\ e=7 \Rightarrow p=7,\ Z=7,\ n=7$ → Nitrogen.
- Row 3: $A=24,\ p=12 \Rightarrow Z=12,\ e=12,\ n=12$ → Magnesium.
- Row 4: $Z=15,\ n=16 \Rightarrow p=15,\ e=15,\ A=31$ → Phosphorus.
- Row 5: $A=1,\ n=0 \Rightarrow p=1,\ Z=1,\ e=1$ → Hydrogen.
- (i) Protons & electrons$p = A – n = 35 – 18 = 17$, so $p = e = 17$
- (ii) Atomic number$Z = p = 17$
- (iii) Identify the element$Z=17$ → Chlorine (Cl)
- (iv) Electronic configuration2, 8, 7
- (v) Valence electrons7 (in the outermost shell)
- (vi) Add 2 neutronsnew $A = 35 + 2 = 37$ (protons unchanged)
- (vii) RelationSame $Z$ (17), different mass numbers (35 and 37) → isotopes
- (i) Atomic number — unchanged (12). Atomic number depends only on the number of protons, which is untouched.
- (ii) Atomic mass — increases. Normal electrons are so light their mass is ignored. Particles 500× heavier are no longer negligible (each is roughly $500/1836 \approx 0.27$ of a proton’s mass), so 12 of them add a noticeable amount of mass.
- (iii) Mass number — unchanged (24). Mass number counts only nucleons: $p + n = 12 + 12 = 24$. Electrons (or their replacements) are not included.
- (iv) Overall charge — unchanged (neutral). The new particles still carry one negative charge each, so 12 of them still balance the 12 protons.
The quest continues…
From Dalton’s indivisible sphere to Bohr’s energy levels and beyond — the story of the atom is still being written.
