Structure of Atom · Lesson 12 of 12

Chapter Summary and Practice

“Connect the experimental evidence, models, equations, and electron arrangements across the whole chapter.”

Learning Objectives

• Reconstruct how experimental evidence progressively changed the model of the atom. • Choose appropriate equations for radiation, transitions, matter waves, and uncertainty. • Connect quantum numbers with orbital shape, nodes, energy, and electron capacity. • Use electron counting and filling rules to explain atoms, ions, and configuration exceptions. • Solve mixed problems while checking signs, units, assumptions, and physical meaning.

One chapter, one connected investigation

The chapter begins with a question about chemical diversity: why do elements behave differently if all are made of atoms? Experiments first reveal electrons and nuclear particles. Scattering locates a nucleus. Radiation then reveals that an electron cannot take an arbitrary bound energy. Matter waves and uncertainty finally lead to orbitals and electronic configurations.

These ideas form a connected argument. Particle counts identify the species. Allowed states determine which radiation it can absorb or emit. Orbital energies and occupation rules determine how its electrons are arranged. That arrangement supplies a foundation for understanding chemical behaviour. A successful revision should connect these levels of explanation, not just recall a separate definition for each word.

Evidence or problem	What it establishes	What still needs explanation
Cathode rays and oil drops	Universal electrons; elementary charge and electron mass	Arrangement of positive and negative particles
Gold-foil scattering	Tiny massive positive nucleus	Stable electronic energies and distribution
Black-body and photoelectric results	Discrete energy exchange; photons	Allowed atomic states
Hydrogen lines and Bohr model	Main one-electron energy and transition pattern	Multi-electron atoms and a consistent electron description
Matter waves and uncertainty	Failure of a definite electron path	Probability distributions and allowed quantum states
Quantum orbitals and filling rules	Electron arrangements in atoms	Detailed later chemistry built on those arrangements

Keep identity, charge, and mass number separate

Atomic number counts protons and fixes the element. Mass number counts protons plus neutrons. Charge records a difference between proton and electron counts. These three pieces of information must not be used interchangeably. Isotopes differ in neutron number; ions differ in electron number; isobars share mass number without sharing elemental identity.

Electron charge has the same magnitude as proton charge but opposite sign. Their masses are very different. A neutron contributes mass without charge. These facts explain how an isotope change can leave the usual electronic arrangement almost unchanged, while gaining or losing an electron can strongly change chemical behaviour.

Example — Combining nuclear composition and electronic arrangement

Problem
A neutral atom has 29 electrons and 35 neutrons. Find its isotope symbol, then its ground-state configuration.

1.Neutrality makes proton count equal to electron count, so Z = 29 identifies copper.
2.Mass number is $29 + 35 = 64$ , giving $_{29}^{64} Cu$ .
3.The ground-state electron arrangement is $[Ar] 3 d^{10} 4 s^{1}$ , not the naive $3 d^{9} 4 s^{2}$ . Nuclear counting identifies the species; total-energy and occupation considerations determine its electron arrangement.

Choose equations by the physical event

Wavelength and frequency describe the same radiation in different ways. Photon energy connects that radiation to an energy exchange. In the photoelectric effect, part of the photon energy frees an electron and the rest becomes kinetic energy. In an atomic transition, the energy difference between two allowed states determines the photon.

Matter-wave wavelength is instead set by a particle’s momentum. The uncertainty relation concerns the spread of possible values, not simply the value of position or speed itself. Ask which event is being described before choosing a formula: light propagation, photon counting, photoelectric escape, atomic transition, matter diffraction, or localisation.

Physical task	Core relationship	Check before substituting
Radiation conversion	$c = ν λ, \tilde{ν} = 1/ λ$	Consistent length units
Photon energy and number	$E_{γ} = h ν, N = E_{total} / E_{γ}$	Per photon versus per mole or total
Photoelectric escape	$K_{m a x} = h ν - W_{0}$	Threshold satisfied; all energies in matching units
One-electron states	$E_{n} = - 2.18 \times 1 0^{- 18} Z^{2} / n^{2} J$	Exactly one electron; Z is nuclear atomic number
Radiative transition	$E_{γ} = ∣ E_{f} - E_{i} ∣$	Photon energy positive; atomic change may be negative
Matter wavelength	$λ = h / p$	Use p = mv only in the appropriate non-relativistic case
Uncertainty bound	$Δ x Δ p_{x} \geq h / (4 π)$	Uncertainty and momentum component, not merely central values

Example — A mixed photon and matter-wave calculation

Problem
A 400 nm photon ejects an electron from a metal with work function 2.00 eV. Find the maximum electron kinetic energy and the corresponding de Broglie wavelength.

1.Photon energy is $h c / λ = 4.9695 \times 1 0^{- 19} J \approx 3.102 eV$ .
2.Subtract the work function: $K_{m a x} \approx 1.102 eV = 1.765 \times 1 0^{- 19} J$ .
3.For the fastest electron, $p = 2 m_{e} K_{m a x} \approx 5.67 \times 1 0^{- 25} kg m s^{- 1}$ .
4.Therefore $λ_{e} = h / p \approx 1.17 \times 1 0^{- 9} m$ . The electron wavelength is not the incoming photon wavelength; the equations refer to different physical objects.

Example — Comparing hydrogen and a one-electron ion

Problem
Find a hydrogen transition matching the wavelength of the $n = 4 \to 2$ transition in $He^{+}$ .

1.For helium’s one-electron ion, the reciprocal-square factor is $Z^{2} (1/ 2^{2} - 1/ 4^{2}) = 4 (1/4 - 1/16) = 3/4$ .
2.For hydrogen, $n = 2 \to 1$ has factor $1 - 1/4 = 3/4$ .
3.The equal factors give equal photon energies and wavelengths under the elementary model, even though the two transitions belong to different species and series.

Connect orbitals, quantum numbers, and configurations

A spatial orbital is labelled by n, l, and magnetic orbital quantum number; electron spin adds the fourth label. The same n can contain several subshell types, and the same l can occur in different shells. An orbital’s shape type is linked to l, while its radial structure depends on n as well.

Nodes make this connection visible. The angular count is l and the radial count is n − l − 1, adding to n − 1. A 4p orbital therefore retains p-like angular structure while having more radial nodes than 2p. A drawn boundary surface illustrates a chosen high-probability region, not a hard atomic edge.

Within hydrogen-like atoms, energies depend on n in the elementary model. In multi-electron atoms, shielding, penetration, and repulsion make subshell type matter. The n + l guide supplies an approximate filling sequence, Pauli limits each spatial orbital to two opposite-spin electrons, and Hund arranges electrons within degenerate orbitals. Only after applying all three should you write a ground-state configuration.

Example — An orbital description and its capacity

Problem
For the 4d subshell, determine its quantum-number possibilities, node counts, and maximum electron capacity.

1.The label gives n = 4 and l = 2. Magnetic orbital values are −2, −1, 0, +1, and +2, so there are five spatial orbitals.
2.Each has one radial node and two angular nodes, for three total nodes.
3.Each spatial orbital allows two opposite spin states, so the full subshell capacity is ten electrons. Five orbitals and ten electrons are different counts.

Example — Reading a configuration rather than memorising it

Problem
Compare ground-state silicon, $[Ne] 3 s^{2} 3 p^{2}$ , and phosphorus, $[Ne] 3 s^{2} 3 p^{3}$ .

1.The shared neon core supplies ten electrons. Adding the written occupations gives Z = 14 and Z = 15 for the neutral atoms.
2.Hund’s rule gives two singly occupied p orbitals for silicon and three for phosphorus. Their unpaired counts are two and three.
3.The added proton in phosphorus is accompanied by an added same-shell electron, which does not perfectly screen it. Comparing configurations helps connect electron count, spin pattern, and changing nuclear attraction.

Misconceptions that change the meaning

An orbit and an orbital answer different kinds of question: a classical path specifies a trajectory, whereas a wave function specifies a quantum state and probability distribution. Spin arrows do not show literal spinning balls. Orbital colours do not divide the electron into positive and negative charges. These distinctions preserve the meaning of the quantum description.

Similarly, a negative atomic energy is a bound-state reference, not negative kinetic energy. Brightness is not photon energy. A full principal-shell capacity is not a statement that it fills completely before the next shell starts. Chromium and copper are reminders that a useful filling mnemonic is not an exact many-electron energy calculation.

Check a result in four ways: does the equation match the event, do the units cancel correctly, does the arithmetic use the appropriate particle count, and does the physical direction make sense? An emitted photon must have positive energy; a shorter wavelength must mean a higher photon energy; a cation must have fewer electrons than its neutral atom.

Common mistake

Some problem statements need more data before a numerical answer exists. Photon number without a duration gives energy, not power; pulse duration and photon number without frequency or wavelength do not fix photon energy. Several practice items below explicitly supply these missing conditions so that the calculation is well posed.

Quick check

A singly positive ion has Z = 12 and A = 24. Its particle counts are:

Quick check

Which change increases maximum photoelectron kinetic energy above threshold?

Quick check

Which transition emits a photon?

Quick check

An electron momentum becomes four times larger. Its de Broglie wavelength becomes:

Quick check

Which has two radial nodes and one angular node?

Quick check

An orbital-box diagram pairs two electrons while an equal-energy p orbital remains empty. Which ground-state rule is not being followed?

Quick check

What is required to convert the energy of a monochromatic pulse into its average power?

Quick check

Which statement correctly connects the chapter’s ideas?

Mixed chapter practice

The following problems revisit the complete chapter. They are grouped by the main reasoning task while retaining a unique question number for checking coverage. Use the constants introduced in the lessons unless an item supplies a different rounded value. Show the electron count, unit conversions, and energy sign convention whenever these choices affect the result.

For a multi-part question, explain why each formula applies before substituting numbers. In qualitative items, connect the conclusion to an observation or principle. Several questions deliberately connect two ideas, such as nuclear identity with configuration or photoelectric energy with matter wavelength. Those links are the purpose of a complete chapter revision.

Particles, isotopes, charge, and atomic scales

1. (i) Calculate how many electrons have a combined mass of 1.00 g. (ii) Calculate the mass and signed charge of one mole of electrons.
2. (i) Count the electrons in one mole of methane. (ii) For 7 mg of carbon-14, find the total neutron count and neutron mass. (iii) For 34 mg of ammonia, find the total proton count and proton mass; explain whether changing temperature or pressure without changing the sample changes the result. Use neutron mass $1.675 \times 1 0^{- 27} kg$ .
3. Count protons and neutrons in $_{6}^{13} C$ , $_{8}^{16} O$ , $_{12}^{24} Mg$ , $_{26}^{56} Fe$ , and $_{38}^{88} Sr$ .
4. Write complete atomic symbols for $(Z, A) = (17, 35), (92, 233), (4, 9)$ .
27. Count the electrons in $H_{2}^{+}$ , $H_{2}$ , and $O_{2}^{+}$ . Use the atomic numbers of the constituent atoms and the overall molecular charge; no molecular-orbital model is needed.
35. A carbon atom has diameter 0.15 nm. Approximately how many could be placed touching side by side along 20 cm?
36. A line of $2.0 \times 1 0^{8}$ touching carbon atoms is 2.4 cm long. Estimate atomic radius.
37. A zinc atom has diameter $2.6 \overset{˚}{A}$ . Find (i) its radius in pm and (ii) the number of touching atoms in a line 1.6 cm long.
38. A particle has excess negative charge of magnitude $2.5 \times 1 0^{- 16} C$ . Estimate the number of excess electrons. Explain why a rounded measured charge need not produce an exact integer before rounding.
39. An oil drop has charge $- 1.282 \times 1 0^{- 18} C$ . Find its number of excess electrons and explain the role of X-rays in charging the drop.
40. Explain qualitatively how alpha-particle scattering would differ for a thin foil of light nuclei such as aluminium compared with heavy nuclei such as gold, under comparable conditions.
41. Explain why standard isotope notation places mass number at the upper left and atomic number at the lower left. For bromine-79, distinguish a standard isotope label, an abbreviated isotope label, and a wrongly interchanged pair of numbers.
42. An atom has mass number 81 and approximately 31.7% more neutrons than protons. Identify the element and isotope, allowing for the rounded percentage.
43. A singly negative ion has mass number 37 and approximately 11.1% more neutrons than electrons. Identify it, allowing for rounding.
44. An ion has mass number 56, charge 3+, and approximately 30.4% more neutrons than electrons. Identify it, allowing for rounding.

Radiation, photon counting, and photoelectric energy

5. A sodium lamp emits at 580 nm. Calculate frequency and wavenumber in both inverse metres and inverse centimetres.
6. Calculate photon energies for (i) frequency $3.0 \times 1 0^{15} Hz$ and (ii) wavelength $0.50 \overset{˚}{A}$ .
7. A light wave has period $2.0 \times 1 0^{- 10} s$ . Find frequency, wavelength, and wavenumber.
8. How many photons of wavelength 4000 pm together carry 1 J?
9. A 400 nm photon strikes a metal with work function 2.13 eV. Find photon energy in eV, maximum photoelectron kinetic energy, and maximum electron speed.
10. Radiation of wavelength 242 nm is just sufficient to ionise sodium atoms. Find the corresponding ionisation energy in $kJ mo l^{- 1}$ .
11. An idealised source radiates 25 W entirely as monochromatic light of wavelength $0.57 μ m$ . Find photons emitted per second.
12. Electrons just emerge with zero maximum speed under light of wavelength $6800 \overset{˚}{A}$ . Find threshold frequency and work function.
45. Arrange microwave-oven radiation, amber traffic-light radiation, FM radio radiation, gamma radiation from a cosmic source, and X-rays in increasing frequency. Explain why the broad phrase “cosmic rays” alone does not specify an electromagnetic band.
46. A nitrogen laser emits at 337.1 nm and sends out $5.6 \times 1 0^{24}$ photons each second. Calculate its radiated power. Then state why a photon count without a time interval would determine energy but not power.
47. Neon emits strongly at 616 nm. Find (i) frequency, (ii) distance travelled in vacuum in 30 s, (iii) energy per photon, and (iv) photon count corresponding to 2 J.
48. A detector receives $3.15 \times 1 0^{- 18} J$ of 600 nm radiation from a distant source. Estimate the photon count.
49. A 2 ns pulse contains $2.5 \times 1 0^{15}$ photons, each of wavelength 500 nm. Find pulse energy and average power during the pulse. Explain why duration and photon number alone would not determine energy if wavelength or frequency were not supplied.
50. A close absorption doublet has wavelengths 589.0 nm and 589.6 nm. Find each frequency. If both transitions begin from the same lower state, find the energy difference between the upper states.
51. Caesium has work function 1.9 eV. Find threshold wavelength and frequency. For 500 nm irradiation, find maximum kinetic energy and speed of emitted electrons.
52. A sodium photoelectric dataset gives wavelengths 500, 450, and 400 nm and corresponding maximum electron speeds $2.55 \times 1 0^{5}$ , $4.35 \times 1 0^{5}$ , and $5.35 \times 1 0^{5} m s^{- 1}$ . Use a graph or slope of kinetic energy against frequency to estimate Planck’s constant and the threshold wavelength. The speed unit here is metres per second; account for rounding in the observations.
53. For 256.7 nm light on silver, a stopping potential of 0.35 V is required. Use $K_{m a x} = e V_{s}$ to find the work function. Explain the energy conversion represented by the stopping potential.
54. A 150 pm photon ejects a bound electron at $1.5 \times 1 0^{7} m s^{- 1}$ . Find the electron’s binding energy from photon energy minus kinetic energy.

Hydrogen energies and spectral connections

13. Find the wavelength emitted when hydrogen makes a transition from n = 4 to n = 2.
14. Find the energy needed to ionise hydrogen from n = 5. Compare it with ionisation from the ground state.
15. For an ensemble of hydrogen atoms initially excited to n = 6, find the maximum number of distinct downward-transition emission lines possible as atoms return to the ground state. Explain why one atom in one decay sequence need not emit every line.
16. (i) Given hydrogen ground-state energy $- 2.18 \times 1 0^{- 18} J$ , find its fifth-state energy. (ii) Find the radius of the fifth Bohr orbit.
17. Find the wavenumber of the longest-wavelength Balmer emission line and explain your choice of levels.
18. Find the energy needed to excite hydrogen from n = 1 to n = 5, then find the wavelength for a direct return to n = 1. Use ground-state energy $- 2.18 \times 1 0^{- 18} J$ ; verify that this equals $- 2.18 \times 1 0^{- 11}$ erg.
19. Calculate the ionisation energy from hydrogen n = 2 and the longest wavelength, in centimetres, that can just cause this ionisation.
33. Which hydrogen transition has the same wavelength as the n = 4 to n = 2 transition in $He^{+}$ ?
34. Find the energy for $He^{+} (g) \to He^{2 +} (g) + e^{-}$ when the ion starts in its ground state. Use hydrogen ground-state ionisation energy $2.18 \times 1 0^{- 18} J$ per atom.
55. A hydrogen Paschen emission line is observed at 1285 nm. Use $ν = 3.29 \times 1 0^{15} (1/ 3^{2} - 1/ n^{2}) Hz$ to estimate the upper n and identify the spectral region.
56. A hydrogen emission transition begins at a Bohr orbit of radius 1.3225 nm and ends at one of radius 211.6 pm. Find the two n values, the emitted wavelength, the series, and the spectral region.

Matter waves and uncertainty

20. Find the de Broglie wavelength of an electron moving at $2.05 \times 1 0^{7} m s^{- 1}$ , using the non-relativistic approximation.
21. An electron of mass $9.1 \times 1 0^{- 31} kg$ has kinetic energy $3.0 \times 1 0^{- 25} J$ . Find its wavelength.
32. Combine the Bohr angular-momentum condition with the de Broglie relation to show that a permitted orbit circumference is an integer multiple of electron wavelength.
57. An electron in a microscope moves at $1.6 \times 1 0^{6} m s^{- 1}$ . Find its de Broglie wavelength and explain the link to imaging fine structure.
58. Neutrons used in diffraction have wavelength 800 pm. Find their speed and explain why diffraction does not depend on the particle carrying electric charge.
59. For an electron speed of $2.19 \times 1 0^{6} m s^{- 1}$ in the first Bohr orbit, calculate its wavelength and compare it with the orbit circumference.
60. A proton and a 0.10 kg hockey ball each move at $4.37 \times 1 0^{5} m s^{- 1}$ . Find both de Broglie wavelengths and compare them. The proton speed may be obtained by acceleration through 1000 V using $e V = m v^{2} /2$ .
61. For $Δ x = 0.002 nm$ , find the lower bound on electron momentum uncertainty. Compare it with a claimed sharply known momentum magnitude $p_{0} = h / [4 π (0.05 nm)]$ . Explain the difficulty of claiming a narrow momentum distribution centred on this value while maintaining the stated localisation.

Quantum numbers, orbital energies, and configurations

22. Group the isoelectronic species among $Na^{+}, K^{+}, Mg^{2 +}, Ca^{2 +}, S^{2 -}, Ar$ .
23. (i) Write configurations of $H^{-}, Na^{+}, O^{2 -}, F^{-}$ . (ii) Identify ground-state neutral atoms whose outer configurations end in $3 s^{1}$ , $2 p^{3}$ , and $3 p^{5}$ . (iii) Identify atoms with configurations $[He] 2 s^{1}$ , $[Ne] 3 s^{2} 3 p^{3}$ , and $[Ar] 3 d^{1} 4 s^{2}$ .
24. What is the smallest principal quantum number allowing a g subshell? Explain using l.
25. For a 3d electron, give n, l, and every permitted magnetic orbital quantum number.
26. A neutral atom contains 29 electrons and 35 neutrons. Find its proton count, isotope symbol, and ground-state electronic configuration.
28. (i) List allowed l and magnetic values for n = 3. (ii) List the permitted spatial quantum-number sets for 3d. (iii) Decide which labels are possible: 1p, 2s, 2p, 3f.
29. Write subshell labels for $(n, l) = (1, 0), (3, 1), (4, 2), (4, 3)$ .
30. Check each set $(n, l, m_{l}, m_{s})$ and explain every invalid one: $(0, 0, 0, + 1/2)$ ; $(1, 0, 0, - 1/2)$ ; $(1, 1, 0, + 1/2)$ ; $(2, 1, 0, - 1/2)$ ; $(3, 3, - 3, + 1/2)$ ; $(3, 1, 0, + 1/2)$ .
31. How many electrons at most can have (i) $n = 4, m_{s} = - 1/2$ and (ii) $n = 3, l = 0$ ? Distinguish spatial orbitals from electron states.
62. Order the following electron labels by the usual multi-electron n + l guide, identifying equal-energy members within a subshell: $(4, 2, - 2, - 1/2)$ , $(3, 2, 1, + 1/2)$ , $(4, 1, 0, + 1/2)$ , $(3, 2, - 2, - 1/2)$ , $(3, 1, - 1, + 1/2)$ , $(4, 1, 0, + 1/2)$ . The repeated final label is a repeated entry to compare, not two permissible simultaneous electrons with identical four-number sets.
63. Among bromine electrons in 2p, 3p, and 4p, which group is most strongly shielded and experiences the lowest effective nuclear attraction in the elementary comparison?
64. Within the same atom, compare qualitative effective nuclear attraction for (i) 2s versus 3s, (ii) 4d versus 4f, and (iii) 3d versus 3p. Explain the roles of shell and penetration.
65. Compare the effective nuclear attraction experienced by the 3p electrons of aluminium and silicon. Explain the effect of the added proton and incomplete same-shell screening.
66. Count ground-state unpaired electrons in phosphorus, silicon, chromium, iron, and krypton. Show the relevant orbital boxes.
67. (i) How many subshells belong to n = 4? (ii) How many electrons at most in that shell can have $m_{s} = - 1/2$ ? Explain why the two answers are different.

Key Takeaways

The nucleus fixes elemental identity, while electron arrangement is central to chemical behaviour. Radiation supplies evidence of quantised energy differences; photon energy and intensity must be distinguished. Bohr’s model captures one-electron energy patterns but a quantum wave description replaces definite orbits. Quantum numbers connect orbital type, orientation, nodes, and electron capacity. Shielding, orbital energies, Pauli exclusion, and Hund’s rule organise electronic configurations, with real total-energy exceptions. Reliable solutions require the correct physical model, consistent units, explicit particle counts, and a check that the question supplies enough information.

Previous · Lesson 11

Writing Electronic Configurations

End of chapter

One chapter, one connected investigation

Keep identity, charge, and mass number separate

Choose equations by the physical event

Connect orbitals, quantum numbers, and configurations

Misconceptions that change the meaning

Quiz

Mixed chapter practice

Practice Problems

Key Takeaways