Is it correct to say: Writing the energy of a bound electron as positive?

E_n = −13.6 Z²/n² eV is negative. The energy of an emitted photon is the positive difference E(n₂) − E(n₁).

Is it correct to say: Swapping n₁ and n₂ in the Rydberg formula and getting a negative wavelength?

n₁ is always the lower level (it names the series); n₂ > n₁.

Is it correct to say: Using the longest-wavelength rule for the shortest wavelength?

Longest wavelength: smallest energy gap, n₂ = n₁ + 1. Shortest wavelength (series limit): n₂ → ∞.

Is it correct to say: Using λ in h⁻¹ with activity in becquerel?

1 Bq = 1 decay per second, so λ must be in s⁻¹ when using A = λN with A in Bq.

Is it correct to say: Thinking all nuclei decay in two half-lives?

Each half-life halves what remains: after 2 half-lives 1/4 remains, after 3 half-lives 1/8, and so on.

Is it correct to say: Assuming bigger nuclei are denser?

R ∝ A^(1/3) makes volume ∝ A, so nuclear density is nearly the same (about 2.3×10¹⁷ kg/m³) for all nuclei.

📄 Formula sheet (PDF)

Structure of Atoms and Nuclei

Bohr's model, radius, speed and energy of electron orbits, hydrogen spectrum, nuclear size and density, mass defect, binding energy, radioactive decay law, half-life, fission and fusion — Maharashtra HSC Physics Ch 15

📐 19 formulas✏️ 4 examples🎯 6 practice🏫 Maharashtra State Board📚 Class 12✓ 2025–26 syllabus

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Board Exam Tips

→Derivations of the radius of the nth Bohr orbit and the energy of the electron in it are standard theory questions. Both start from the same two equations: Coulomb force = centripetal force, and mvr = nh/2π.
→Learn the proportionalities: r ∝ n²/Z, v ∝ Z/n, E ∝ −Z²/n². With these, ratio questions need no constants at all.
→For spectral lines, first name the series from the lower level (Lyman 1, Balmer 2, Paschen 3, Brackett 4, Pfund 5). The longest wavelength uses the next level up; the shortest (series limit) uses n₂ → ∞.
→In binding-energy numericals, keep the mass defect in u and multiply by 931.5 MeV. Converting to kg first and using E = mc² gives the same answer but takes longer and invites arithmetic slips.
→For radioactive decay, count half-lives first: if t is a whole number of half-lives, N = N₀/2ⁿ needs no logarithms.
→Distinguish half-life (T½ = 0.693/λ) from mean life (τ = 1/λ). Mean life is always longer: τ ≈ 1.44 T½.

📐 Formulas(19)

Bohr's Quantisation Condition

L = m v r = \frac{nh}{2 π}, n = 1, 2, 3, \dots

Symbol	Meaning
$m$	Mass of electron = 9.1×10⁻³¹ kg
$v$	Speed of electron in the orbit (m s⁻¹)
$r$	Radius of the orbit (m)
$h$	Planck's constant = 6.63×10⁻³⁴ J s

Radius of the nth Bohr Orbit★ Board fav

r_{n} = \frac{ε _{0} h ^{2} n ^{2}}{π m Z e ^{2}}, r_{1} \approx 0.53 \times 1 0^{- 10} m (hydrogen)

Symbol	Meaning
$ε_{0}$	Permittivity of free space = 8.85×10⁻¹² C² N⁻¹ m⁻²
$Z$	Atomic number (Z = 1 for hydrogen)
$e$	Electronic charge = 1.6×10⁻¹⁹ C

Speed of the Electron in the nth Orbit

v_{n} = \frac{Z e ^{2}}{2 ε _{0} nh}, v_{1} \approx 2.18 \times 1 0^{6} m s^{- 1} (hydrogen)

Energy of the Electron in the nth Orbit★ Board fav

E_{n} = - \frac{m Z ^{2} e ^{4}}{8 ε _{0}^{2} h ^{2} n ^{2}} = - \frac{13.6 Z ^{2}}{n ^{2}} eV

Kinetic, Potential and Total Energy

K = - E, U = 2 E, U = - 2 K

Bohr's Frequency Condition

h ν = E_{n_{2}} - E_{n_{1}} (n_{2} > n_{1})

Wavelength of a Spectral Line (Rydberg Formula)★ Board fav

\frac{1}{λ} = R Z^{2} (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}}), R = \frac{m e ^{4}}{8 ε _{0}^{2} h ^{3} c} = 1.097 \times 1 0^{7} m^{- 1}

Symbol	Meaning
$R$	Rydberg constant = 1.097×10⁷ m⁻¹
$n_{1}$	Lower energy level
$n_{2}$	Higher energy level (n₂ = n₁ + 1, n₁ + 2, …)

Spectral Series of Hydrogen

Lyman: n_{1} = 1; Balmer: n_{1} = 2; Paschen: n_{1} = 3; Brackett: n_{1} = 4; Pfund: n_{1} = 5

Nuclear Radius

R = R_{0} A^{1/3}, R_{0} \approx 1.2 \times 1 0^{- 15} m

Symbol	Meaning
$R$	Radius of the nucleus (m)
$A$	Mass number

Nuclear Density

ρ = \frac{A m _{N}}{\frac{4}{3} π R _{0}^{3} A} = \frac{3 m _{N}}{4 π R _{0}^{3}} \approx 2.3 \times 1 0^{17} kg m^{- 3}

Mass Defect

Δ m = [Z m_{p} + (A - Z) m_{n}] - M

Symbol	Meaning
$m_{p}$	Mass of proton = 1.007276 u
$m_{n}$	Mass of neutron = 1.008665 u
$M$	Mass of the nucleus (u)

Binding Energy★ Board fav

B . E . = Δ m c^{2}, 1 u \equiv 931.5 MeV

Binding Energy per Nucleon

B ˉ = \frac{B . E .}{A}

Law of Radioactive Decay★ Board fav

\frac{d N}{d t} = - λ N, N = N_{0} e^{- λ t}

Symbol	Meaning
$N_{0}$	Number of nuclei at t = 0
$N$	Number of undecayed nuclei at time t
$λ$	Decay constant (s⁻¹)

Half-Life★ Board fav

T_{1/2} = \frac{l n 2}{λ} = \frac{0.693}{λ}, N = N_{0} (\frac{1}{2})^{t / T_{1/2}}

Mean Life

τ = \frac{1}{λ} = \frac{T _{1/2}}{0.693} \approx 1.44 T_{1/2}

Activity

A = \frac{d N}{d t} = λ N = A_{0} e^{- λ t}

Alpha and Beta Decay

_{Z}^{A} X \to_{Z - 2}^{A - 4} Y +_{2}^{4} He,_{Z}^{A} X \to_{Z + 1}^{A} Y + e^{-} + ν ˉ

Energy Released in a Nuclear Reaction (Q-value)

Q = (\sum m_{r e a c t an t s} - \sum m_{p r o d u c t s}) c^{2}

✏️ Solved Examples

1Solved Exampleeasy3 steps

Find the total energy, kinetic energy and potential energy of the electron in the third orbit of a hydrogen atom.

Energy in the nth orbit of hydrogen (Z = 1).

E_{3} = - \frac{13.6}{3 ^{2}} = - \frac{13.6}{9} \approx - 1.51 eV

2Solved Exampleboard3 steps

Calculate the wavelength of the first line (longest wavelength) of the Balmer series of hydrogen and the series limit. (R = 1.097×10⁷ m⁻¹)

Balmer series: n₁ = 2. The first line uses n₂ = 3.

\frac{1}{λ} = R (\frac{1}{2 ^{2}} - \frac{1}{3 ^{2}}) = R \times \frac{5}{36}

3Solved Exampleboard4 steps

Find the binding energy and the binding energy per nucleon of a helium-4 nucleus. Mass of ⁴He nucleus = 4.001506 u, mₚ = 1.007276 u, mₙ = 1.008665 u, 1 u = 931.5 MeV.

Helium-4 has Z = 2 protons and A − Z = 2 neutrons.

Z m_{p} + (A - Z) m_{n} = 2 (1.007276) + 2 (1.008665) = 4.031882 u

4Solved ExampleHOTS4 steps

The activity of a radioactive sample falls from 8000 Bq to 1000 Bq in 12 hours. Find its half-life, decay constant and mean life, and the number of radioactive nuclei present initially.

The activity fell by a factor of 8 = 2³, so 3 half-lives have passed.

3 T_{1/2} = 12 h \Rightarrow T_{1/2} = 4 h

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Writing the energy of a bound electron as positive
✓E_n = −13.6 Z²/n² eV is negative. The energy of an emitted photon is the positive difference E(n₂) − E(n₁).
2
Swapping n₁ and n₂ in the Rydberg formula and getting a negative wavelength
✓n₁ is always the lower level (it names the series); n₂ > n₁.
3
Using the longest-wavelength rule for the shortest wavelength
✓Longest wavelength: smallest energy gap, n₂ = n₁ + 1. Shortest wavelength (series limit): n₂ → ∞.
4
Using λ in h⁻¹ with activity in becquerel
✓1 Bq = 1 decay per second, so λ must be in s⁻¹ when using A = λN with A in Bq.
5
Thinking all nuclei decay in two half-lives
✓Each half-life halves what remains: after 2 half-lives 1/4 remains, after 3 half-lives 1/8, and so on.
6
Assuming bigger nuclei are denser
✓R ∝ A^(1/3) makes volume ∝ A, so nuclear density is nearly the same (about 2.3×10¹⁷ kg/m³) for all nuclei.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Taking the radius of the first Bohr orbit of hydrogen as 0.53 Å, find the radius of the third orbit.
Q2
Find the energy needed to remove the electron from the ground state of He⁺ (Z = 2).
Q3
Calculate the shortest wavelength in the Lyman series of hydrogen. (R = 1.097×10⁷ m⁻¹)
Q4
Find the ratio of the nuclear radii of ²⁷Al and ¹²⁵Te.
Q5
The half-life of a radioactive isotope is 1600 years. Find its decay constant in s⁻¹. (Take 1 year = 365 days.)
Q6
In a nuclear reaction the total mass of the products is 0.2 u less than that of the reactants. How much energy is released?

📝 Notes

Structure of Atoms and Nuclei — Maharashtra HSC Overview

Chapter 15 of the Maharashtra Board Std XII Physics textbook goes from the atom to its nucleus. It traces the models of Thomson, Rutherford (after the Geiger–Marsden experiment) and Bohr, explains the hydrogen spectrum, and then turns to nuclear size, binding energy and radioactivity.

Bohr's model in two equations

Everything about a Bohr orbit follows from two equations, $mv^2/r = Ze^2/(4\pi\varepsilon_0 r^2)$ and $mvr = nh/2\pi$ . Solve them together and you get $r_n \propto n^2/Z$ , $v_n \propto Z/n$ and $E_n = -13.6Z^2/n^2$ eV. Add Bohr's frequency condition $h\nu = E_{n_2} - E_{n_1}$ and the Rydberg formula follows, with $R$ expressed entirely in fundamental constants. Know the limitations too: the model works only for one-electron systems and cannot explain the relative intensities of spectral lines.

The nucleus

Size: $R = R_0A^{1/3}$ , so all nuclei have nearly the same density.
Stability: the mass defect $\Delta m$ appears as binding energy $\Delta m c^2$ . With $\Delta m$ in u, multiply by 931.5 MeV.
The binding energy per nucleon curve explains nuclear energy. Its peak near iron means that splitting heavy nuclei (fission) and joining light ones (fusion) both move towards more tightly bound nuclei and release energy.

Radioactivity

The decay law $N = N_0e^{-\lambda t}$ leads to half-life $T_{1/2} = 0.693/\lambda$ , mean life $\tau = 1/\lambda$ and activity $A = \lambda N$ . In α decay the mass number falls by 4 and the atomic number by 2. In β⁻ decay the atomic number rises by 1 and the mass number is unchanged. Balance both numbers on each side of every nuclear equation you write.