Multiple Choice & Assertion-Reason

PHYExam Practice

This is a whole-chapter revision set: the one-mark MCQs and Assertion-Reason items sweep across both halves of the chapter — Bohr's atom (orbit radius $r_n \propto n^2/Z$ , energy $E_n \propto -Z^2/n^2$ , spectral series) and the nucleus (size $R \propto A^{1/3}$ , binding-energy-per-nucleon curve, the mass-energy unit $1\,\text{u} = 931.5\,\text{MeV}$ , and the decay law $N = N_0 e^{-\lambda t}$ ).

They reward knowing the structure of each relation — what is proportional to what — far more than heavy arithmetic. Examiners favour these because a single line such as 'nuclear density is independent of $A$ ' or 'mean life $\tau = T_{1/2}/0.693$ ' lets them test cause-and-effect reasoning, which is exactly what Assertion-Reason questions probe.

Bohr orbit radius and energy (hydrogen-like)

r_n = \dfrac{n^2}{Z}\,a_0, \qquad E_n = -13.6\,\dfrac{Z^2}{n^2}\ \text{eV}

n

is the principal quantum number,

Z

the atomic number,

a_0 = 0.53\,\text{Å}

the Bohr radius. Radius grows as

n^2

(so

r \propto n^2

for hydrogen,

Z=1

); energy is negative (bound), rising toward

0

n \to \infty

. For

n=2

in hydrogen,

E_2 = -13.6/4 = -3.4\,\text{eV}

Number of spectral lines on de-excitation

\text{lines} = \dfrac{n(n-1)}{2}

n

is the highest level the atoms are excited to (de-exciting to the ground state). For

n=4

this gives

\dfrac{4\times 3}{2} = 6

distinct lines. The total counts every allowed downward transition, spread across the Lyman, Balmer and Paschen series.

Nuclear radius and density

R = R_0 A^{1/3}, \qquad \rho = \dfrac{3 m_N}{4\pi R_0^{3}}

R_0 \approx 1.2\,\text{fm}

A

the mass number,

m_N

the mass of one nucleon. Since mass

\propto A

and volume

\propto R^3 \propto A

, the density

\rho

is independent of

A

— the same for all nuclei (

\sim 2.3\times 10^{17}\,\text{kg m}^{-3}

Mass-energy equivalence and the atomic mass unit

E = \Delta m\,c^2, \qquad 1\,\text{u} = 931.5\,\text{MeV}/c^2

\Delta m

is the mass defect. The conversion

1\,\text{u} \to 931.5\,\text{MeV}

(MeV, not eV or keV) is the single most-used number in the nuclear half of the chapter; binding-energy-per-nucleon peaks near

8.8\,\text{MeV}

for iron (

A \approx 56

Radioactive decay: remaining fraction, half-life and mean life

\dfrac{N}{N_0} = \left(\dfrac{1}{2}\right)^{t/T_{1/2}}, \qquad T_{1/2} = \dfrac{\ln 2}{\lambda} = 0.693\,\tau

T_{1/2}

is the half-life,

\tau = 1/\lambda

the mean (average) life,

\lambda

the decay constant. After

n

half-lives the surviving fraction is

(1/2)^n

(so

1/8

after

3

). Mean life is longer than half-life:

\tau = T_{1/2}/0.693

Radius scales as $n^2$ and energy as $-1/n^2$ for hydrogen; restore the $Z$ -dependence for hydrogen-like ions as $r \propto n^2/Z$ and $E \propto -Z^2/n^2$ . Speed in an orbit goes as $v \propto Z/n$ , so it decreases for higher orbits.
Spectral series are fixed by the lower level: Lyman ends on $n=1$ (UV), Balmer on $n=2$ (the only series wholly in the visible), Paschen on $n=3$ (IR), Brackett on $n=4$ (IR). 'Visible' on an MCQ almost always means Balmer.
The line count $n(n-1)/2$ is the number of distinct wavelengths emitted when a gas of atoms (not a single atom) de-excites from level $n$ ; a single atom makes only one downward jump at a time.
$R \propto A^{1/3}$ is the master nuclear-size relation: it makes nuclear volume proportional to $A$ , which is why nuclear density is constant and why packing nucleons does not dilute the nucleus.
On the binding-energy-per-nucleon curve, the peak near $A \approx 56$ (iron) marks the most stable nuclei; light nuclei release energy by fusion and heavy nuclei by fission, both moving toward the peak.
Always quote the unit conversion as $1\,\text{u} = 931.5\,\text{MeV}$ — the trap options replace MeV with eV, keV or GeV, which are wrong by factors of $10^6$ , $10^3$ and $10^3$ respectively.
Surviving fraction after $n$ half-lives is $(1/2)^n$ : $1/2, 1/4, 1/8, \dots$ — never $1/n$ or $1/n^2$ . Decay is exponential, not linear.
Assertion-Reason method: judge the Assertion and the Reason as independently true or false first, then ask whether the Reason actually explains the Assertion. For 'nuclear density is constant' the Reason ' $R \propto A^{1/3}$ so volume $\propto A$ ' is true and is the correct explanation, giving the 'both true, Reason explains Assertion' option.

Where the marks go

Confusing the $n$ -scalings: writing $r \propto n$ or $E \propto -1/n$ instead of $r \propto n^2$ and $E \propto -1/n^2$ . Energy of the $n=2$ state is $-3.4\,\text{eV}$ ( $=-13.6/4$ ), not $-6.8\,\text{eV}$ .
Using $1/n$ for the surviving fraction after $n$ half-lives: after $3$ half-lives the answer is $1/8$ , not $1/3$ .
Inverting the mean-life relation, writing $\tau = 0.693\,T_{1/2}$ . It is the other way round: $\tau = T_{1/2}/0.693 \approx 1.44\,T_{1/2}$ , so mean life exceeds half-life.
Picking $931.5\,\text{keV}$ or $\text{eV}$ for $1\,\text{u}$ , or thinking nuclear density depends on $A$ — both follow from forgetting $R \propto A^{1/3}$ and the MeV scale of nuclear energies.

How the board asks it

Assertion–Reason $R \propto A^{1/3}$ and nuclear density
Assertion: The density of a nucleus is independent of its mass number $A$ . Reason: The nuclear radius is given by $R = R_0 A^{1/3}$ , so nuclear volume is proportional to $A$ . Choose: (a) both true and Reason explains Assertion; (b) both true but Reason does not explain Assertion; (c) Assertion true, Reason false; (d) Assertion false, Reason true.
Assertion–Reasonmean life vs half-life
Assertion: The mean life of a radioactive sample is greater than its half-life. Reason: Mean life and half-life are related by $\tau = T_{1/2}/0.693 \approx 1.44\,T_{1/2}$ . Choose: (a) both true and Reason explains Assertion; (b) both true but Reason does not explain Assertion; (c) Assertion true, Reason false; (d) Assertion false, Reason true.
Assertion–ReasonBohr $n$ -scalings for orbital speed
Assertion: The orbital speed of the electron in a hydrogen atom decreases as it moves to higher orbits. Reason: For a hydrogen-like atom $v \propto Z/n$ , so speed falls as $n$ rises. Choose: (a) both true and Reason explains Assertion; (b) both true but Reason does not explain Assertion; (c) Assertion true, Reason false; (d) Assertion false, Reason true.
Multiple choicesurviving fraction $(1/2)^n$
The fraction of a radioactive sample that remains undecayed after $3$ half-lives is: (a) $1/3$ (b) $1/6$ (c) $1/8$ (d) $1/9$ . Choose the correct option.
Multiple choice $1\,\text{u} = 931.5\,\text{MeV}$ and spectral series
Which one of the following statements is correct? (a) $1\,\text{u} = 931.5\,\text{keV}$ (b) the Balmer series lies in the visible region (c) the Lyman series ends on $n = 2$ (d) nuclear density increases with $A$ . Choose the correct option.
Multiple choicebinding-energy-per-nucleon curve
On the binding-energy-per-nucleon curve, the most stable nuclei lie near the peak at: (a) $A \approx 4$ (b) $A \approx 56$ (c) $A \approx 120$ (d) $A \approx 238$ . Choose the correct option.

PreviousMass Defect & Binding Energy

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.