IMO Practice Test: Atoms and Nuclei | Class 12 Physics

Question 1 of 14

The ratio of the radii of the third and first Bohr orbits of hydrogen is:

$3 : 1$

$9 : 1$

$1 : 9$

$1 : 3$

Explanation: $r_n \propto n^2$, so $r_3 : r_1 = 9 : 1$.

Question 2 of 14

The ratio of the longest wavelength of the Lyman series to that of the Balmer series is:

$5 : 27$

$27 : 5$

$1 : 3$

$3 : 1$

Explanation: Lyman ($2\to1$): $1/\lambda = R(3/4)$. Balmer ($3\to2$): $1/\lambda = R(5/36)$. Ratio of wavelengths $= \frac{4/3}{36/5} = \frac{20}{108} = \frac{5}{27}$.

Question 3 of 14

The shortest wavelength (series limit) of the Lyman series corresponds to a transition from:

$n = 2 \to 1$

$n = 3 \to 1$

$n = \infty \to 1$

$n = \infty \to 2$

Explanation: The series limit is the $n = \infty \to 1$ transition, giving the shortest $\lambda$.

Question 4 of 14

If the radius of a nucleus of mass number 27 is $3.6\ \text{fm}$, the radius of a nucleus of mass number 64 is:

$2.7\ \text{fm}$

$5.4\ \text{fm}$

$7.2\ \text{fm}$

$4.8\ \text{fm}$

Explanation: $R \propto A^{1/3}$: $R = 3.6 \times (64/27)^{1/3} = 3.6 \times \frac{4}{3} = 4.8\ \text{fm}$.

Question 5 of 14

A nucleus emits one $\alpha$ and two $\beta^-$ particles. Compared with the parent, the daughter has:

the same atomic number

atomic number lower by 2

atomic number higher by 1

mass number lower by 2

Explanation: $Z$: $-2$ (from $\alpha$) $+2$ (from two $\beta^-$) $= 0$ net; $A$ falls by 4. Same $Z$ means it is an isotope of the parent.

Question 6 of 14

A sample decays to $\frac{1}{16}$ of its initial value in 8 hours. Its half-life is:

1 hour

4 hours

2 hours

8 hours

Explanation: $\frac{1}{16} = (1/2)^4$, so 4 half-lives in 8 hr means $T_{1/2} = 2\ \text{hr}$.

Question 7 of 14

Two radioactive samples have half-lives in the ratio $1 : 2$ and the same number of initial nuclei. After a time equal to two half-lives of the first, the ratio of remaining nuclei (first : second) is:

$1 : 1$

$2 : 1$

$1 : 4$

$1 : 2$

Explanation: In time $2T$ (with $T_2 = 2T$): first undergoes 2 half-lives ($1/4$), second undergoes 1 half-life ($1/2$). Ratio $= (1/4):(1/2) = 1 : 2$.

Question 8 of 14

The binding energy of a deuteron ${}^{2}_{1}\text{H}$ is $2.2\ \text{MeV}$. Its binding energy per nucleon is:

$2.2\ \text{MeV}$

$1.1\ \text{MeV}$

$4.4\ \text{MeV}$

$0.55\ \text{MeV}$

Explanation: $\frac{E_b}{A} = \frac{2.2}{2} = 1.1\ \text{MeV per nucleon}$.

Question 9 of 14

When a hydrogen atom is excited from $n=1$ to $n=4$, the number of distinct spectral lines that can be emitted is:

6

4

3

10

Explanation: Number of lines $= \frac{n(n-1)}{2} = \frac{4 \times 3}{2} = 6$.

Question 10 of 14

If $1\ \text{u}$ of mass is fully converted to energy, the energy obtained is closest to:

$931.5\ \text{keV}$

$93.15\ \text{MeV}$

$931.5\ \text{MeV}$

$9.315\ \text{MeV}$

Explanation: $1\ \text{u} \equiv 931.5\ \text{MeV}$ by $E = mc^2$.

Question 11 of 14

The activity of a radioactive sample falls from $8000\ \text{Bq}$ to $1000\ \text{Bq}$ in 9 days. The half-life is:

1 day

9 days

4.5 days

3 days

Explanation: $\frac{1000}{8000} = \frac{1}{8} = (1/2)^3$, so 3 half-lives in 9 days $\Rightarrow T_{1/2} = 3\ \text{days}$.

Question 12 of 14

In the fusion reaction $4\,{}^{1}_{1}\text{H} \to {}^{4}_{2}\text{He} + 2e^{+} + \text{energy}$, the energy comes from:

the kinetic energy of protons only

the increase in binding energy per nucleon

the charge of the positrons

the gravitational field of the Sun

Explanation: Helium has a higher binding energy per nucleon than hydrogen, so the surplus is released as energy ($\Delta m\, c^2$).

Question 13 of 14

The wavelength of the $\text{H}_\alpha$ line ($n=3 \to 2$) is $656\ \text{nm}$. The energy of this photon is about:

$3.4\ \text{eV}$

$13.6\ \text{eV}$

$1.89\ \text{eV}$

$12.1\ \text{eV}$

Explanation: $E = \frac{1240}{656} \approx 1.89\ \text{eV}$, matching $E_3 - E_2$.

Question 14 of 14

A heavy nucleus splits into two equal fragments. Compared with the parent, the radius of each fragment is:

$\frac{1}{2^{1/3}}$ times the parent radius

half the parent radius

the same

twice the parent radius

Explanation: Each fragment has $A/2$, and $R \propto A^{1/3}$, so $R_f = R_p (1/2)^{1/3} = \frac{R_p}{2^{1/3}}$.

IMO Practice Test — Atoms and Nuclei