11. (JEE Main 2019 (Online) 8th April Evening Slot)

The ratio of mass densities of nuclei of ⁴⁰Ca and ¹⁶O is close to :-

A. 1

B. 5

C. 0.1

D. 2

The correct answer is option (A)

Densities of nucleus happens to be constant, irrespective of mass number.

12.(JEE Main 2018 (Online) 15th April Evening Slot)

An unstable heavy nucleus at rest breaks into two nuclei which move away with velocities in the ratio of 8 : 27. The ratio of the radii of the nuclei (assumed to be spherical) is :

A. 8 : 27

B. 4 : 9

C. 3 : 2

D. 2 : 3

The correct answer is option (C)

The two nuclei have velocity in ratio 8 : 27. By conservation of momentum, we have

$m_{1} v_{1} = m_{2} v_{2} \Rightarrow \frac{v_{1}}{v_{2}} = \frac{m_{2}}{m_{1}} \Rightarrow \frac{m_{2}}{m_{1}} = \frac{8}{27}$

Now, since $m = ρ \frac{4}{3} π r^{3}$

Therefore, $\frac{m_{2}}{m_{1}} = \frac{ρ \frac{4}{3} π r_{2}^{3}}{ρ \frac{4}{3} π r_{1}^{3}} \Rightarrow \frac{m_{2}}{m_{1}} = {(\frac{r_{2}}{r_{1}})}^{3} \Rightarrow {(\frac{r_{2}}{r_{1}})}^{3} = \frac{8}{27}$

$\Rightarrow \frac{r_{2}}{r_{1}} = \frac{2}{3}$

Thus, ratio of radii of nuclei $r_{1} : r_{2} = 3 : 2$ .

13.(AIEEE 2005)

If radius of the $\begin{matrix} 27 \\ 13 \end{matrix}$ $A l$ nucleus is estimated to be $3.6$ fermi then the radius of $\begin{matrix} 125 \\ 52 \end{matrix} T e$ nucleus is estimated to be nearly

A. $8$

B. $6$

C. $5$ fermi

D. $4$

The correct answer is option (B)

KEY CONCEPT : $R = R_{0} {(A)}^{1 / 3}$

Here A = Mass number

$∴$ $\frac{R_{1}}{R_{2}} = {(\frac{A_{1}}{A_{2}})}^{1 / 3}$

$= {(\frac{27}{125})}^{1 / 3} = \frac{3}{5}$

$R_{2} = \frac{5}{3} \times 3.6 = 6$ fermi

14.(AIEEE 2004)

A nucleus disintegrated into two nuclear parts which have their velocities in the ratio of $2 : 1.$ The ratio of their nuclear sizes will be

A. $3^{1 / 2} : 1$

B. $1 : 2^{1 / 3}$

C. $2^{1 / 3} : 1$

D. $1 : 3^{1 / 2}$

The correct answer is option (B)

From conservation of momentum $m_{1} v_{1} = m_{2} v_{2}$

$\Rightarrow (\frac{m_{1}}{m_{2}}) = (\frac{v_{2}}{v_{1}})$ given $\frac{v_{1}}{v_{2}} = 2$

$\Rightarrow \frac{m_{1}}{m_{2}} = \frac{1}{2}$

$\Rightarrow \frac{r_{1}^{3}}{r_{2}^{3}} = \frac{1}{2}$

$\Rightarrow (\frac{r_{1}}{r_{2}}) = {(\frac{1}{2})}^{1 / 3}$

15.(AIEEE 2003)

The wavelengths involved in the spectrum of deuterium $(_{1}^{2} D)$ are slightly different from that of hydrogen spectrum, because

A. the size of the two nuclei are different

B. the nuclear forces are different in the two cases

C. the masses of the two nuclei are different

D. the attraction between the electron and the nucleus is different in the two cases

The correct answer is option (C)

The wavelength of spectrum is given by

$\frac{1}{λ} = R z^{2} (\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}})$

where $R = \frac{1.097 \times 10^{7}}{1 + \frac{m}{M}}$ $m =$ mass of electron

$M =$ mass of nucleus.

For different $M, R$ is different and therefore $λ$ is different

Nuclei