Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R. Assertion A : The nuclear density of nuclides ${}_{5}^{10} \mathrm{~B},{}_{3}^{6} \mathrm{Li},{}_{26}^{56} \mathrm{Fe},{}_{10}^{20} \mathrm{Ne}$ and ${}_{83}^{209} \mathrm{Bi}$ can be arranged as $\rho_{\mathrm{Bi}}^{\mathrm{N}}>\rho_{\mathrm{Fe}}^{\mathrm{N}}>\rho_{\mathrm{Ne}}^{\mathrm{N}}>\rho_{\mathrm{B}}^{\mathrm{N}}>\rho_{\mathrm{Li}}^{\mathrm{N}}$. Reason R : The radius R of nucleus is related to its mass number $A$ as $R=R_{0} A^{1 / 3}$, where $R_{0}$ is a constant. In the light of the above statement, choose the correct answer from the options given below :

(B) $\mathbf{A}$ is false but $\mathbf{R}$ is true

(A) Both $\mathbf{A}$ and $\mathbf{R}$ are true and R is the correct explanation of A

(C) $\mathbf{A}$ is true but $\mathbf{R}$ is false

(D) Both $\mathbf{A}$ and $\mathbf{R}$ are true but $\mathbf{R}$ is NOT the correct explanation of A

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Modern Physics

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A : The nuclear density of nuclides

_{5}^{10} B,_{3}^{6} Li,_{26}^{56} Fe,_{10}^{20} Ne

and

_{83}^{209} Bi

can be arranged as

ρ_{Bi}^{N} > ρ_{Fe}^{N} > ρ_{Ne}^{N} > ρ_{B}^{N} > ρ_{Li}^{N}

.
Reason R : The radius R of nucleus is related to its mass number

A

R = R_{0} A^{1/3}

, where

R_{0}

is a constant.
In the light of the above statement, choose the correct answer from the options given below :

(A)Both $A$ and $R$ are true and R is the correct explanation of A
(B) $A$ is false but $R$ is true✓ Correct
(C) $A$ is true but $R$ is false
(D)Both $A$ and $R$ are true but $R$ is NOT the correct explanation of A

Answer: (B)

Solution

The nuclear radius obeys

R = R_{0} A^{1/3}

, so the nuclear volume

\frac{4}{3} π R^{3} \propto A

. The nuclear mass is also proportional to

A

, hence the density

ρ = \frac{A m _{u}}{\frac{4}{3} π R _{0}^{3} A}

cancels its dependence on the mass number and is the same for every nucleus. The ordering of densities claimed in the assertion therefore cannot hold, while the radius relation stated in the reason is correct; the assertion is false but the reason is true.

JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Single correct option

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JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Modern Physics

Solution

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