Question

The radius of the nucleus is proportional to, (if A is the atomic mass number)
$A.\quad A$
$B.\quad { A }^{ 3 }$
$C.\quad { A }^{ \dfrac { 1 }{ 3 } }$
$D.\quad { A }^{ \dfrac { 2 }{ 3 } }$

Answer 1

Hint: Use result from Rutherford’s experiment stating relation between volume of nucleus and atomic mass number. Substitute value for volume of nucleus and find the relation between radius of nucleus and atomic mass number.

Formula used:
$Volume\quad of\quad nucleus\quad \propto \quad Atomic\quad mass\quad number$

Complete step-by-step answer:
According to Rutherford’s Experiment, volume of nucleus is proportional to atomic mass number.

$Volume\quad of\quad nucleus\quad \propto \quad Atomic\quad mass\quad number$
$\therefore \quad V\quad \propto \quad A$

But, volume of nucleus is $\dfrac { 4 }{ 3 } \pi { R }^{ 3 }$

$\therefore \dfrac { 4 }{ 3 } \pi { R }^{ 3 }\quad \propto \quad A$
$\therefore \quad { R }^{ 3 }\quad \propto \quad A$
$\therefore \quad { R }\quad \propto \quad { A }^{ \dfrac { 1 }{ 3 } }$ …(1)
$\therefore \quad R={ R }_{ 0 }\quad { A }^{ \dfrac { 1 }{ 3 } }$
where, ${ R }_{ 0 }$ is a constant and ${ R }_{ 0 }=\quad 1.2\times { 10 }^{ -15 }m$= 1.2 fm.
The value of ${ R }_{ 0 }$ is the same for every nucleus.

From equation (1), we can say ${ R\quad }\propto \quad { A }^{ \dfrac { 1 }{ 3 } }$

So, the correct answer is “Option C”.

Note:
This relation between the radius of nucleus and atomic mass number can be used to find the unknown radius of one nucleus if radius and atomic mass number of another nucleus is given. And similarly unknown atomic mass number can be calculated from the known radius and atomic mass number of another nucleus.