Solution:

Ans. We have

$\therefore$ Density,

$$ \begin{align*} R & =R_{0} A^{1 / 3} \ \rho & =\frac{m A}{\frac{4}{3} \pi\left(R_{0} A^{1 / 3}\right)^{3}} \ & =\frac{m}{\frac{4}{3} \pi R_{0}^{3}} \end{align*} $$

Hence $\rho$ is independent of $A$.

(Here $m$ is the mass of the nucleus.)

$1 / 2$

[CBSE Marking Scheme 2013]

Detailed Answer :

mass of nucleus $M=$ volume of nucleus $\times$ nucleus

$$ \begin{aligned} M= & V \times \rho \ M= & \frac{4}{3} \pi R^{3} \rho \ & (R=\text { radius of the nucleus }) \ R^{3}= & \frac{3 M}{4 \pi \rho} \end{aligned} $$

$$ \begin{equation*} R=\left(\frac{3}{4 \pi \rho}\right)^{\frac{1}{3}} M^{\frac{1}{3}} \tag{i} \end{equation*} $$

If $m=$ mass of one nucleon then mass of nucleus $M=m A$ where $A=$ mass number $(Z+N)$

Putting the value of $M$ in eq. (i)

$$ R=\left(\frac{3}{4 \pi \rho}\right)^{\frac{1}{3}}(m A)^{\frac{1}{3}} $$

We know that

$$ R=R_{0} A^{\frac{1}{3}} $$

Hence,

$$ R_{0} A^{\frac{1}{3}}=\left(\frac{3}{4 \pi \rho}\right)^{\frac{1}{3}}(m A)^{\frac{1}{3}} $$

Cubing both sides

$$ R_{0}^{3} A=\frac{3}{4 \pi \rho} m A $$

Or $\rho=\frac{3 m}{4 \pi R_{0}^{3}}$

Hence proved that Nuclear density $\rho$ over a wide range of nuclei is constant and independent of mass number $A$.

(i) The mass of a nucleus in its ground state is always less than the total mass of its constituents neutrons and protons. Explain.

(ii) Plot a graph showing variation of potential energy of a pair of nucleons as a function of their separation.

U] + R [OD 2009]

Ans. (i) Protons and neutrons have to come very near to bind together and form a nucleus. This distance is of the order of $10^{-14} \mathrm{~m}$. In order to achieve this distance a lot of energy is required. The required energy is provided by the nucleon at the expense of some portion of their masses. This is the reason that the mass of a nucleus in its ground state is always less than the total mass of its constituents neutrons and protons.

(ii)

$1+1=2$

Commonly Made Error

• Most of the students couldn’t write the correct reasoning.
• They wrote that mass of a nucleus is always less than the total mass of its constituents and energy is released if it is converted to a nucleus of high $B E / A$.