Question

A vertical column $50\, cm$ long at $50^{\circ} C$ balances another column of same liquid $60 \,cm$ along at $100^{\circ} C$. The coefficient of absolute expansion of the liquid is

• A. $0.002 /{ }^{\circ} C$
• B. $0.003 /{ }^{\circ} C$
• C. $0.004 /{ }^{\circ} C$
• D. $0.005 /{ }^{\circ} C$

Answer 1

Answer: (D)

Given, $h_1=50 \,cm , T_1=50^{\circ} C$
$h_2=60\,cm , T_2=100^{\circ} C$
Let the density of the given liquid at STP be $\rho_0$, if both vertical columns balance each other, then their pressure should be equal.
i.e., $p=\rho g h$
$ \Rightarrow \rho_1 g h_1=\rho_2 g h_2 $
$ \Rightarrow \frac{\rho_1}{\rho_2}=\frac{h_1}{h_2}$
If $r$ be the coefficient of absolute expansion of liquid, then, $\rho_1=\frac{p_0}{1+r T_1}$ and $\rho_2=\frac{\rho_0}{1+r T_2}$
$\therefore$ From Eq. (i) we have
$ \frac{\frac{\rho_0}{1+r T_1}}{\frac{\rho_0}{1+r T_2}}=\frac{h_1}{h_2}=\frac{60}{50} $
$ \Rightarrow \frac{1+r T_1}{1+r T_2}=\frac{6}{5} \Rightarrow 5 r T_2-6 r T_1=1$
$ \Rightarrow r=\frac{1}{200}=0.005 /{ }^{\circ} C$