Question

A vertical column $50cm$ long at $50^{\circ }C$ balances another column of same liquid $60cm$ 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, $h_1=50cm,T_1=50^{\circ }C$
$h_2=60cm,T_2=100^{\circ }C$
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., $p=\rho gh$
$\Rightarrow {\rho }_{1}g{h}_1={\rho }_{2}g{h}_2$
$\Rightarrow \frac{{\rho }_1}{{\rho }_2}=\frac{h_1}{h_2}$
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
$\frac{\frac{{\rho }_0}{1+r{T}_1}}{\frac{{\rho }_0}{1+r{T}_2}}=\frac{h_1}{h_2}=\frac{60}{50}$
$\Rightarrow \frac{1+r{T}_1}{1+r{T}_2}=\frac{6}{5}\Rightarrow 5r{T}_2-6r{T}_1=1$
$\Rightarrow r=\frac{1}{200}=0.005/{}^{\circ }C$