Question

The coefficient of apparent expansion of a liquid when determined using two different vessels A and B are ${\gamma }_1$ and ${\gamma }_2$ respectively. If the coefficient of linear expansion of vessel A is $\alpha$, the coefficient of linear expansion of vessel B is

• A. $\frac{\alpha {\gamma }_1{\gamma }_2}{{\gamma }_1+{\gamma }_2}$
• B. $\frac{{\gamma }_1-{\gamma }_2}{2\alpha }$
• C. $\frac{{\gamma }_1-{\gamma }_2+\alpha }{3}$
• D. $\frac{{\gamma }_1-{\gamma }_2}{3}+\alpha$

Answer 1

The correct option is D

$\frac{{\gamma }_1-{\gamma }_2}{3}+\alpha$

We know that, coefficient of real expansion of a liquid = coefficient of apparent expansion of the liquid + coefficient of volume expansion of the vessel
$\Rightarrow {\gamma }_r={\gamma }_a+{\gamma }_v={\gamma }_a+3{\alpha }_v$
For vessel A: ${\gamma }_r={\gamma }_1+3{\alpha }_1={\gamma }_1+3\alpha (\because {\alpha }_1=\alpha )$
For vessel B: ${\gamma }_r={\gamma }_2+3{\alpha }_2$
Since the liquid is the same, ${\gamma }_r$ is same.
Hence, ${\gamma }_1+3\alpha ={\gamma }_2+3{\alpha }_2$
$\Rightarrow {\alpha }_2=\frac{{\gamma }_1-{\gamma }_2}{3}+\alpha$, which is choice (d).