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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
$\frac{{\alpha {\gamma _1}{\gamma _2}{\kern 1pt} }}{{{\gamma _1} + {\gamma _2}{\kern 1pt} }}$
$\frac{{{\gamma _1} - {\gamma _2}}}{{2\alpha }}$
$\frac{{{\gamma _1} - {\gamma _2} + \alpha }}{3}$
$\frac{{{\gamma _1} - {\gamma _2}}}{3} + \alpha $
Solution
$\gamma_{\text {real}}=\gamma_{\text {app}}+\gamma_{\text {vessel}} ; \gamma_{\text {vessle}}=3 \alpha$
$ForvesselA$ $=\gamma_{\text {real}}=\gamma_{1}+3 \alpha$
$ForvesselB$ $=\gamma_{\text {real}}=\gamma_{2}+3 \alpha_{B}$
Hence, $\gamma_{1}+3 \alpha=\gamma_{2}+3 \alpha_{B}$
$\Rightarrow \alpha_{B}=\frac{\gamma_{1}-\gamma_{2}}{3}+\alpha$
