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 the vessel B is?

We know that coefficient of real expansion for liquid$\left({\gamma }_r\right)$=coefficient of apparent expansion of the liquid$\left({\gamma }_a\right)$+coefficient of volume or cubical expansion of the vessel$\left({\gamma }_v\right)$

i.e., ${\gamma }_r={\gamma }_a+{\gamma }_v={\gamma }_a+3\alpha$

since the liquid is same in both the vessel, so value of$\left({\gamma }_r\right)$ is same.

From vessel A,

${\gamma }_r={\gamma }_1+3\alpha .......\left(1\right)$

For vessel B

${\gamma }_r={\gamma }_2+3{\alpha }_2 ......\left(2\right)$

From (1) and (2)

${\gamma }_1+3\alpha ={\gamma }_2+3{\alpha }_2$

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