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 the vessel A is $\alpha$ , the coefficient of linear expansion of the vessel B is
(A) $\dfrac{{\alpha {\gamma _1}{\gamma _2}}}{{{\gamma _1} + {\gamma _2}}}$
(B) $\dfrac{{{\gamma _1} - {\gamma _2}}}{{2\alpha }}$
(C) $\dfrac{{{\gamma _1} - {\gamma _2} + \alpha }}{3}$
(D) $\dfrac{{{\gamma _1} - {\gamma _2}}}{3} + \alpha$

Answer 1

Hint
Since the given liquid is same, we know that coefficient of real expansion $\left( {{\gamma _{real}}} \right)$ is equal for the same liquid i.e., ${\gamma _{real}} = {\gamma _{app}} + {\alpha _v}$ ; where ${\alpha _v}$ is coefficient of volume expansion . So, we equal the coefficients of real expansions for the two vessels and find the coefficient of linear expansion of the vessel B.

Complete step by step answer
Now, For vessel A,
${\gamma _A}_{real} = {\gamma _1} + {\alpha _{vA}}$
Since, ${\alpha _{vA}} = 3\alpha$
$\Rightarrow {\gamma _A}_{real} = {\gamma _1} + 3\alpha$ …(i)
And for vessel B
$\Rightarrow {\gamma _{Breal}} = {\gamma _2} + {\alpha _{vB}}$
Since, ${\alpha _{vB}} = 3{\alpha _B}$
$\Rightarrow {\gamma _B}_{real} = {\gamma _2} + 3{\alpha _B}$ …(ii)
Where, ${\alpha _B}$ is coefficient of linear expansion of the vessel B.
Now, we equal the coefficients of real expansions for the two vessels
From (i) and (ii), we get
${\gamma _2} + 3{\alpha _B} = {\gamma _1} + 3\alpha$
$\Rightarrow 3{\alpha _B} = {\gamma _1} - {\gamma _2} + 3\alpha$
$\Rightarrow {\alpha _B} = \dfrac{{{\gamma _1} - {\gamma _2} + 3\alpha }}{3}$
$\Rightarrow {\alpha _B} = \dfrac{{{\gamma _1} - {\gamma _2}}}{3} + \alpha$
Therefore, the coefficient of linear expansion of the vessel (B) is $\dfrac{{{\gamma _1} - {\gamma _2}}}{3} + \alpha$
Hence, option (D) is correct.

Note
Here the coefficient of real expansion is same and it is the sum of coefficient of apparent expansion and the coefficient of volume expansion but not the coefficient of linear expansion and the relation between coefficient of linear expansion and coefficient of volume expansion is ${\alpha _v} = 3\alpha$