Question

The coefficient of apparent volume expansion of a liquid in a copper vessel is $C$ and in a silver vessel is $S$. The coefficient of volume expansion of copper is ${\gamma _c}$. What is the coefficient of linear expansion of silver?
(A) $\dfrac{{\left( {C + {\gamma _c} + S} \right)}}{3}$
(B) $\dfrac{{\left( {C - {\gamma _c} + S} \right)}}{3}$
(C) $\dfrac{{\left( {C + {\gamma _c} - S} \right)}}{3}$
(D) $\dfrac{{\left( {C - {\gamma _c} - S} \right)}}{3}$

Answer 1

Hint We are considering a liquid that is heated in a copper vessel. The coefficient of apparent volume expansion of the liquid in the copper vessel is given. The coefficient of apparent volume expansion when the liquid is heated in a silver vessel is also given. From this, we have to find the coefficient of linear expansion of silver.
Formula used:
The actual expansion of the liquid
${\gamma _L} = {\gamma _{app}} + {\gamma _V}$
where ${\gamma _L}$ is the actual expansion of the liquid, ${\gamma _{app}}$ stands for the apparent expansion of the liquid, and ${\gamma _V}$ stands for the volume expansion of the solid container.

Complete Step by step solution
The apparent expansion will always be slightly less than the actual expansion of the liquid.
Therefore, we can write the actual expansion as,
${\gamma _L} = {\gamma _{app}} + {\gamma _V}$
The coefficient of apparent expansion of the liquid in a copper vessel is given as $C$
The coefficient of volume expansion of the copper vessel be ${\gamma _c}$
The actual expansion of the liquid can be written as,
${\gamma _L} = C + {\gamma _c}$
The coefficient of apparent expansion of the liquid in a silver vessel is given as $S$
Let the coefficient of volume expansion of the copper vessel be ${\gamma _s}$
The actual expansion of the liquid can be written as,
${\gamma _L} = S + {\gamma _s}$
Since the L.H.S of the two equations are the same, we can equate them
We get
$C + {\gamma _c} = S + {\gamma _s}$
From this, we can write the volume expansion coefficient of silver as,
${\gamma _s} = C + {\gamma _c} - S$
The volume expansion coefficient will be three times that of the linear expansion coefficient. i.e.
${\gamma _s} = 3{\alpha _s}$
Substituting this value in the above equation,
$3{\alpha _s} = C + {\gamma _c} - S$
From this, we can write the linear expansion coefficient of silver as,
${\alpha _s} = \dfrac{{C + {\gamma _c} - S}}{3}$
The answer is:

Option (C): $\dfrac{{\left( {C + {\gamma _c} - S} \right)}}{3}$

Note
The apparent expansion of liquid means that, when a liquid is heated in a container, the heat will be transmitted to the liquid through the solid container. Therefore, the container will expand first. This will cause the level of liquid to drop slightly. When the liquid gets heated up, it will start rising and will reach a level beyond its original level. This is known as the apparent expansion of the liquid.