The coefficient of apparent expansion of a liquid in a copper vessel is C and in a silver vessel S. The coefficient of volume expansion of copper is γc . What is the coefficient of linear expansion of silver

The correct option is (C + γc -S)/3.

To find the coefficient of linear expansion of silver, we use the relationship between the real expansion coefficient of a liquid and its apparent expansion coefficient in a vessel.

The real coefficient of volume expansion of a liquid (${\gamma }_r$) is equal to the sum of the coefficient of apparent expansion of the liquid (${\gamma }_a$) and the coefficient of volume expansion of the vessel (${\gamma }_v$):
${\gamma }_r={\gamma }_a+{\gamma }_v$

Step 1: Write the equation for the liquid in the copper vessel
For the copper vessel, the apparent expansion of the liquid is $C$ and the volume expansion of copper is ${\gamma }_c$. Therefore, the real volume expansion coefficient of the liquid is:
${\gamma }_r=C+{\gamma }_c$        --- (Equation 1)

Step 2: Write the equation for the liquid in the silver vessel
For the silver vessel, the apparent expansion of the liquid is $S$. Let ${\gamma }_s$ be the coefficient of volume expansion of silver. The real volume expansion coefficient of the same liquid is:
${\gamma }_r=S+{\gamma }_s$        --- (Equation 2)

Step 3: Equate the two expressions for the real expansion of the liquid
Since the liquid is the same in both cases, its real expansion coefficient (${\gamma }_r$) remains constant. Equating Equation 1 and Equation 2 gives:
$C+{\gamma }_c=S+{\gamma }_s$

Rearranging the equation to solve for the coefficient of volume expansion of silver (${\gamma }_s$):
${\gamma }_s=C+{\gamma }_c-S$

Step 4: Find the coefficient of linear expansion of silver
The coefficient of volume expansion ($\gamma$) of a material is three times its coefficient of linear expansion ($\alpha$). Thus, for silver:
${\gamma }_s=3{\alpha }_s$

Substituting this relation into the expression for ${\gamma }_s$:
$3{\alpha }_s=C+{\gamma }_c-S$

${\alpha }_s=\frac{C+{\gamma }_c-S}{3}$

Hence, the coefficient of linear expansion of silver is $\frac{C+{\gamma }_c-S}{3}$.