Question

The apparent coefficient of expansion of liquid when heated in copper vessel is C and when heated in a silver vessel is S. If A is the coefficient of linear expansion of copper, coefficient of linear expansion of silver is:
A. \[\dfrac{{C + S - 3A}}{3}\]
B. \[\dfrac{{C + 3A - S}}{3}\]
C. \[\dfrac{{S + 3A - C}}{3}\]
D. \[\dfrac{{C + S + 3A}}{3}\]

Answer 1

Hint: The real expansion coefficient of the liquid is the sum of apparent expansion coefficient of the liquid and coefficient of expansion of the vessel. Express the real expansion coefficients of liquid in both cases. The real coefficient of expansion of liquid remains the same in both the cases.

Formula used:
\[{\gamma _r} = {\gamma _{app}} + {\gamma _V}\]
Here, \[{\gamma _r}\] is the real coefficient of expansion, \[{\gamma _{app}}\] is the apparent expansion coefficient of the liquid and \[{\gamma _V}\]is the coefficient of expansion of the vessel.

Complete step by step answer:
We have given the apparent coefficient of expansion of a certain liquid when heat in a copper vessel is C and when it is heat in a silver vessel its apparent coefficient of expansion is S. We know the expansion of the liquid is volume expansion.
We know that the real expansion coefficient of the liquid is the sum of apparent expansion coefficient of the liquid and coefficient of expansion of the vessel. Therefore, let’s express the real expansion coefficient of liquid when heated in the copper vessel as follows,
\[{\gamma _r} = {\gamma _{app}} + {\gamma _C}\]
Here, \[{\gamma _{app}}\] is the apparent expansion coefficient of the liquid and \[{\gamma _C}\]is the coefficient of expansion of copper.
Substituting C for \[{\gamma _{app}}\] and 3A for \[{\gamma _C}\] in the above equation, we get,
\[{\gamma _r} = C + 3A\] ……. (1)
Now, let’s express the real coefficient of expansion of liquid when heated in the silver vessel as follows,
\[{\gamma _r} = {\gamma _{app}} + {\gamma _S}\]
Here, \[{\gamma _{app}}\] is the apparent expansion coefficient of the liquid when heated in the silver vessel and \[{\gamma _S}\]is the coefficient of expansion of silver.
Substituting S for \[{\gamma _{app}}\] and \[3{\alpha _S}\] for \[{\gamma _S}\] in the above equation, we get,
\[{\gamma _r} = S + 3{\alpha _S}\] …….. (2)
We know that the real expansion coefficient of the liquid is the same in both the vessels. Therefore, equating equation (1) and (2), we get,
\[C + 3A = S + 3{\alpha _S}\]
\[ \Rightarrow {\alpha _S} = \dfrac{{C + 3A - S}}{3}\]

So, the correct answer is “Option B”.

Note:
The coefficient of volume expansion is always thrice the coefficient of linear expansion. That is why we have substituted 3A for \[{\gamma _C}\] in equation (1). We have given the apparent coefficient of expansion of the liquid since as we heat the liquid, the copper vessel also expands.