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 `gamma_C`. What is the coefficient of linear expansion of silver

Options

A. `(C+gamma_C+S)//3`
B. `(C-gamma_C+S)//3`
C. `(C+gamma_C+S)//3`
D. `(C-gamma_C-S)//3`

Explanation

To solve the problem, we need to find the coefficient of linear expansion of silver, given the coefficients of apparent expansion of a liquid in copper and silver vessels, and the coefficient of volume expansion of copper.

### Step-by-Step Solution:

1. **Understanding the Coefficient of Apparent Expansion**:
The coefficient of apparent expansion of a liquid in a vessel is given by the formula:
\[ \gamma_{\text{apparent}} = \gamma_{\text{liquid}} - \gamma_{\text{vessel}} \]
where \(\gamma_{\text{liquid}}\) is the coefficient of volume expansion of the liquid and \(\gamma_{\text{vessel}}\) is the coefficient of volume expansion of the vessel material.

2. **Setting Up the Equations**:
For the liquid in a copper vessel, we have:
\[ C = \gamma_{\text{liquid}} - \gamma_C \]
Rearranging gives:
\[ \gamma_{\text{liquid}} = C + \gamma_C \tag{1} \]

For the liquid in a silver vessel, we have:
\[ S = \gamma_{\text{liquid}} - \gamma_S \]
Rearranging gives:
\[ \gamma_{\text{liquid}} = S + \gamma_S \tag{2} \]

3. **Equating the Two Expressions for \(\gamma_{\text{liquid}}\)**:
From equations (1) and (2), we set them equal to each other:
\[ C + \gamma_C = S + \gamma_S \]

4. **Solving for \(\gamma_S\)**:
Rearranging the equation to isolate \(\gamma_S\):
\[ \gamma_S = C + \gamma_C - S \]

5. **Finding the Coefficient of Linear Expansion**:
The coefficient of linear expansion (\(\alpha\)) is related to the coefficient of volume expansion (\(\gamma\)) by the formula:
\[ \alpha = \frac{1}{3} \gamma \]
Therefore, the coefficient of linear expansion of silver (\(\alpha_S\)) can be expressed as:
\[ \alpha_S = \frac{1}{3} \gamma_S \]
Substituting \(\gamma_S\) from the previous step:
\[ \alpha_S = \frac{1}{3}(C + \gamma_C - S) \]

### Final Answer:
The coefficient of linear expansion of silver is:
\[ \alpha_S = \frac{1}{3}(C + \gamma_C - S) \]