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 find the coefficient of linear expansion of silver (denoted as α_S) given the coefficients of apparent expansion of a liquid in copper (C) and silver (S) vessels, as well as the coefficient of volume expansion of copper (γ_C), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Apparent Expansion**:
The coefficient of apparent expansion of a liquid in a vessel is defined as the difference between the coefficient of volume expansion of the liquid (γ_L) and the coefficient of volume expansion of the vessel (γ_vessel). This can be expressed as:
\[ \gamma_{apparent} = \gamma_L - \gamma_{vessel} \]

2. **Setting Up the Equations**:
For the copper vessel, we have:
\[ \gamma_L = C + \gamma_C \]
For the silver vessel, we have:
\[ \gamma_L = S + \gamma_S \]

3. **Equating the Expressions**:
Since both expressions represent the same liquid (with the same coefficient of volume expansion γ_L), we can equate them:
\[ C + \gamma_C = S + \gamma_S \]

4. **Rearranging the Equation**:
Rearranging the above equation to isolate γ_S gives us:
\[ \gamma_S = C + \gamma_C - S \]

5. **Relating Volume Expansion to Linear Expansion**:
The coefficient of volume expansion (γ) is related to the coefficient of linear expansion (α) by the relation:
\[ \gamma = 3\alpha \]
Therefore, for silver, we can write:
\[ \gamma_S = 3\alpha_S \]

6. **Substituting for γ_S**:
Substituting the expression for γ_S into the linear expansion equation gives:
\[ 3\alpha_S = C + \gamma_C - S \]

7. **Solving for α_S**:
Finally, we can solve for α_S:
\[ \alpha_S = \frac{C + \gamma_C - S}{3} \]

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