The apparent coefficient of expansion of liquid, when heated in a copper vessel is `C` and when heated in a silver vessel is `S`. If `A` is the linear coefficient of expansion of Copper, linear expansion coefficient of silver is

Options

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

Answer

To solve the problem, we need to find the linear coefficient of expansion of silver (denoted as \( \alpha_{Ag} \)) based on the apparent coefficients of expansion of a liquid in copper and silver vessels.

### Step-by-Step Solution:

1. **Understanding Apparent Coefficient of Expansion**:
- The apparent coefficient of expansion of a liquid in a vessel is given by:
\[ C = \gamma - 3A \]
where \( \gamma \) is the true coefficient of volume expansion of the liquid and \( A \) is the linear coefficient of expansion of the material of the vessel (in this case, copper).

2. **Equation for Copper Vessel**:
- For the copper vessel, we can write:
\[ C = \gamma - 3A \]
- Rearranging gives:
\[ \gamma = C + 3A \quad \text{(Equation 1)} \]

3. **Equation for Silver Vessel**:
- Similarly, for the silver vessel, the apparent coefficient of expansion is:
\[ S = \gamma - 3\alpha_{Ag} \]
- Rearranging gives:
\[ \gamma = S + 3\alpha_{Ag} \quad \text{(Equation 2)} \]

4. **Equating the Two Expressions for \( \gamma \)**:
- From Equation 1 and Equation 2, we can equate the two expressions for \( \gamma \):
\[ C + 3A = S + 3\alpha_{Ag} \]

5. **Solving for \( \alpha_{Ag} \)**:
- Rearranging the equation to isolate \( \alpha_{Ag} \):
\[ 3\alpha_{Ag} = C + 3A - S \]
- Dividing both sides by 3 gives:
\[ \alpha_{Ag} = \frac{C + 3A - S}{3} \]

### Final Answer:
The linear coefficient of expansion of silver is given by:
\[ \alpha_{Ag} = \frac{C + 3A - S}{3} \]