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

Options

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

Explanation

To solve the problem, we need to derive the relationship between the coefficients of apparent expansion of a liquid in different vessels and the coefficients of linear expansion of the materials of those vessels.

### Step-by-Step Solution:

1. **Understanding Apparent Expansion**:
The coefficient of apparent expansion of a liquid when heated in a vessel can be expressed as:
\[ C = \gamma - 3A \]
where \(C\) is the coefficient of apparent expansion in the copper vessel, \(\gamma\) is the coefficient of volume expansion of the liquid, and \(A\) is the coefficient of linear expansion of copper.

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

3. **Setting Up the Equation for Silver**:
Similarly, for the silver vessel, the apparent expansion coefficient can be expressed as:
\[ S = \gamma - 3\alpha_{Ag} \]
where \(S\) is the coefficient of apparent expansion in the silver vessel and \(\alpha_{Ag}\) is the coefficient of linear expansion of silver.

4. **Setting Up the Second Equation**:
For silver, we can write:
\[ S = \gamma - 3\alpha_{Ag} \quad \text{(Equation 2)} \]

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

6. **Rearranging to Find \(\alpha_{Ag}\)**:
Rearranging the equation to solve for \(\alpha_{Ag}\):
\[ 3\alpha_{Ag} = C + 3A - S \]
\[ \alpha_{Ag} = \frac{C - S + 3A}{3} \]

7. **Final Expression**:
Thus, the coefficient of linear expansion of silver is:
\[ \alpha_{Ag} = \frac{C - S + 3A}{3} \]