The coefficient of apparent expansion of mercury in a glass vessel is `153xx10^(-6)//^(@)C` and in a steel vessel is `114xx10^(-6)//^(@)C`. If `alpha` for steel is `12xx10^(-6)//^(@)C`, then that of glass is

Options

A. `9xx10^(-6).^@C^(-1)`
B. `6xx10^(-6).^@C^(-1)`
C. `36xx10^(-6).^@C^(-1)`
D. `27xx10^(-6).^@C^(-1)`

Answer

To find the coefficient of linear expansion of glass (αg), we can use the relationship between the apparent expansion coefficients of mercury in different vessels and the actual expansion coefficients of the materials involved.

### Step-by-Step Solution:

1. **Understand the relationship**: The coefficient of apparent expansion (γ) of a liquid in a container is given by the formula:
\[ \gamma = \alpha_L - \alpha_C \]
where:
- \( \gamma \) = coefficient of apparent expansion of the liquid,
- \( \alpha_L \) = coefficient of volume expansion of the liquid,
- \( \alpha_C \) = coefficient of volume expansion of the container.

2. **Write down the equations for both vessels**:
- For the glass vessel:
\[ \gamma_{glass} = \alpha_L - \alpha_g \quad (1) \]
- For the steel vessel:
\[ \gamma_{steel} = \alpha_L - \alpha_s \quad (2) \]
where \( \alpha_g \) is the coefficient of volume expansion of glass and \( \alpha_s \) is the coefficient of volume expansion of steel.

3. **Substitute the known values**:
- Given:
- \( \gamma_{glass} = 153 \times 10^{-6} \, ^(@)C \)
- \( \gamma_{steel} = 114 \times 10^{-6} \, ^(@)C \)
- \( \alpha_s = 12 \times 10^{-6} \, ^(@)C \)

4. **Set up the equations**:
- From equation (1):
\[ 153 \times 10^{-6} = \alpha_L - \alpha_g \quad (3) \]
- From equation (2):
\[ 114 \times 10^{-6} = \alpha_L - 12 \times 10^{-6} \quad (4) \]

5. **Solve equation (4) for αL**:
\[ \alpha_L = 114 \times 10^{-6} + 12 \times 10^{-6} = 126 \times 10^{-6} \, ^(@)C \]

6. **Substitute αL into equation (3)**:
\[ 153 \times 10^{-6} = 126 \times 10^{-6} - \alpha_g \]
Rearranging gives:
\[ \alpha_g = 126 \times 10^{-6} - 153 \times 10^{-6} \]
\[ \alpha_g = -27 \times 10^{-6} \, ^(@)C \]

7. **Final answer**: The coefficient of volume expansion of glass is:
\[ \alpha_g = 27 \times 10^{-6} \, ^(@)C \]