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`
B. `6xx10^(-6)//^(@)C`
C. `36xx10^(-6)//^(@)C`
D. `27xx(10^(-6)//^(@)C`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Apparent Expansion**:
The coefficient of apparent expansion (β) of a liquid in a vessel is given by the formula:
\[ \beta = \alpha_{liquid} - \alpha_{vessel} \]
where:
- \(\beta\) is the coefficient of apparent expansion,
- \(\alpha_{liquid}\) is the coefficient of linear expansion of the liquid,
- \(\alpha_{vessel}\) is the coefficient of linear expansion of the vessel.

2. **Setting Up the Equations**:
For mercury in a glass vessel:
\[ \beta_{glass} = \alpha_{mercury} - \alpha_{glass} \]
For mercury in a steel vessel:
\[ \beta_{steel} = \alpha_{mercury} - \alpha_{steel} \]

3. **Substituting the Given Values**:
Given:
- \(\beta_{glass} = 153 \times 10^{-6} \, ^(@)C\)
- \(\beta_{steel} = 114 \times 10^{-6} \, ^(@)C\)
- \(\alpha_{steel} = 12 \times 10^{-6} \, ^(@)C\)

From the second equation:
\[ 114 \times 10^{-6} = \alpha_{mercury} - 12 \times 10^{-6} \]
Rearranging gives:
\[ \alpha_{mercury} = 114 \times 10^{-6} + 12 \times 10^{-6} = 126 \times 10^{-6} \, ^(@)C \]

4. **Finding α_glass**:
Now substitute \(\alpha_{mercury}\) into the first equation:
\[ 153 \times 10^{-6} = 126 \times 10^{-6} - \alpha_{glass} \]
Rearranging gives:
\[ \alpha_{glass} = 126 \times 10^{-6} - 153 \times 10^{-6} \]
\[ \alpha_{glass} = -27 \times 10^{-6} \, ^(@)C \]

5. **Calculating the Linear Expansion Coefficient**:
Since we are looking for the volumetric expansion coefficient, we can use the relationship:
\[ \alpha_{glass} = \frac{1}{3} \times (-27 \times 10^{-6}) = -9 \times 10^{-6} \, ^(@)C \]

### Final Answer:
The coefficient of linear expansion of glass (α_glass) is:
\[ \alpha_{glass} = -9 \times 10^{-6} \, ^(@)C \]