A weight thermometer contains `52g` of liquid at `10^(0)C`. When it is heated to `110^(0)C,2 g` of the liquid is expelled. The coefficient of real expansion of the liquid is `[alpha_(g I) is 9 xx 10^(-6//0)C]`

Options

A. `27 xx 10^(-6//0)C`
B. `427 xx 10^(-6//0)C`
C. `373 xx 10^(-6//0)C`
D. `473 xx 10^(-6//0)C`

Explanation

To find the coefficient of real expansion of the liquid in the weight thermometer, we can follow these steps:

### Step 1: Identify Given Data
- Mass of liquid initially, \( m = 52 \, \text{g} \)
- Initial temperature, \( T_1 = 10^\circ C \)
- Final temperature, \( T_2 = 110^\circ C \)
- Mass of liquid expelled, \( m_e = 2 \, \text{g} \)
- Coefficient of linear expansion of the liquid, \( \alpha = 9 \times 10^{-6} \, \text{°C}^{-1} \)

### Step 2: Calculate the Change in Temperature
The change in temperature, \( \Delta T \), is given by:
\[ \Delta T = T_2 - T_1 = 110^\circ C - 10^\circ C = 100^\circ C \]

### Step 3: Calculate the Mass of Liquid Remaining
The mass of liquid remaining in the container after expulsion is:
\[ m_r = m - m_e = 52 \, \text{g} - 2 \, \text{g} = 50 \, \text{g} \]

### Step 4: Calculate the Coefficient of Apparent Expansion
The coefficient of apparent expansion, \( \beta \), can be calculated using the formula:
\[ \beta = \frac{m_e}{m_r \cdot \Delta T} \]
Substituting the values:
\[ \beta = \frac{2 \, \text{g}}{50 \, \text{g} \cdot 100 \, \text{°C}} = \frac{2}{5000} = 4 \times 10^{-4} \, \text{°C}^{-1} \]

### Step 5: Calculate the Coefficient of Real Expansion
The coefficient of real expansion, \( \gamma \), is given by the relation:
\[ \gamma = \beta + 3\alpha \]
Substituting the values of \( \beta \) and \( \alpha \):
\[ \gamma = 4 \times 10^{-4} \, \text{°C}^{-1} + 3 \times (9 \times 10^{-6} \, \text{°C}^{-1}) \]
Calculating \( 3 \times 9 \times 10^{-6} \):
\[ 3 \times 9 \times 10^{-6} = 27 \times 10^{-6} = 0.000027 \, \text{°C}^{-1} \]
Now, convert \( 4 \times 10^{-4} \) to the same power of ten:
\[ 4 \times 10^{-4} = 400 \times 10^{-6} \, \text{°C}^{-1} \]
Now, adding these:
\[ \gamma = 400 \times 10^{-6} + 27 \times 10^{-6} = 427 \times 10^{-6} \, \text{°C}^{-1} \]

### Final Answer
Thus, the coefficient of real expansion of the liquid is:
\[ \gamma = 427 \times 10^{-6} \, \text{°C}^{-1} \]