A glass flask of volume `200cm^(3)` is just filled with mercury at `20^(@)C`. The amount of mercury that will overflow when the temperature of the system is raised to `100^(@)C` is `(gamma_(glass)=1.2xx10^(-5)//C^(@),gamma_(mercury)=1.8xx10^(-4)//C^(@))`

Options

A. 2.15 `cm^(3)`
B. 2.69 `cm^(3)`
C. 2.52 `cm^(3)`
D. 2.25 `cm^(3)`

Answer

To solve the problem, we need to determine the amount of mercury that overflows when the temperature of the system is raised from \(20^\circ C\) to \(100^\circ C\). We will use the coefficients of volume expansion for both the glass flask and the mercury.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Volume of the glass flask, \( V = 200 \, \text{cm}^3 \)
- Initial temperature, \( T_1 = 20^\circ C \)
- Final temperature, \( T_2 = 100^\circ C \)
- Coefficient of volume expansion for glass, \( \gamma_{\text{glass}} = 1.2 \times 10^{-5} \, \text{C}^{-1} \)
- Coefficient of volume expansion for mercury, \( \gamma_{\text{mercury}} = 1.8 \times 10^{-4} \, \text{C}^{-1} \)

2. **Calculate the Change in Temperature:**
\[ \Delta T = T_2 - T_1 = 100^\circ C - 20^\circ C = 80^\circ C \]

3. **Calculate the Increase in Volume of the Glass Flask:**
The change in volume of the glass flask due to temperature change can be calculated using:
\[ \Delta V_{\text{glass}} = V \cdot \gamma_{\text{glass}} \cdot \Delta T \]
Substituting the values:
\[ \Delta V_{\text{glass}} = 200 \, \text{cm}^3 \cdot 1.2 \times 10^{-5} \, \text{C}^{-1} \cdot 80 \, \text{C} \]
\[ \Delta V_{\text{glass}} = 200 \cdot 1.2 \cdot 80 \times 10^{-5} = 1920 \times 10^{-5} \, \text{cm}^3 = 0.192 \, \text{cm}^3 \]

4. **Calculate the Increase in Volume of Mercury:**
Similarly, the change in volume of mercury is:
\[ \Delta V_{\text{mercury}} = V \cdot \gamma_{\text{mercury}} \cdot \Delta T \]
Substituting the values:
\[ \Delta V_{\text{mercury}} = 200 \, \text{cm}^3 \cdot 1.8 \times 10^{-4} \, \text{C}^{-1} \cdot 80 \, \text{C} \]
\[ \Delta V_{\text{mercury}} = 200 \cdot 1.8 \cdot 80 \times 10^{-4} = 28800 \times 10^{-4} \, \text{cm}^3 = 2.88 \, \text{cm}^3 \]

5. **Calculate the Overflow Volume:**
The amount of mercury that overflows is given by the difference in the increase in volume of mercury and the increase in volume of the glass flask:
\[ V_{\text{overflow}} = \Delta V_{\text{mercury}} - \Delta V_{\text{glass}} \]
Substituting the values:
\[ V_{\text{overflow}} = 2.88 \, \text{cm}^3 - 0.192 \, \text{cm}^3 = 2.688 \, \text{cm}^3 \]

### Final Answer:
The amount of mercury that will overflow when the temperature is raised to \(100^\circ C\) is approximately \(2.688 \, \text{cm}^3\).

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