if quantity of heat 1163.4 J supplied to one mole of nitrogen gas , at room temperature at constant pressure , then the rise in temperature is

Options

A. 54 K
B. 28 K
C. 65 K
D. 40 K

Answer

To solve the problem of finding the rise in temperature when 1163.4 J of heat is supplied to one mole of nitrogen gas at constant pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Molar Specific Heat at Constant Pressure (Cp)**:
For a diatomic gas like nitrogen (N₂), the molar specific heat at constant pressure (Cp) can be calculated using the formula:
\[ C_p = C_v + R \]
where \(C_v\) is the molar specific heat at constant volume and \(R\) is the universal gas constant.

2. **Calculate Cv for Nitrogen**:
The degrees of freedom (F) for a diatomic gas is 5. Therefore, we can calculate \(C_v\) as:
\[ C_v = \frac{F}{2} R = \frac{5}{2} R \]

3. **Substitute Cv into the Cp Formula**:
Now substituting \(C_v\) into the equation for \(C_p\):
\[ C_p = \frac{5}{2} R + R = \frac{7}{2} R \]

4. **Use the Heat Equation**:
The heat added to the gas is related to the change in temperature (\(\Delta T\)) by the equation:
\[ Q = n C_p \Delta T \]
where \(Q\) is the heat supplied, \(n\) is the number of moles, and \(\Delta T\) is the change in temperature.

5. **Substituting Known Values**:
Given that \(Q = 1163.4 J\) and \(n = 1\) mole, we can rewrite the equation:
\[ 1163.4 = 1 \cdot \left(\frac{7}{2} R\right) \Delta T \]

6. **Substituting the Value of R**:
The value of the gas constant \(R\) is approximately \(8.314 J/(mol \cdot K)\). Thus, we have:
\[ 1163.4 = \left(\frac{7}{2} \cdot 8.314\right) \Delta T \]

7. **Calculate Cp**:
Calculate \(C_p\):
\[ C_p = \frac{7}{2} \cdot 8.314 = 29.1 J/(mol \cdot K) \]

8. **Solving for \(\Delta T\)**:
Rearranging the equation to solve for \(\Delta T\):
\[ \Delta T = \frac{1163.4}{29.1} \]

9. **Final Calculation**:
Performing the calculation:
\[ \Delta T \approx 40 K \]

### Conclusion:
The rise in temperature when 1163.4 J of heat is supplied to one mole of nitrogen gas at constant pressure is approximately **40 K**.