If `C_(v)=4.96cal//"mole"K,` , then increase in internal energy when temperature of 2 moles of this gas is increased from 340 K to 342 K

Options

A. 27.80 cal
B. 19.84 cal
C. 13.90 cal
D. 9.92 cal

Answer

To find the increase in internal energy when the temperature of 2 moles of a gas is increased from 340 K to 342 K, we can use the formula related to the change in internal energy at constant volume.

### Step-by-Step Solution:

1. **Identify the given values**:
- Specific heat at constant volume, \( C_v = 4.96 \, \text{kcal/mole K} \)
- Number of moles, \( N = 2 \, \text{moles} \)
- Initial temperature, \( T_1 = 340 \, \text{K} \)
- Final temperature, \( T_2 = 342 \, \text{K} \)

2. **Calculate the change in temperature (\( \Delta T \))**:
\[ \Delta T = T_2 - T_1 = 342 \, \text{K} - 340 \, \text{K} = 2 \, \text{K} \]

3. **Use the formula for change in internal energy**:
The change in internal energy (\( \Delta U \)) at constant volume can be calculated using the formula:
\[ \Delta U = N C_v \Delta T \]

4. **Substitute the known values into the formula**:
\[ \Delta U = 2 \, \text{moles} \times 4.96 \, \text{kcal/mole K} \times 2 \, \text{K} \]

5. **Perform the calculation**:
\[ \Delta U = 2 \times 4.96 \times 2 = 19.84 \, \text{kcal} \]

6. **Conclusion**:
The increase in internal energy when the temperature of 2 moles of the gas is increased from 340 K to 342 K is \( 19.84 \, \text{kcal} \).