The temperature of 5 moles of a gas at constant volume is changed from `100^(@)C` to `120^(@)C`. The change in internal energy is 80 J. The total heat capacity of the gas at constant volume will be in `J/K.`

Options

A. 8
B. 4
C. 0.8
D. 0.4

Answer

To find the total heat capacity of the gas at constant volume, we can use the relationship between the change in internal energy, the number of moles, the specific heat capacity at constant volume, and the change in temperature.

### Step-by-step Solution:

1. **Identify the Given Values**:
- Number of moles (n) = 5 moles
- Change in internal energy (ΔU) = 80 J
- Initial temperature (T1) = 100°C
- Final temperature (T2) = 120°C

2. **Calculate the Change in Temperature (ΔT)**:
\[ \Delta T = T2 - T1 = 120°C - 100°C = 20°C \]
Since the change in temperature in Celsius is the same as in Kelvin, we can use ΔT = 20 K.

3. **Use the Formula for Change in Internal Energy**:
The formula relating the change in internal energy to heat capacity at constant volume is:
\[ \Delta U = n C_v \Delta T \]
Where:
- \( C_v \) is the heat capacity at constant volume.

4. **Rearrange the Formula to Solve for \( C_v \)**:
\[ C_v = \frac{\Delta U}{n \Delta T} \]

5. **Substitute the Known Values**:
\[ C_v = \frac{80 \, \text{J}}{5 \, \text{moles} \times 20 \, \text{K}} \]

6. **Calculate \( C_v \)**:
\[ C_v = \frac{80}{100} = 0.8 \, \text{J/K} \]

### Final Answer:
The total heat capacity of the gas at constant volume is \( C_v = 0.8 \, \text{J/K} \).

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