The temperature of `5 mol` of gas which was held at constant volume was change from `100^(@)C` to `120^(@)C`. The change in internal energy was found to ve `80 J`. The total heat capacity of the gas at constant volume will be equal to

Options

A. `8 JK^(-1)`
B. `0.8 JK^(-1)`
C. `4 JK^(-1)`
D. `0.4 JK^(-1)`

Explanation

To find the total heat capacity of the gas at constant volume, we can follow these steps:

### Step 1: Identify the given values
- Number of moles (N) = 5 mol
- Initial temperature (T1) = 100°C
- Final temperature (T2) = 120°C
- Change in internal energy (ΔU) = 80 J

### Step 2: Calculate the change in temperature (ΔT)
\[ \Delta T = T2 - T1 = 120°C - 100°C = 20°C \]
Since temperature change in Celsius is equivalent to Kelvin, we can use ΔT = 20 K.

### Step 3: Use the formula for change in internal energy
The change in internal energy (ΔU) is related to the number of moles, the heat capacity at constant volume (Cv), and the change in temperature by the equation:
\[ \Delta U = N \cdot C_v \cdot \Delta T \]

### Step 4: Rearrange the equation to solve for Cv
\[ C_v = \frac{\Delta U}{N \cdot \Delta T} \]

### Step 5: Substitute the known values into the equation
\[ C_v = \frac{80 \, \text{J}}{5 \, \text{mol} \cdot 20 \, \text{K}} = \frac{80}{100} = 0.8 \, \text{J/mol·K} \]

### Step 6: Calculate the total heat capacity (C)
The total heat capacity at constant volume (C) is given by:
\[ C = N \cdot C_v \]
Substituting the values:
\[ C = 5 \, \text{mol} \cdot 0.8 \, \text{J/mol·K} = 4 \, \text{J/K} \]

### Final Answer
The total heat capacity of the gas at constant volume is **4 J/K**.

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