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. (a) 8JK^-1
B. (b) 0.8JK^-1
C. (c) 4JK^-1
D. (d) 0.4JK^-1

Answer

To solve the problem, we need to find the total heat capacity of the gas at constant volume. Let's go through the solution step by step.

### Step 1: Identify the Given Data
- Number of moles of gas, \( n = 5 \, \text{mol} \)
- Initial temperature, \( T_i = 100^\circ C \)
- Final temperature, \( T_f = 120^\circ C \)
- Change in internal energy, \( \Delta U = 80 \, \text{J} \)

### Step 2: Calculate the Change in Temperature
The change in temperature (\( \Delta T \)) can be calculated as:
\[ \Delta T = T_f - T_i = 120^\circ C - 100^\circ C = 20^\circ C \]
(Note: \( 20^\circ C \) is equivalent to \( 20 \, \text{K} \) since the size of the degree is the same in Celsius and Kelvin.)

### Step 3: Use the Formula for Change in Internal Energy
The change in internal energy at constant volume is given by:
\[ \Delta U = n C_v \Delta T \]
Where:
- \( C_v \) is the molar heat capacity at constant volume.

### Step 4: Rearrange the Formula to Solve for \( C_v \)
We can rearrange the equation to solve for \( C_v \):
\[ C_v = \frac{\Delta U}{n \Delta T} \]

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

### Step 6: Calculate the Total Heat Capacity at Constant Volume
The total heat capacity (\( C_{V, \text{total}} \)) at constant volume is given by:
\[ C_{V, \text{total}} = n C_v \]
Substituting the values we have:
\[ C_{V, \text{total}} = 5 \, \text{mol} \times 0.8 \, \text{J/mol K} = 4 \, \text{J/K} \]

### Final Answer
The total heat capacity of the gas at constant volume is:
\[ \boxed{4 \, \text{J/K}} \]