Five moles of hydrogen `(gamma = 7//5)`, initially at `STP`, is compressed adiabatically so that its temperature becomes `400^(@)C`. The increase in the internal energy of the gas in kilojules is `(R = 8.30 J//mol-K)`

Options

A. 21.55
B. 41.5
C. 65.55
D. 80.55

Explanation

To solve the problem of finding the increase in internal energy of five moles of hydrogen gas compressed adiabatically, we can follow these steps:

### Step 1: Identify the given values
- Number of moles (n) = 5 moles
- Specific heat ratio (γ) = 7/5
- Final temperature (T_final) = 400°C = 400 + 273.15 = 673.15 K
- Initial temperature (T_initial) = 0°C = 0 + 273.15 = 273.15 K
- Universal gas constant (R) = 8.30 J/(mol·K)

### Step 2: Calculate the change in temperature (ΔT)
\[ \Delta T = T_{\text{final}} - T_{\text{initial}} = 673.15 \, \text{K} - 273.15 \, \text{K} = 400 \, \text{K} \]

### Step 3: Calculate the molar specific heat at constant volume (C_v)
Using the relation between specific heats:
\[ C_v = \frac{R}{\gamma - 1} \]
Substituting the values:
\[ C_v = \frac{8.30 \, \text{J/(mol·K)}}{\frac{7}{5} - 1} = \frac{8.30}{0.4} = 20.75 \, \text{J/(mol·K)} \]

### Step 4: Calculate the increase in internal energy (ΔU)
The change in internal energy can be calculated using the formula:
\[ \Delta U = n C_v \Delta T \]
Substituting the known values:
\[ \Delta U = 5 \, \text{moles} \times 20.75 \, \text{J/(mol·K)} \times 400 \, \text{K} \]
\[ \Delta U = 5 \times 20.75 \times 400 = 41500 \, \text{J} \]

### Step 5: Convert the internal energy from Joules to kilojoules
\[ \Delta U = \frac{41500 \, \text{J}}{1000} = 41.5 \, \text{kJ} \]

### Final Answer
The increase in the internal energy of the gas is **41.5 kJ**.

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