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, we need to calculate the increase in internal energy of the hydrogen gas during the adiabatic compression. Here’s the step-by-step solution:

### Step 1: Identify the initial and final temperatures
- The initial temperature (T_initial) at STP (Standard Temperature and Pressure) is 0°C, which is equivalent to 273.15 K.
- The final temperature (T_final) is given as 400°C, which is equivalent to 400 + 273.15 = 673.15 K.

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

### Step 3: Determine the value of \( C_v \)
The specific heat at constant volume \( C_v \) for a diatomic gas (like hydrogen) can be calculated using the formula:
\[ C_v = \frac{R}{\gamma - 1} \]
Where:
- \( R = 8.31 \, J/(mol \cdot K) \) (the gas constant)
- \( \gamma = \frac{7}{5} = 1.4 \)

Calculating \( C_v \):
\[ C_v = \frac{8.31}{\frac{7}{5} - 1} = \frac{8.31}{1.4 - 1} = \frac{8.31}{0.4} = 20.775 \, J/(mol \cdot K) \]

### Step 4: Calculate the increase in internal energy (ΔU)
The increase in internal energy can be calculated using the formula:
\[ \Delta U = N C_v \Delta T \]
Where:
- \( N = 5 \, moles \)
- \( C_v = 20.775 \, J/(mol \cdot K) \)
- \( \Delta T = 400 \, K \)

Calculating \( \Delta U \):
\[ \Delta U = 5 \times 20.775 \times 400 \]
\[ \Delta U = 5 \times 20.775 \times 400 = 41550 \, J \]

### Step 5: Convert ΔU to kilojoules
To convert from joules to kilojoules, divide by 1000:
\[ \Delta U = \frac{41550}{1000} = 41.55 \, kJ \]

### Final Answer
The increase in the internal energy of the gas is approximately **41.55 kJ**.

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