Ten mole of hydrogen at `N.T.P` is compressed adiabatically so that it temperature becomes `400^(@)C`. How much work is done on the gas? Also, Calculate the increase in internal energy of the gas. Take `R=8.4J mol e^(-1)K^(-1) and gamma= 1.4`.

Answer

To solve the problem step by step, we will calculate the work done on the gas during the adiabatic compression and the increase in internal energy of the gas.

### Step 1: Identify the given values
- Number of moles, \( N = 10 \, \text{moles} \)
- Initial temperature at NTP, \( T_1 = 0^\circ C = 273 \, \text{K} \)
- Final temperature, \( T_2 = 400^\circ C = 673 \, \text{K} \)
- Gas constant, \( R = 8.4 \, \text{J mol}^{-1} \text{K}^{-1} \)
- Heat capacity ratio, \( \gamma = 1.4 \)

### Step 2: Calculate the work done on the gas during adiabatic compression
The formula for work done on an ideal gas during an adiabatic process is given by:

\[ W = \frac{N R}{1 - \gamma} (T_2 - T_1) \]

Substituting the values into the formula:

\[ W = \frac{10 \, \text{moles} \times 8.4 \, \text{J mol}^{-1} \text{K}^{-1}}{1 - 1.4} (673 \, \text{K} - 273 \, \text{K}) \]

Calculating \( T_2 - T_1 \):

\[ T_2 - T_1 = 673 \, \text{K} - 273 \, \text{K} = 400 \, \text{K} \]

Now substituting this back into the work formula:

\[ W = \frac{10 \times 8.4}{1 - 1.4} \times 400 \]

Calculating \( 1 - 1.4 = -0.4 \):

\[ W = \frac{10 \times 8.4}{-0.4} \times 400 \]

Calculating \( \frac{10 \times 8.4}{-0.4} = -210 \):

\[ W = -210 \times 400 = -84000 \, \text{J} \]

Thus, the work done on the gas is:

\[ W = -84000 \, \text{J} \quad \text{(or } -8.4 \times 10^4 \, \text{J)} \]

### Step 3: Calculate the increase in internal energy of the gas
In an adiabatic process, the first law of thermodynamics states:

\[ dQ = dU + dW \]

Since \( dQ = 0 \) for an adiabatic process, we have:

\[ dU = -dW \]

Substituting the value of \( dW \):

\[ dU = -(-84000 \, \text{J}) = 84000 \, \text{J} \]

Thus, the increase in internal energy of the gas is:

\[ dU = 84000 \, \text{J} \quad \text{(or } 8.4 \times 10^4 \, \text{J)} \]

### Final Results
- Work done on the gas: \( W = -84000 \, \text{J} \)
- Increase in internal energy: \( dU = 84000 \, \text{J} \)