Question

In a piston cylinder assembly, one kmol of an ideal gas is compressed from an initial state of 200 kPa and 400 K to a final state of 1 MPa and 400 K. If the surroundings are at 400 K, the minimum amount of work (in kJ/kmol) required for the compression process is .............{(rounded off to two decimal places).}
Use: Universal gas constant \( R_u = 8.314 \, {kJ/kmol·K} \)

Hint:

For isothermal reversible compression of an ideal gas, use the formula: \( W = nRT \ln(P_2/P_1) \), where \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the absolute temperature.

Answer 1

Step 1: Since the process is isothermal and we are looking for the minimum work, we use the formula for reversible isothermal compression: \[ W_{{rev}} = nRT \ln \left( \frac{P_2}{P_1} \right) \] Given: \[ n = 1 { kmol}, R = 8.314 { kJ/kmol·K}, T = 400 { K} \] \[ P_1 = 200 { kPa}, P_2 = 1000 { kPa} \] \[ W = 1 \cdot 8.314 \cdot 400 \cdot \ln\left( \frac{1000}{200} \right) = 3325.6 \cdot \ln(5) \] \[ \ln(5) \approx 1.6094 \Rightarrow W \approx 3325.6 \cdot 1.6094 \approx 5365.28 \, {kJ/kmol} \]