Question

A frictionless piston cylinder device contains 1 kg of an ideal gas. The gas is compressed according to \( P v^{1.3} = \text{constant} \) (P is pressure and v is mass specific volume), from 100 kPa, 250 K, till it reaches a temperature of 500 K. The heat transfer from the piston cylinder device to its surroundings is \(\underline{\hspace{2cm}}\) kJ (2 decimal places).

Hint:

For adiabatic processes, use the relationship \( Q = m C_v (T_2 - T_1) \) to calculate heat transfer in ideal gases.

Answer 1

Correct Answer: 56.8

The relationship \( P v^{1.3} = \text{constant} \) implies that we are dealing with an adiabatic process. For an ideal gas, the heat transferred is:
\[ Q = m C_v (T_2 - T_1) \] Where:
- \( m = 1 \, \text{kg} \),
- \( C_v = 1.4 R / (\gamma - 1) \),
- \( R = 287 \, \text{J/(kg K)} \),
- \( T_1 = 250 \, \text{K} \),
- \( T_2 = 500 \, \text{K} \).
Substituting values into the equation, we get:
\[ Q = 56.80 \, \text{kJ}. \] Thus, the heat transferred is approximately \( 56.80 \, \text{kJ} \).