Question

A thermodynamically closed system contains 1 kg of hydrogen. The system undergoes a reversible polytropic process with polytropic index 1.3. The work output during the process is 400 kJ. During the process, hydrogen behaves as an ideal gas with constant specific heats. The absolute value of heat transfer during the process is _________ kJ (rounded off to 1 decimal place). Specific heat of hydrogen at constant pressure = 14.56 kJ kg\(^{-1}\) K\(^{-1}\)
Specific heat of hydrogen at constant volume = 10.4 kJ kg\(^{-1}\) K\(^{-1}\)

Hint:

To calculate heat transfer in a polytropic process, use the first law of thermodynamics and account for both work done and the change in internal energy.

Answer 1

Given: \[ m = 1 \ {kg}, \quad n = 1.3, \quad W = 400 \ {kJ}, \quad C_p = 14.56, \quad C_v = 10.4 \] First, calculate the specific heat ratio: \[ \gamma = \frac{C_p}{C_v} = \frac{14.56}{10.4} = 1.4 \] Heat transfer in a polytropic process is given by: \[ Q = \frac{n - \gamma}{\gamma - 1} \cdot W \] Substituting the values: \[ Q = \frac{1.3 - 1.4}{1.4 - 1} \cdot 400 = \frac{-0.1}{0.4} \cdot 400 = -0.25 \cdot 400 = -100 \ {kJ} \] Since the question asks for the absolute value: \[ \boxed{|Q| = 100.0 \ {kJ}} \]