Question

A piston-cylinder arrangement contains an ideal gas mixture of 4 kg of hydrogen and 13 kg of nitrogen at 250 K and atmospheric pressure. On heat addition, the mixture expands at constant pressure until the temperature rises to 350 K. The average isobaric specific heats (\(c_p\)) for hydrogen and nitrogen are 14.239 kJ/kg-K and 1.040 kJ/kg-K, respectively. The amount of heat (in MJ) added to the cylinder is ............... (rounded off to three decimal places).

Hint:

For mixture problems at constant pressure, always calculate heat contribution species-wise using: \[ Q = \sum m_i c_{p,i} \Delta T \] This avoids errors compared to using average \(c_p\).

Answer 1

Correct Answer: 6.9

Step 1: Heat addition at constant pressure.
At constant pressure, the heat added is: \[ Q = m \cdot c_p \cdot \Delta T \] For a gas mixture, \[ Q = \sum (m_i \, c_{p,i} \, \Delta T) \]

Step 2: Data given.
\[ m_{H_2} = 4 \, kg, c_{p,H_2} = 14.239 \, kJ/kg\!-\!K \] \[ m_{N_2} = 13 \, kg, c_{p,N_2} = 1.040 \, kJ/kg\!-\!K \] \[ \Delta T = (350 - 250) = 100 \, K \]

Step 3: Heat by hydrogen.
\[ Q_{H_2} = m_{H_2} \cdot c_{p,H_2} \cdot \Delta T = (4)(14.239)(100) = 5695.6 \, kJ \]

Step 4: Heat by nitrogen.
\[ Q_{N_2} = m_{N_2} \cdot c_{p,N_2} \cdot \Delta T = (13)(1.040)(100) = 1352 \, kJ \]



Step 5: Total heat.
\[ Q_{total} = Q_{H_2} + Q_{N_2} = 5695.6 + 1352 = 7047.6 \, kJ \] \[ Q_{total} = 7.048 \, MJ \] Final Answer:
\[ \boxed{7.048 \, MJ} \]