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).
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Solution and Explanation
- \( Q \) is the heat added (in kJ),
- \( m \) is the mass of the gas (in kg),
- \( c_p \) is the specific heat at constant pressure (in kJ/kg-K),
- \( \Delta T \) is the change in temperature (in K).
Step 1: For the hydrogen (H\(_2\)) gas, the mass is \( m_H = 4 \, {kg} \), the specific heat is \( c_p(H_2) = 14.239 \, {kJ/kg-K} \), and the temperature change is: \[ \Delta T_H = 350 \, {K} - 250 \, {K} = 100 \, {K} \] Thus, the heat added to hydrogen is: \[ Q_H = m_H \cdot c_p(H_2) \cdot \Delta T_H = 4 \cdot 14.239 \cdot 100 = 5695.6 \, {kJ} \] Step 2: For the nitrogen (N\(_2\)) gas, the mass is \( m_N = 13 \, {kg} \), the specific heat is \( c_p(N_2) = 1.040 \, {kJ/kg-K} \), and the temperature change is: \[ \Delta T_N = 350 \, {K} - 250 \, {K} = 100 \, {K} \] Thus, the heat added to nitrogen is: \[ Q_N = m_N \cdot c_p(N_2) \cdot \Delta T_N = 13 \cdot 1.040 \cdot 100 = 1352 \, {kJ} \] Step 3: The total heat added to the mixture is the sum of the heat added to hydrogen and nitrogen: \[ Q_{{total}} = Q_H + Q_N = 5695.6 \, {kJ} + 1352 \, {kJ} = 7047.6 \, {kJ} \] Step 4: Converting the heat from kJ to MJ: \[ Q_{{total}} = \frac{7047.6}{1000} = 7.048 \, {MJ} \] Step 5: Therefore, the amount of heat added to the cylinder is approximately \( 7.048 \, {MJ} \), which lies between 6.9 and 7.2, and is closest to 7.048 MJ. So, the correct answer is within the specified range.
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