H₂/N₂ Mixture, Constant-Pressure Heating: 04-BS-10, May 2018, Q9

Question 9: H₂/N₂ Mixture, Constant-Pressure Heating (15 marks)

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Given. $m_{H_2}=6$ kg, $m_{N_2}=21$ kg, $T_1=160$ K, $T_2=200$ K, constant pressure ($P=5$ MPa, does not enter the ideal-gas enthalpy calculation). Ideal-gas mixture.

Find. $Q$ [kJ].

Approach

A piston-cylinder at constant pressure with no other work mode gives $Q=\Delta H$ for the whole mixture. Because each component of an ideal-gas mixture behaves as if it alone occupied the volume at the mixture temperature, the total enthalpy change is simply the mass-weighted sum of each pure component's enthalpy change between $T_1$ and $T_2$ — evaluated here from each gas's own ideal-gas enthalpy function (temperature-only dependence, so the actual 5 MPa is irrelevant to the ideal-gas result).

Per-component ideal-gas enthalpy change, H₂. From 160 K to 200 K: $$h_{H_2}(200)-h_{H_2}(160)=2557.13-2028.67=528.46\ \text{kJ/kg}.$$ $$Q_{H_2}=m_{H_2}\times528.46=6\times528.46=\boxed{3170.8\ \text{kJ}}.$$
Per-component ideal-gas enthalpy change, N₂. From 160 K to 200 K: $$h_{N_2}(200)-h_{N_2}(160)=207.00-165.20=41.80\ \text{kJ/kg}.$$ $$Q_{N_2}=m_{N_2}\times41.80=21\times41.80=\boxed{877.8\ \text{kJ}}.$$
Total heat transfer. Constant-pressure, piston-cylinder (boundary work only): $$Q=\Delta H=Q_{H_2}+Q_{N_2}=3170.8+877.8=\boxed{4048.6\ \text{kJ}}.$$ H₂'s small mass fraction (6 of 27 kg, 22%) nonetheless supplies about 78% of the total heat transfer, because hydrogen's per-kilogram specific heat is roughly 14× nitrogen's (its molar mass is about 14× smaller for a similar molar $c_p$).

Quantity	Result
$Q$	4048.6 kJ

Concept & Theory

Topic. Constant-pressure heating of a mass-basis ideal-gas mixture (H₂/ N₂) in a piston-cylinder device.

Key relations. Ideal-gas mixture enthalpy is additive by mass: $\Delta H=\sum_i m_i\,\Delta h_i(T)$, with each $\Delta h_i$ evaluated from the pure component's own ideal-gas enthalpy function. Constant-pressure closed-system energy balance: $Q=\Delta H$ (boundary work absorbs the $P\,\Delta V$ term).

Why this works. Gibbs' ideal-gas mixture rule states each component's internal energy/enthalpy depends only on its own temperature (not on the mixture composition or the other components' properties), so the mixture's total enthalpy change is exactly the mass-weighted sum of each pure gas's own enthalpy change over the same temperature interval — no mixing correction is needed for ideal gases.

Common pitfall. Using a single "mixture" specific heat computed only from the mass fractions without checking each component's actual per-kilogram $c_p$ — here H₂'s tiny mass fraction hides a per-kilogram specific heat about 14× larger than N₂'s (a consequence of its much smaller molar mass for a comparable molar $c_p$), so it dominates the total heat transfer despite being the minority species by mass.

Source. Cengel & Boles, Thermodynamics: An Engineering Approach, 8th ed., Ch. 13 (Gas Mixtures, ideal-gas mixture properties).

Question 9 of 9: H₂/N₂ Mixture, Constant-Pressure Heating

Question 9: H₂/N₂ Mixture, Constant-Pressure Heating (15 marks)

Approach