Determine the total apparent specific heat at constant volume  (c_v,mix in kJ/kg·K) for a fuel–air reactant mixture containing 1 kmol CH₄, 2.5 kmol O₂, and 9.4 kmol N₂ at 500 K and 1 atm. What heat transfer is required to cool the mixture from 500 K to 300 K? Use specific heat values for each species from Table E.1.

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TABLE E.1 Critical Constants and Specific Heats for Selected Gases* P. (105 Pa) 62.4 37.7 Substance Acetylene (C₂H₂) Air (equivalent) Ammonia (NH3) Benzene (CH) n-Butane (C4H10) Carbon dioxide (CO₂) Carbon monoxide (CO) Refrigerant 134a (C₂F4H₂) Ethane (C₂H) Ethylene (C₂H₁) Helium (He) m (kg/kmol) 26.04 28.97 17.04 78.11 58.12 44.01 28.01 102.03 30.07 28.05 4.003 2.016 Tc (K) 309 133 406 562 16.04 28.01 32.00 425.2 304.2 133 374.3 305.4 283 5.2 33.2 190.7 126.2 154.4 370 431 647.1 112.8 48.3 37.9 73.9 35.0 40.6 48.8 51.2 2.3 13.0 46.4 33.9 50.5 42.5 78.7 220.6 vc (m³/kmol) 0.112 0.0829 0.0723 0.256 0.257 0.0941 0.0928 0.200 0.148 0.128 0.0579 Ze cy (kJ/kg-K) 0.272 1.37 0.284 0.718 0.242 1.66 0.274 0.67 1.56 0.657 0.274 0.276 0.294 0.262 0.285 Hydrogen (H₂) 0.0648 Methane (CH₂) 0.0991 Nitrogen (N₂) 0.0897 Oxygen (0₂) 0.0741 Propane (C₂H₂) 44.09 0.200 Sulfur dioxide (SO₂) 64.06 0.124 Water (H₂O) 18.02 0.0558 * Adapted from Wark, K, Jr., and Richards, D. E., Thermodynamics, 6th edn, McGraw-Hill, New York, 1999. 0.279 1.23 0.300 3.12 0.744 0.76 1.48 0.304 0.290 10.2 1.70 0.291 0.743 0.290 0.658 0.278 1.48 0.268 0.471 0.230 1.40 Cp (kJ/kg-K) 1.69 1.005 2.15 0.775 1.71 0.846 1.04 0.85 1.75 1.53 5.19 14.3 2.22 1.04 0.918 1.67 0.601 1.86