Methane and 30% excess air are to be fed to a combustion reactor. An inexperienced technician mistakes his instructions and charges the gases together in the required proportion into an evacuated closed tank. (The gases were supposed to be fed directly into the reactor.) The contents of the charged tank are at 25°C and 4.00 atm absolute.

(a) Calculate the standard internal energy of combustion of the methane combustion reaction, ΔU°_c (kJ/mol), taking CO₂ (g) and H₂O (v) as the presumed products. Then prove that if the constant-pressure heat capacity of an ideal gas species is independent of temperature the specific internal energy of that species at temperature T(°C) relative to the same species at 25°C is given by the expression U  = (C_p – R_g) (T – 25°C) where R_g is the gas constant. Use this formula in the next part of the problem.

(b) You wish to calculate the maximum temperature, T _max (°C). and corresponding pressure, P_max (atm), that the tank would have to withstand if the mixture it contains were to be accidentally ignited. Taking molecular species at 25°C as references and treating all species as ideal gases, prepare an inlet—outlet internal energy table for the closed system combustion process In deriving expressions for each U, at the final reactor condition (T _max, P _max), use the following approximate values for C_p [kJ/ (mol∙°C)]: 0.033 for O₂, 0.032 for N₂, 0.052 for CO₂ and 0.040 for H₂O (v). Then use an energy balance and the ideal gas equation of state to perform the required calculations.

(c) Why would the actual temperature and pressure attained in a real tank be less than the values calculated in part (a)? (State several reasons.)