Analyzing Combustion of Methane with Oxygen at Constant Volume
A mixture of 1 kmol of gaseous methane and 2 kmol of oxygen initially at 25°C and 1 atm burns completely in a closed, rigid container. Heat transfer occurs until the products are cooled to 900 K. If the reactants and products each form ideal gas mixtures, determine (a) the amount of heat transfer, in kJ, and (b) the final pressure, in atm.
Known A mixture of gaseous methane and oxygen, initially at 25°C and 1 atm, burns completely within a closed rigid container. The products are cooled to 900 K.
Find Determine the amount of heat transfer, in kJ, and the final pressure of the combustion products, in atm.
Schematic and Given Data:
Engineering Model
1. The contents of the closed, rigid container are taken as the system.
2. Kinetic and potential energy effects are absent, and W = 0.
3. Combustion is complete.
4. The initial mixture and the products of combustion each form ideal gas mixtures.
5. The initial and final states are equilibrium states.
Analysis The chemical reaction equation for the complete combustion of methane with oxygen is
CH _{4}+2 O _{2} \rightarrow CO _{2}+2 H _{2} O ( g )
a. With assumptions 2 and 3, the closed system energy balance takes the form
U_{ P }-U_{ R }=Q- W ^{\nearrow0}
or
Q=U_{ P }-U_{ R }=\left(1 \bar{u}_{ CO _{2}}+2 \bar{u}_{ H _{2} O ( g )}\right)-\left(1 \bar{u}_{ CH _{4}( g )}+2 \bar{u}_{ O _{2}}\right)
Each coefficient in this equation is the same as the corresponding term of the balanced chemical equation.
Since each reactant and product behaves as an ideal gas, the respective specific internal energies can be evaluated as \bar{u}=\bar{h}-\bar{R} T. The energy balance then becomes
\begin{aligned}Q=&\left[1\left(\bar{h}_{ CO _{2}}-\bar{R} T_{2}\right)+2\left(\bar{h}_{ H _{2} O ( g )}-\bar{R} T_{2}\right)\right] \\&-\left[1\left(\bar{h}_{ CH _{4}( g )}-\bar{R} T_{1}\right)+2\left(\bar{h}_{ O _{2}}-\bar{R} T_{1}\right)\right]\end{aligned}
where T_{1} \text { and } T_{2} denote, respectively, the initial and final temperatures. Collecting like terms
Q=\left(\bar{h}_{ CO _{2}}+2 \bar{h}_{ H _{2} O ( g )}-\bar{h}_{ CH _{4}( g )}-2 \bar{h}_{ O _{2}}\right)+3 \bar{R}\left(T_{1}-T_{2}\right)
The specific enthalpies are evaluated in terms of the respective enthalpies of formation to give
1 \begin{aligned}Q=&\left[\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}\right)_{ CO _{2}}+2\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}\right)_{ H _{2} O ( g )}\right.\\&\left.-\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}^{\nearrow0}\right)_{ CH _{4}( g )}-2\left(\bar{h}_{ f }^{\circ\nearrow0}+\Delta \bar{h}^{\nearrow0}\right)_{ O _{2}}\right]+3 \bar{R}\left(T_{1}-T_{2}\right)\end{aligned}
Since the methane and oxygen are initially at 25°C, \Delta \bar{h}=0 for each of these reactants. Also, \bar{h}_{ f }^{\circ}=0 for oxygen.
With enthalpy of formation values for CO _{2}, H _{2} O ( g ) and CH _{4}( g ) from Table A-25 and enthalpy values for H _{2} O \text { and } CO _{2} from Table A-23
\begin{aligned}Q=&[-393,520+(37,405-9364)]+2[-241,820\\&+(31,828-9904)]-(-74,850) \\&+3(8.314)(298-900)\end{aligned}
= -745,436 kJ
b. By assumption 4, the initial mixture and the products of combustion each form ideal gas mixtures. Thus, for the reactants
p_{1} V=n_{ R } \bar{R} T_{1}
where n_{ R } is the total number of moles of reactants and p_{1} is the initial pressure. Similarly, for the products
p_{2} V=n_{ P } \bar{R} T_{2}
where n_{ P } is the total number of moles of products and p_{2} is the final pressure.
Since n_{ R }=n_{ P }=3 and volume is constant, these equations combine to give
p_{2}=\frac{T_{2}}{T_{1}} p_{1}=\left(\frac{900 K }{298 K }\right)(1 atm )=3.02 atm
1 This expression corresponds to Eq. 13.17b.
\begin{aligned}Q-W &=\sum_{ P } n\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}-\bar{R} T_{ P }\right)-\sum_{ R } n\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}-\bar{R} T_{ R }\right) \\&=\sum_{ P } n\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}\right)-\sum_{ R } n\left(\bar{h}_{ f }^{\circ}+\Delta \bar{h}\right)-\bar{R} T_{ P } \sum_{ P } n+\bar{R} T_{ R } \sum_{ R } n\end{aligned} (13.17b)
Skills Developed
Ability to…
• apply the closed system energy balance to a reacting system.
• evaluate property data appropriately.
• apply the ideal gas equation of state.
Quick Quiz
Calculate the volume of the system, in m ^{3}. Ans. 73.36 m ^{3}.
