Intermediate Thermodynamics Assignment 9 — Chemical Thermodynamics  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Derive the species conservation equations in a closed system undergoing two- and three-body chemical reactions. Specifically, starting from $$ T=\frac{m \overline{q^2}}{3 k_{\rm B}}$$ do the following:
(a)  Show that for the two-body chemical reaction ${\rm N} + {\rm O} \rightarrow {\rm NO}$, the rate of change of the various species corresponds to: $$ \frac{{\rm d} }{{\rm d} t} N_{\rm NO} = k_{\rm NO}^\prime N_{\rm O} N_{\rm N} =-\frac{{\rm d} }{{\rm d} t} N_{\rm N} =-\frac{{\rm d} }{{\rm d} t} N_{\rm O} $$ where $k_{\rm NO}^\prime=f(T_{\rm O},~T_{\rm N})$.
(b)  Show that for the three-body chemical reaction ${\rm e^-} + {\rm N_2}+{\rm O_2} \rightarrow {\rm O}_2^-+{\rm N_2}$, the rate of change of the various species corresponds to: $$ \frac{{\rm d} }{{\rm d} t} N_{\rm O_2^-} =k_{\rm O_2^-}^\prime N_{\rm O_2} N_{\rm N_2} N_{\rm e} =-\frac{{\rm d} }{{\rm d} t} N_{\rm O_2} =-\frac{{\rm d} }{{\rm d} t} N_{\rm e} $$ where $k_{\rm O_2^-}^\prime=f(T_{\rm e},~T_{\rm O_2},~T_{\rm N_2})$.
Question #2
Consider the steady-state combustion of gaseous octane C$_8$H$_{18}$ with dry air:
(a)  Determine the air-fuel ratio on both a molar and mass basis for the complete combustion of octane with the theoretical amount of air (stoichiometric air-fuel ratio), and 150% theoretical air (50% excess air).
(b)  Gaseous octane at $(125+2\times\xi_2)^\circ$C, 1 atm enters a well-insulated reactor and reacts with air entering at the same temperature and pressure. For steady-state operation and negligible effects of kinetic and potential energy, determine the temperature of the combustion products for complete combustion (i.e., the adiabatic flame temperature) with the theoretical amount of air.
Question #3
Consider the steady-state combustion of gaseous methane CH$_4$ with dry air:
(a)  Methane, CH$_4$, is burned with dry air at a pressure of 1 atm. The molar analysis of the products on a dry basis is CO$_2$, 9.7%; CO, 0.5%; O$_2$, 2.95%; and N$_2$, 86.85%. Determine the air-fuel ratio on both a molar and a mass basis and the percent theoretical air.
(b)  Methane gas at 400 K and 1 atm enters a combustion chamber, where it is mixed with air entering at $500$ K and 1 atm. The products of combustion exit at 1800 K and 1 atm with the mixture composition as determined in part (a). For operation at steady-state, determine the rate of heat transfer from the combustion chamber in kJ per kmol of fuel. Neglect kinetic and potential energy effects. The average value for the specific heat $\bar{C}_P$ of methane between 298 K and 400 K is 38 kJ/kmolK.
Question #4
A natural gas has the following molar analysis: $\rm CH_4$, 80.62%; $\rm C_2 H_6$, 5.41%; $\rm C_3 H_8$, 1.87%; $\rm C_4 H_{10}$, 1.60%; $\rm N_2$, 10.50%. The gas is burned with dry air, giving products having a molar analysis on a dry basis: $\rm CO_2$, 7.8%; CO, 0.2%; $\rm O_2$, 7%; $\rm N_2$, 85%. Do the following:
(a)  Determine the air-fuel ratio on a molar basis
(b)  Assuming ideal gas behavior for the fuel mixture, determine the amount of products in kmol that would be formed from $\rm 100~m^3$ of fuel mixture at $(300+5\times\xi_2)$ K and 1 bar
(c)  Determine the percent of theoretical air
Question #5
A mixture of 1 kmol of gaseous methane and 2 kmol of oxygen initially at 25$^\circ$C and 1 atm burns completely in a closed, rigid container. Heat transfer occurs until the products are cooled to $(900-2\times\xi_2)$ K. If the reactants and products each form ideal gas mixtures, determine (i) the amount of heat transfer, and (ii) the final pressure.
Question #6
A gaseous mixture of butane (C$_4$H$_{10}$) and 80% excess air at 25$^\circ$C, 3 atm enters a reactor fitted with a cooling water jacket. Complete combustion occurs, and the products exit as a mixture at 1200 K, 3 atm. Water enters the jacket as a saturated liquid at 1 atm and saturated vapor exits at essentially the same pressure. No significant heat transfer occurs from the outer surface of the jacket, and kinetic and potential energy effects are negligible. Knowing that the mass flow rate of the gaseous mixture of butane and air is of 1 kg/s, determine (i) the mass flow rate of the cooling water in kg/s, and (ii) the rate of entropy production in kW/K (including both the entropy production in the cooling jacket and in the reactor).
Question #7
Hydrogen is mixed with oxygen and nitrogen and stored in a rigid container at a pressure of 1 atm and $25^\circ$C. A spark ignites the mixture and combustion occurs. After the chemical process has reached equilibrium, an analysis of the products reveals that they are composed of 1 kg of $\rm H_2 O$, 4 kg of $\rm N_2$, and no oxygen or hydrogen. Knowing that the volume of the container is not altered by the combustion process, determine the following:
(a)  the mass of hydrogen, oxygen, and nitrogen prior to combustion
(b)  the temperature of the products
(c)  the pressure of the products
(d)  the specific entropy change between the mixture of the products and the mixture of the reactants (in kJ/kgK)
7.  4 kg, 0.111 kg, 0.889 kg, 3360 K, 9.91 atm, 1.66 kJ/kgK
Due on Saturday June 15th at 16:00. Do Questions #3 and #7 only.
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