# Calculation of enthalpy and internal energy or air in chemical equilibrium

1. Sep 28, 2011

### roldy

1. The problem statement, all variables and given/known data
Consider air in chemical equilibrium at 0.1 atm and T=4500 K. The chemical species are O2, O, N2, N (Ignore NO). Calculate the enthalpy and internal energy per unit mass of the mixture. Neglect electronic excitation in your calculations.

$$K_{p,O_2}=12.19 atm$$
$$K_{p,N_2}=0.7899*10^{-4}atm$$

$$(\Delta H_f^o)_O=2.47*10^8 J/(kg*mol)$$
$$(\Delta H_f^o)_N=4.714*10^8 J/(kg*mol)$$

$$(\Theta_v)_{N_2}=3390 K$$
$$(\Theta_v)_{O_2}=2270 K$$

2. Relevant equations
$$(1)h=\Sigma ^n_{i=1}c_ih_i + \Sigma ^n_{i=1}c_i \Delta h_{f_i}$$

$$c_i=X_i\frac{\mu_i}{\mu}$$
$$X_i=\frac{P_i}{P}$$

$$R_{air}=\frac{R_u}{\mu_{air}}$$
$$\mu_{air}=X_{O_2}*\mu_{O_2} + X_O*\mu_O + X_N*\mu_N + X_{N_2}*\mu_{N_2}$$

Diatomic Gas:
$$h=e_{sens}+RT$$
$$e_{sens}=3/2RT + RT +\frac{\frac{\Theta_v}{T}}{e^{\frac{\Theta_v}{T}}-1}RT$$

Monatomic Gas:
$$h=5/2RT$$
$$=e_{sens}=3/2RT$$

3. The attempt at a solution

The weight of each species is as follows:
$$O=16kg/(kg*mol)$$
$$N=14kg(kg*mol)$$
$$O_2=32kg/(kg*mol)$$
$$N_2=28kg/(kg*mol)$$

I then found what the gas constant was for each species with Ru=8314 J/(kg*K).
$$R_O=519.625$$
$$R_N=593.8571$$
$$R_{O_2}=259.8125$$
$$R_{N_2}=296.9286$$

I then calculated the sensible internal enthalpy for each species.
$$e_{sense,O}=3/2(519.625)(4500)=3.5075*10^6$$
$$e_{sense,N}=3/2(593.8571)(4500)=4.0085*10^6$$
$$e_{sense,O_2}=3/2(259.8125)(4500) + (259.8125)(4500) +\frac{\frac{2270}{4500}}{e^{\frac{2270}{4500}}-1}(259.8125)(4500)=3.8218*10^6$$
$$e_{sense,N_2}=3/2(296.9286)(4500) + (296.9286)(4500) +\frac{\frac{3390}{4500}}{e^{\frac{3390}{4500}}-1}(296.9286)(4500)=4.2359*10^6$$

Next the enthalpy per unit mass of each species was calculated.
$$h_O=5/2(519.625)(4500)=5.8458*10^6$$
$$h_N=5/2(593.8571)(4500)=6.6809*10^6$$
$$h_{O_2}=3.8218*10^6+259.8125(4500)=4.9910*10^6$$
$$h_{N_2}=4.2359*10^6+296.9286(4500)=5.5721*10^6$$

This is where I get stuck. I believe I use equation (1) in some way but I do not see how since I am not given the partial pressures to find $$c_i$$. I'm not sure how the equilibrium constants work into this problem.