How would one calculate the Heat Capacity Ratio for a multiple compound Gas?

Question

So I'm attempting to calculate the heat capacity ratio (known also as adiabatic index) of a gas with several concentrations of different compounds such as N2, H2O, CO2, CH4 etc. And while I have an idea (I show below) I haven't actually found any confirmation that this is how the ratios work. Perhaps there is some special way the gases interact changing the heat capacity ratio of the entire gas that someone knows of?

What I'm assuming I should do is take the ratios of each and multiply them by the percentage of the weights within the gas. For example if my gas was 20% CO2 and 80% N2, and CO2 had a ratio of K=1.2, and N2 had a ratio of K=1.33 (just as placeholder values), then my calculation would be

Total ratio of my gas : (0.2)(1.2) + (0.8)(1.33) = 1.304

I'm operating at about temperature ranges of 1000°K - 3000°K, and pressures from 455 Megapascal and under. However I'm still basically assuming ideal gas (I really don't know how you would do this if not anyway, if anyone has an idea I am open to suggestions)

Separate portion of my question, does anyone know what the heat capacity ratios of the compounds I named above would actually be? Maybe someone here has a detailed chart of heat capacity ratios for different compounds as a function of temperature? The wiki has a table but it's not enough for my purposes.

I have them down as the following using the algorithm from the degrees of freedom wiki (including vibrational modes) and plugging them into

heat capacity ratio = 1 + [2/(degrees of freedom)]

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N2 : k = 1+ 2/[3 + 2 + (3x2 - 5)] = 1.33 (linear algorithm)

H2O : k = 1+ 2/[3 + 3 + (3x3 - 6)] = 1.22 (non-linear algorithm)

CO : k = 1+ 2/[3 + 2 + (3x2 - 5)] = 1.33 (linear algorithm)

CH4 : k = 1+ 2/[3 + 3 + (5x3 - 6)] = 1.13 (non-linear algorithm)

(k is heat capacity ratio)

EDIT : I just found out that vibrational degrees of freedom contribute twice according to the equipartition theorem? And thus twice for the specific heat ratio? Meaning I would have multiply the vibrational brackets by 2? So my new ratios would be (?) :

N2 : k = 1+ 2/[3 + 2 + 2(3x2 - 5)] = 1.285 (linear algorithm)

H2O : k = 1+ 2/[3 + 3 + 2(3x3 - 6)] = 1.167 (non-linear algorithm)

CO : k = 1+ 2/[3 + 2 + 2(3x2 - 5)] = 1.285 (linear algorithm)

CH4 : k = 1+ 2/[3 + 3 + 2(5x3 - 6)] = 1.083 (non-linear algorithm)

Would love if anyone can confirm any of this or answer my question

Answer 1

The heat capacity is a measure of translational, vibrational and rotational energy in molecule. Heat capacity is $U/T$. The translational energy is always $U=3RT/2$ ($R$ is gas constant, $T$ temperature). A diatomic molecule has rotational energy $RT$ and vibrational energy $RT$. This means that the constant volume heat capacity for a diatomic molecule is $C_V=7RT/2$ provided the temperature is high enough.

For a polyatomic the rotational energy is $3RT/2$ and vibrational energy $xRT$ where $x$ is the number of vibrations, $3N-6$ for a molecule of $N$ atoms and $3N-5$ if it is linear. This means that the heat capacity is $C_V=(x+3)R$. The constant pressure heat capacity is given by $C_P-C_V=R$ and the ratio $\gamma=C_P/C_V$ is often quoted as adiabatic index, $1.66\cdot$ for an atom, $1.29$ for a diatomic and $\displaystyle \frac{x+4}{x+3}$ for a polyatomic.

At $1000$K the temperature is probably so high that many vibrational levels are populated and these expressions will be accurate ( take their limiting values) if not you will need to consult a phys. chem. text book and work out the partition functions $Z$ for the molecules and this means knowing the frequencies vibrational motion. These are tabulated and the calculation is not hard just a bit tedious. You can then calculate how the heat capacity varies with temperature. If you cannot find the equations I can post them. (The translational and rotational energies are so small that their limiting values can be assumed without error at $1000$ K and also usually at room temperature.)

Edit:

In general the partition function for energies $E_i$ is

$$\displaystyle Z=\sum_i g_ie^{-E_i/k_BT}$$

The partition functions you need are for vibrations. Each vibration's energy is $E_i= hc\omega_i(n_i+1/2)$ assuming harmonic motion and frequency $\omega$ in wavenumbers. ($h$ is the Planck constant and $c$ the speed of light, $k_B$ Boltzmann constant, $g_i$ is the degeneracy, $1$ for most vibrations). For a molecule with many vibrations with frequencies $\omega_1$ having quanta $n_1$ etc.

$$\displaystyle Z=\sum_{n_1}\sum_{n_2}\cdots e^{-(u_1n_1+u_2n_2\cdots)}$$

where $u_i=hc\omega_i/k_BT=1.4387\omega_i/T$.

This can be simplified to

$$\displaystyle Z=\prod_i\sum_{n_i} e^{-u_i n_i}$$

and as all thermodynamic functions depend on $\ln(Z)$ this becomes

$$\ln(Z)=\sum_i\ln\left( \sum_{n_i}e^{-u_i n_i}\right)$$

Luckily the exponential summation has a closed form as

$$\displaystyle \sum_{n_i}e^{-u_i n_i}=\frac{1}{1-e^{-u_i} }$$

The final thing you need is the connection for an ideal gas between the partition function and the heat capacity and this is

$$\displaystyle C_V=RT\left(2\frac{d \ln(Z)}{dT} +T\frac{d^2 \ln(Z)}{dt^2}\right)$$

You can probably get accurate enough values for the derivatives numerically but an algebraic solution looks straightforwards but messy.

I get for one vibration $$\displaystyle C_V=\frac{R s^{2} e^{\frac{s}{T}}}{T^{2} \left(e^{\frac{2 s}{T}} - 2 e^{\frac{s}{T}} + 1\right)}$$

with $u=s/T$ but you should check this.