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At a given point, I know for each species the specific heat ratio, ##\gamma_i##, and number density, ##n_i##. The thermal energy density is

$$\epsilon = \frac{p}{\gamma-1},$$

where ##p## is the thermal pressure. The thermal energy density is equal to the sums of the values for each individual species

$$\epsilon = \sum_i \epsilon_i = \sum_i \frac{p_i}{\gamma_i - 1}.$$

Inserting ##p = n k_B T## (where ##n = \sum_i n_i##) and ##p_i = n_i k_B T## (therefore assuming all species have the same temperature) gives

$$\frac{n k_B T}{\gamma - 1} = \sum_i \frac{n_i k_B T}{\gamma_i - 1},$$

which can be rearranged to give

$$\gamma = \frac{n}{\sum_i \left( \frac{n_i}{\gamma_i - 1} \right) } + 1.$$

This ##\gamma## is the value that I want. Am I correct here or have I made a mistake somewhere?