I am reading Rocket Propulsion Elements by George P. Sutton & Oscar Biblarz, 9th Edition. In the fifth chapter, I was introduce to the specific heat ratio k for the perfect gas mixture, Eq. 5-7. Without explanation the Equation was stated as follows:

(5-7) $k_{\text{mix}} = \frac{(C_{p})_{\text{mix}}}{(C_{p})_{\text{mix}}-R'}$

From Thermodynamics I know,

(a) $k_{j} = \frac{(C_{p})_{\text{j}}}{(C_{v})_{\text{j}}}$ ; Species specific heat ratio

(b) $R_{j} = \frac{R'}{\mathfrak{M}_{j}}; $ Species Gas constant equal to Univ. Gas constant over Molar Mass of species $\text{&}$

(c) $R_{j} = (C_{p})_{\text{j}} - (C_{v})_{\text{j}}$ ;Mayer's Formula for species

So, it seems to me that Eq. (5-7) should instead include the mixture gas constant $R_{\text{mix}}$ in the place of the Universal gas constant, $R'$.

I tried to derive Eq. (5-7) from Eqs. a, b, and c without success. Can someone clarify the derivation of Eq. (5-7) to me?

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    $\begingroup$ It seems to me, Eqn. 5-7 is referring to molar quantities. $\endgroup$ – Chet Miller Sep 17 '20 at 20:05

In general for a perfect or ideal gas,

$$C_p=C_V + R'$$

(using your notation) where the heat capacities are molar quantities.

It follows that for a perfect gas mixture $(C_p)_\text{mix}=(C_V)_\text{mix} + R'$.

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    $\begingroup$ I agree. I believe my issue was due to a misunderstanding. $C_{p}$ is the molar specific heat, and $c_{p}$ is the mass-basis specific heat. Thank you! $\endgroup$ – John Ortiz Sep 17 '20 at 20:01

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