# Molar mass in Mayer's relation

$$C_p$$ and $$C_v$$ are specific heats at constant pressure and constant volume respectively. It is observed that for hydrogen gas $$C_p - C_v = a$$ and for nitrogen gas $$C_p - C_V = b$$ . The correct relation between a and b is? [JEE mains 2017]

I applied Mayers relation and used $$C_p - C_v=R$$ for all gases, hence $$a=b$$ but that formula seems to be incorrect here. Supposedly, the correct mayer's relation is given as:

$$C_p -C_v = \frac{R}{M}$$

Where $$M$$ is molar mass of gas in consideration.

I don't get how this equation came about. I had written a derivation for this Mayer's relation as a self answer (see here) in one of my old posts but I can't see where in the derivation missed a factor of molecular mass.

So, my question is where does molar mass of gas come into the formula of Mayer's relation from?

• Is there a difference between the terms molar heat capacity and specific heat capacity? Commented Mar 9, 2021 at 21:24
• Oh @ChetMiller I think I can self answer now Commented Mar 9, 2021 at 21:27

The thing to notice is that in the regular Mayer relation, the coefficients given are of specific heats (see here). So, if call the original molar quantites as $$C_{p'}$$ and $$C_{v'}$$, then:

$$C_p = \frac{C_{p'}}{M}$$

And,

$$C_v = \frac{C_{v'}}{M}$$

Hence, putting these back into the 'regular' mayer relation , we get the result.

Thanks to @Chet Miller for the hint