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$C_p$ 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?

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  • $\begingroup$ Is there a difference between the terms molar heat capacity and specific heat capacity? $\endgroup$
    – Chet Miller
    Mar 9, 2021 at 21:24
  • $\begingroup$ Oh @ChetMiller I think I can self answer now $\endgroup$
    – tryst with freedom
    Mar 9, 2021 at 21:27

1 Answer 1

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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

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