Explain the difference between the expressions $C_P - C_V = R$,$C_P - C_V = \frac{R}{J}$,and $C_P - C_V = \frac{r}{J}$.

(N/A) The expression $C_P - C_V = R$ is Mayer's relation where $C_P$ and $C_V$ are molar specific heats measured in units of energy (Joules per mole per Kelvin) and $R$ is the universal gas constant $(8.314 \ J \ mol^{-1} K^{-1})$.
The expression $C_P - C_V = \frac{R}{J}$ is used when $C_P$ and $C_V$ are expressed in thermal units (calories per mole per Kelvin). Here,$J$ is the mechanical equivalent of heat $(J \approx 4.18 \ J/cal)$,which converts the energy unit $R$ into calories.
The expression $C_P - C_V = \frac{r}{J}$ is used when $C_P$ and $C_V$ are specific heats per unit mass (specific heat capacity) rather than molar specific heats. Here,$r$ is the specific gas constant,defined as $r = \frac{R}{M}$,where $M$ is the molar mass of the gas. This equation gives the difference in units of calories per gram per Kelvin.