C_p and C_v are the specific heats at constant pressure and...

For a given gas, the molar heat capacities at constant pressure and constant volume are related by Mayer's relation:

$C_{P,m} - C_{V,m} = R$

The specific heat is defined as the heat capacity per unit mass. Therefore, the difference between the specific heats at constant pressure ($C_p$) and constant volume ($C_v$) is given by:

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

where $M$ is the molar mass of the gas.

For hydrogen gas ($H_2$), the molar mass is $M_{H_2} = 2 \times 1 = 2$ u. Thus:

$a = \dfrac{R}{2}$

For carbon dioxide gas ($CO_2$), the molar mass is $M_{CO_2} = 12 + 2 \times 16 = 44$ u. Thus:

$b = \dfrac{R}{44}$

Taking the ratio of $a$ to $b$:

$\dfrac{a}{b} = \dfrac{\dfrac{R}{2}}{\dfrac{R}{44}} = \dfrac{44}{2} = 22$

$\Rightarrow a = 22 b$

Answer: $a = 22\ b$