-18.22 (a) If you apply the same amount of heat to 1.00 mol of an ideal monatomic gas and 1.00 mol of an ideal diatomic gas, which one (if any) will increase more in temperature? (b) Physically, why do diatomic gases have a greater molar heat capacity than mono-atomic gases?

Molar heat capacity is the amount of heat required to raise the temperature of 1 mole of a substance by$\mathbf{1}\mathbf{°}\mathbf{C}$.

Molar heat capacity is related to heat intake as, $Q=n{C}_v∆T$where $Q$is the heat intake, $n$is the number of moles, $C_v$is the molar specific heat capacity at constant volume and $∆T$is the change in temperature due to heat absorbed.

Finding the increase in temperature by rearranging the equation for heat capacity as-

$∆T=\frac{Q}{n{C}_v}$

The temperature is thus inversely proportional to the specific heat capacity, provided the amount of heat transferred remains constant. The substance with more specific heat capacity will experience lesser change in temperature.

Simplifying,

$∆T\propto \frac{1}{C_v}$

For a monoatomic gas, molar specific heat capacity is $C_v=\frac{3}{2}R$and for a diatomic gas, $C_v=\frac{5}{2}R$.

Which implies that the diatomic gas has more molar specific heat capacity, and thus the change in temperature $DT$will be greater for the monoatomic gas.

Hence the monoatomic will increase more in temperature if we apply the same heat to the same quantity of monoatomic gas and diatomic gas.

Heat capacity is dependent on the translational kinetic energy itself.

For a monoatomic gas, the heat supplied is entirely consumed for increasing the translational kinetic energy.

For a diatomic gas, there are additional degrees of freedom, like rotational and vibrational. These additional degrees do consume a part of the supplied heat energy.

So, more heat is required to raise the translational kinetic energy of diatomic gases to the same amount as in the monoatomic gases.