Ratio of Specific Heats of Diatomic and Monoatomic Gases

Concept:
The ratio of the specific heats of a gas is defined as the ratio of the heat capacities of the gas at constant pressure and constant volume. For a diatomic gas, the ratio of specific heats is 7/5, and for a monoatomic gas, the ratio is 5/3.

Explanation:
Given that two moles of diatomic gas and one mole of monoatomic gas are mixed. Let us assume that the diatomic gas is nitrogen (N2) and the monoatomic gas is helium (He).

The total number of moles of gas is 2 + 1 = 3. We can find the ratio of specific heats of the mixture using the formula:

γmix = Cp,mix/Cv,mix

where Cp,mix is the heat capacity of the mixture at constant pressure, and Cv,mix is the heat capacity of the mixture at constant volume.

The heat capacity of a mixture of gases can be calculated using the law of mixtures:

Cp,mix = ∑niCpi
Cv,mix = ∑niCvi

where ni is the number of moles of gas i in the mixture, and Cpi and Cvi are the heat capacities of gas i at constant pressure and constant volume, respectively.

For nitrogen, Cp/N2 = 7/2 R and Cv/N2 = 5/2 R
For helium, Cp/He = 5/2 R and Cv/He = 3/2 R

Using the law of mixtures, we can find the heat capacities of the mixture:

Cp,mix = (2 mol)(7/2 R) + (1 mol)(5/2 R) = 20.5 R
Cv,mix = (2 mol)(5/2 R) + (1 mol)(3/2 R) = 9.5 R

Therefore, the ratio of specific heats of the mixture is:

γmix = Cp,mix/Cv,mix = 20.5 R / 9.5 R = 41/19 = 19/13

Hence, the correct answer is option (C).

Conclusion:
The ratio of specific heats of a mixture of gases can be calculated using the law of mixtures and the individual specific heats of the gases in the mixture. In this problem, we found the ratio of specific heats of a mixture of two moles of diatomic gas and one mole of monoatomic gas to be 19/13.