Answer to Question #87738 in Molecular Physics | Thermodynamics for Sridhar

Question

Consider a mixture of equal number of non-interacting single atomic gas and diatomic gas. The internal energy per particle is given by :
(A) 2kT
(B) 4kT
(C) 3/2kT
(D)5/2kT

Answer 1

In case of a mixture of equal number of non-interacting single atomic gas and diatomic gas the number of molecules of each gas satisfies the condition:

$$N_1 = N_2 = \frac{1}{2} N,$$

where N is the total number of molecules in the mixture. Dividing this expression over the Avogadro constant, the similar expression can be obtained for the molar amount of substance:

$$\nu_1 = \nu_2 = \frac{1}{2}\nu$$

The general expression for the internal energy of a gas can be written as follows:

$$U = \frac{i}{2} \nu R T,$$

where i is the number of degrees of freedom of a single molecule of a gase (i=3 for monoatomic gas and i=5 for diatomic gas).

Hence, the total internal energy of a mixture can be calculated as:

$$U = U_1 + U_2 = \frac{3}{2} \frac{\nu}{2} RT + \frac{5}{2} \frac{\nu}{2} RT = 2 \nu R T = 2 N k T,$$

where we take into account the relation:

$$R = k N_A$$

Finally, dividing the expression for the total internal energy over the total number of molecules, we obtain

$$\left< K \right> = \frac{U}{N} = 2kT$$

Answer: a) 2kT.