Question

A gas mixture contains monoatomic and diatomic molecules of 2 moles each. The mixture has a total internal energy of (symbols have usual meanings)

• A. 3 RT
• B. 5 RT
• C. 8 RT
• D. 9 RT

Answer 1

The Correct Option is C

Step 1: Recall the formula for the internal energy of a gas
The internal energy of an ideal gas depends on its temperature and the number of moles. For a gas with $n$ moles, the internal energy for a monoatomic ideal gas is given by:

$U_{\text{mono}} = \frac{3}{2} n R T$

For a diatomic ideal gas, the internal energy is:

$U_{\text{diatomic}} = \frac{5}{2} n R T$

where:
- $n$ is the number of moles,
- $R$ is the universal gas constant,
- $T$ is the temperature.

Step 2: Identify the number of moles for each gas type
The mixture contains:
- 2 moles of monoatomic gas,
- 2 moles of diatomic gas.

Step 3: Calculate the internal energy for each type of gas
For the monoatomic gas:

$U_{\text{mono}} = \frac{3}{2} \times 2 \times R T = 3 R T$

For the diatomic gas:

$U_{\text{diatomic}} = \frac{5}{2} \times 2 \times R T = 5 R T$

Step 4: Calculate the total internal energy of the mixture
The total internal energy $U_{\text{total}}$ is the sum of the internal energies of both gases:

$U_{\text{total}} = U_{\text{mono}} + U_{\text{diatomic}} = 3 R T + 5 R T = 8 R T$

Final Answer: The total internal energy of the mixture is $8 R T$, which matches option (C).

Answer 2

The internal energy of a monatomic gas is given by $U_\text{mono} = \frac{3}{2}nRT$, and the internal energy of a diatomic gas is given by $U_\text{di} = \frac{5}{2}nRT$, where:

• $n$ is the number of moles,
• $R$ is the universal gas constant, and
• $T$ is the absolute temperature.

We are given a mixture containing 2 moles each of monatomic and diatomic molecules. Thus, $n_\text{mono} = n_\text{di} = 2$. The total internal energy of the mixture is the sum of the internal energies of the individual gases:

$U_\text{total} = U_\text{mono} + U_\text{di}$

$U_\text{total} = \frac{3}{2}n_\text{mono}RT + \frac{5}{2}n_\text{di}RT$

$U_\text{total} = \frac{3}{2}(2)RT + \frac{5}{2}(2)RT$

$U_\text{total} = 3RT + 5RT$

$U_\text{total} = 8RT$

The correct answer is (C) 8 RT.