Calculate the average molar heat capacity at constant volume of a mixture containing 2 moles of monoatomic and 3 moles of diatomic ideal gas.

Options

A. R
B. 2.1 R
C. 3.2 R
D. 4 R

Answer

To calculate the average molar heat capacity at constant volume (Cv) of a mixture of gases, we can follow these steps:

### Step 1: Identify the heat capacities for the gases
For monoatomic ideal gas:
- Cv1 = \( \frac{3}{2} R \)

For diatomic ideal gas:
- Cv2 = \( \frac{5}{2} R \)

### Step 2: Determine the number of moles of each gas
- Number of moles of monoatomic gas (n1) = 2 moles
- Number of moles of diatomic gas (n2) = 3 moles

### Step 3: Use the formula for average molar heat capacity at constant volume
The average molar heat capacity (Cv,avg) for the mixture can be calculated using the formula:
\[ C_{v,avg} = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ C_{v,avg} = \frac{(2 \cdot \frac{3}{2} R) + (3 \cdot \frac{5}{2} R)}{2 + 3} \]

### Step 5: Simplify the expression
Calculating the numerator:
\[ = \frac{(3R) + (7.5R)}{5} \]
\[ = \frac{10.5R}{5} \]

### Step 6: Final calculation
\[ C_{v,avg} = 2.1 R \]

### Conclusion
The average molar heat capacity at constant volume for the mixture is:
\[ C_{v,avg} = 2.1 R \]

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