Calculate average molar heat capacity at constant volume of gaseous mixture contained 2 mole of each of two ideal gases `A(C_(v,m)=(3)/(2)R)` and `B(C_(v,m)=(5)/(2)R) :`

Options

A. R
B. 2R
C. 3R
D. 8R

Explanation

To calculate the average molar heat capacity at constant volume of a gaseous mixture containing 2 moles of each of two ideal gases A and B, we will follow these steps:

### Step 1: Identify the given values
We are given:
- For gas A:
- Number of moles, \( n_A = 2 \)
- Molar heat capacity at constant volume, \( C_{v,m,A} = \frac{3}{2}R \)

- For gas B:
- Number of moles, \( n_B = 2 \)
- Molar heat capacity at constant volume, \( C_{v,m,B} = \frac{5}{2}R \)

### Step 2: Write the formula for average molar heat capacity
The average molar heat capacity at constant volume for a mixture of two gases can be calculated using the formula:

\[ C_{v,m,\text{mix}} = \frac{n_A \cdot C_{v,m,A} + n_B \cdot C_{v,m,B}}{n_A + n_B} \]

### Step 3: Substitute the values into the formula
Now, substituting the values we have:

\[ C_{v,m,\text{mix}} = \frac{2 \cdot \left(\frac{3}{2}R\right) + 2 \cdot \left(\frac{5}{2}R\right)}{2 + 2} \]

### Step 4: Simplify the numerator
Calculating the numerator:

\[ = 2 \cdot \frac{3}{2}R + 2 \cdot \frac{5}{2}R \]
\[ = 3R + 5R = 8R \]

### Step 5: Calculate the denominator
The denominator is:

\[ n_A + n_B = 2 + 2 = 4 \]

### Step 6: Combine the results
Now, substituting back into the formula:

\[ C_{v,m,\text{mix}} = \frac{8R}{4} = 2R \]

### Conclusion
The average molar heat capacity at constant volume of the gaseous mixture is:

\[ C_{v,m,\text{mix}} = 2R \]

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