Two moles of oxygen are mixed with eight moles of helium. The effective specific heat of the mixture at constant volume is

Options

A. 1.3 R
B. 1.4 R
C. 1.7 R
D. 1.9 R

Explanation

To find the effective specific heat of the mixture at constant volume for two moles of oxygen mixed with eight moles of helium, we can follow these steps:

### Step 1: Identify the number of moles and types of gases
We have:
- \( N_1 = 2 \) moles of oxygen (O₂)
- \( N_2 = 8 \) moles of helium (He)

### Step 2: Determine the specific heat capacities at constant volume for each gas
- For oxygen (a diatomic gas), the specific heat capacity at constant volume \( C_{v1} \) is given by:
\[ C_{v1} = \frac{5}{2} R \]

- For helium (a monatomic gas), the specific heat capacity at constant volume \( C_{v2} \) is given by:
\[ C_{v2} = \frac{3}{2} R \]

### Step 3: Use the formula for the effective specific heat of the mixture
The effective specific heat at constant volume \( C_{v, \text{mix}} \) for the mixture can be calculated using the formula:
\[ C_{v, \text{mix}} = \frac{N_1 C_{v1} + N_2 C_{v2}}{N_1 + N_2} \]

### Step 4: Substitute the values into the formula
Substituting the values we have:
\[ C_{v, \text{mix}} = \frac{2 \left(\frac{5}{2} R\right) + 8 \left(\frac{3}{2} R\right)}{2 + 8} \]

### Step 5: Simplify the expression
Calculating the numerator:
\[ = \frac{2 \cdot \frac{5}{2} R + 8 \cdot \frac{3}{2} R}{10} \]
\[ = \frac{5 R + 12 R}{10} \]
\[ = \frac{17 R}{10} \]

### Step 6: Final result
Thus, the effective specific heat of the mixture at constant volume is:
\[ C_{v, \text{mix}} = 1.7 R \]

### Conclusion
The effective specific heat of the mixture at constant volume is \( 1.7 R \).

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