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. `1.4R`
C. `1.7R`
D. `1.9R`

Explanation

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

### Step 1: Identify the number of moles and specific heats
- Let \( N_1 = 2 \) moles of oxygen.
- Let \( N_2 = 8 \) moles of helium.
- The molar specific heat at constant volume for oxygen (\( Cv_1 \)) is \( \frac{5R}{2} \) (since oxygen is a diatomic molecule).
- The molar specific heat at constant volume for helium (\( Cv_2 \)) is \( \frac{3R}{2} \) (since helium is a monoatomic molecule).

### Step 2: Use the formula for effective specific heat
The effective specific heat at constant volume for the mixture can be calculated using the formula:
\[ Cv = \frac{N_1 \cdot Cv_1 + N_2 \cdot Cv_2}{N_1 + N_2} \]

### Step 3: Substitute the values into the formula
Substituting the values we have:
\[ Cv = \frac{2 \cdot \frac{5R}{2} + 8 \cdot \frac{3R}{2}}{2 + 8} \]

### Step 4: Simplify the equation
Calculating the numerator:
\[ = \frac{2 \cdot \frac{5R}{2} + 8 \cdot \frac{3R}{2}}{10} = \frac{5R + 12R}{10} = \frac{17R}{10} \]

### Step 5: Final calculation
Now, we can simplify:
\[ Cv = \frac{17R}{10} \]
This simplifies to:
\[ Cv = 1.7R \]

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