Two moles of oxygen are mixed with eight moles of helium. The effective specific heat of the mixture at constant volume is a) 1.3 R b) 1.4 R c) 1.7 R d) 1.9 R

Options

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

Explanation

To find the effective specific heat of a mixture of gases at constant volume, we can use the formula:

\[ C_v = \frac{n_1 C_{v1} + n_2 C_{v2}}{n_1 + n_2} \]

where:
- \( n_1 \) and \( n_2 \) are the number of moles of the two gases,
- \( C_{v1} \) is the specific heat at constant volume for the first gas,
- \( C_{v2} \) is the specific heat at constant volume for the second gas.

### Step 1: Identify the moles and specific heats of the gases
- We have 2 moles of oxygen (\( n_1 = 2 \)).
- We have 8 moles of helium (\( n_2 = 8 \)).
- The specific heat at constant volume for oxygen (\( C_{v1} \)) is \( \frac{5}{2} R \) (since oxygen is a diatomic gas).
- The specific heat at constant volume for helium (\( C_{v2} \)) is \( \frac{3}{2} R \) (since helium is a monatomic gas).

### Step 2: Substitute the values into the formula
Now we can substitute these values into the formula:

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

### Step 3: Simplify the equation
Calculating the numerator:

\[ C_v = \frac{(5R) + (12R)}{10} \]

Combine the terms in the numerator:

\[ C_v = \frac{17R}{10} \]

### Step 4: Final calculation
Now simplify the fraction:

\[ C_v = 1.7 R \]

### Conclusion
Thus, the effective specific heat of the mixture at constant volume is:

**Answer: 1.7 R (Option c)**

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