Three moles of oxygen ar mixed with two moles of helium. What will be the ratio of specific heats at constant pressure and constant volume for the mixture ?

Options

A. 6.7
B. 1.5
C. 4
D. none of the above

Explanation

To find the ratio of specific heats at constant pressure and constant volume (denoted as γ) for a mixture of gases, we can follow these steps:

### Step 1: Identify the gases and their moles
We have:
- Oxygen (O₂) with 3 moles (n₁ = 3)
- Helium (He) with 2 moles (n₂ = 2)

### Step 2: Determine the specific heats for each gas
For the gases, we use the following values:
- For Oxygen (O₂, a diatomic gas):
- Specific heat at constant volume (Cᵥ₁) = \( \frac{5}{2} R \)
- Specific heat at constant pressure (Cₚ₁) = \( \frac{7}{2} R \)
- Ratio (γ₁) = \( \frac{Cₚ₁}{Cᵥ₁} = \frac{\frac{7}{2} R}{\frac{5}{2} R} = \frac{7}{5} \)

- For Helium (He, a monatomic gas):
- Specific heat at constant volume (Cᵥ₂) = \( \frac{3}{2} R \)
- Specific heat at constant pressure (Cₚ₂) = \( \frac{5}{2} R \)
- Ratio (γ₂) = \( \frac{Cₚ₂}{Cᵥ₂} = \frac{\frac{5}{2} R}{\frac{3}{2} R} = \frac{5}{3} \)

### Step 3: Calculate the weighted average ratio for the mixture
To find the overall ratio (γ) for the mixture, we use the formula:
\[ \gamma = \frac{n₁ \gamma₁ + n₂ \gamma₂}{n₁ + n₂} \]

Substituting the values:
\[ \gamma = \frac{3 \cdot \frac{7}{5} + 2 \cdot \frac{5}{3}}{3 + 2} \]

Calculating the numerator:
- First term: \( 3 \cdot \frac{7}{5} = \frac{21}{5} \)
- Second term: \( 2 \cdot \frac{5}{3} = \frac{10}{3} \)

Finding a common denominator (15):
- Convert \( \frac{21}{5} \) to \( \frac{63}{15} \)
- Convert \( \frac{10}{3} \) to \( \frac{50}{15} \)

Now, add these two fractions:
\[ \frac{63}{15} + \frac{50}{15} = \frac{113}{15} \]

Now, divide by the total number of moles:
\[ \gamma = \frac{\frac{113}{15}}{5} = \frac{113}{75} \]

### Step 4: Simplify the result
Calculating \( \frac{113}{75} \) gives approximately \( 1.5 \).

### Final Answer
Thus, the ratio of specific heats for the mixture is \( \gamma \approx 1.5 \).

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