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. `2.5`
B. `3.5`
C. `1.5`
D. 1

Explanation

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

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

### Step 2: Determine the specific heats for each gas
1. **For Oxygen (O2)**:
- Cv (O2) = \( \frac{5}{2} R \)
- Cp (O2) = \( \frac{7}{2} R \)

The ratio of specific heats (γ) for oxygen:
\[ \gamma_1 = \frac{C_p}{C_v} = \frac{\frac{7}{2} R}{\frac{5}{2} R} = \frac{7}{5} \]

2. **For Helium (He)**:
- Cv (He) = \( \frac{3}{2} R \)
- Cp (He) = \( \frac{5}{2} R \)

The ratio of specific heats (γ) for helium:
\[ \gamma_2 = \frac{C_p}{C_v} = \frac{\frac{5}{2} R}{\frac{3}{2} R} = \frac{5}{3} \]

### Step 3: Calculate the effective specific heat ratio for the mixture
Using the formula for the effective specific heat ratio (γ_net) for a mixture:
\[ \gamma_{net} = \frac{n_1 \gamma_1 + n_2 \gamma_2}{n_1 + n_2} \]
where:
- \( n_1 = 3 \) (moles of O2)
- \( n_2 = 2 \) (moles of He)

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

### Step 4: Simplify the expression
Calculating the numerator:
\[ 3 \cdot \frac{7}{5} = \frac{21}{5} \]
\[ 2 \cdot \frac{5}{3} = \frac{10}{3} \]

Finding a common denominator (15):
\[ \frac{21}{5} = \frac{63}{15} \]
\[ \frac{10}{3} = \frac{50}{15} \]

Adding these:
\[ \frac{63}{15} + \frac{50}{15} = \frac{113}{15} \]

Now, the denominator:
\[ 3 + 2 = 5 \]

So:
\[ \gamma_{net} = \frac{\frac{113}{15}}{5} = \frac{113}{75} \]

### Step 5: Final calculation
Calculating:
\[ \gamma_{net} = \frac{113}{75} \approx 1.51 \]

### Conclusion
The ratio of specific heats at constant pressure and constant volume for the mixture is approximately \( 1.51 \).