A gaseous mixture consists of 16 g of helium and 16 g of oxygen. Find `gamma` for the mixture.

Options

A. `23/29`
B. `47/23`
C. `29/23`
D. `47/29`

Explanation

To find the value of gamma (γ) for a gaseous mixture consisting of 16 g of helium and 16 g of oxygen, we can follow these steps:

### Step 1: Calculate the number of moles of each gas
- **For Helium (He)**:
- Molar mass of Helium = 4 g/mol
- Number of moles of Helium (n₁) = mass / molar mass = 16 g / 4 g/mol = 4 moles

- **For Oxygen (O₂)**:
- Molar mass of Oxygen = 32 g/mol
- Number of moles of Oxygen (n₂) = mass / molar mass = 16 g / 32 g/mol = 0.5 moles

### Step 2: Determine the degrees of freedom for each gas
- **For Helium** (monoatomic gas):
- Degrees of freedom (f₁) = 3

- **For Oxygen** (diatomic gas):
- Degrees of freedom (f₂) = 5

### Step 3: Calculate the equivalent degrees of freedom for the mixture
Using the formula for the equivalent degrees of freedom (f_equivalent):
\[ f_{equivalent} = \frac{n₁ f₁ + n₂ f₂}{n₁ + n₂} \]
Substituting the values:
\[ f_{equivalent} = \frac{(4 \text{ moles} \times 3) + (0.5 \text{ moles} \times 5)}{4 + 0.5} \]
Calculating the numerator:
\[ = \frac{12 + 2.5}{4.5} = \frac{14.5}{4.5} \]
Now simplifying:
\[ = \frac{29}{9} \]

### Step 4: Calculate gamma (γ) for the mixture
The formula for gamma is:
\[ \gamma = 1 + \frac{2}{f_{equivalent}} \]
Substituting the value of f_equivalent:
\[ \gamma = 1 + \frac{2}{\frac{29}{9}} = 1 + \frac{2 \times 9}{29} = 1 + \frac{18}{29} \]
Now, converting 1 to a fraction:
\[ = \frac{29}{29} + \frac{18}{29} = \frac{47}{29} \]

### Final Answer
Thus, the value of gamma (γ) for the mixture is:
\[ \gamma = \frac{47}{29} \]