Calculate the molar specific heat of oxygen gas at constant volume. `(R=8.314" J "mol^(-1)K^(-1))`

Explanation

To calculate the molar specific heat of oxygen gas at constant volume, we can follow these steps:

### Step 1: Identify the type of gas
Oxygen gas (O₂) is a diatomic gas.

### Step 2: Determine the degrees of freedom (F)
For a diatomic gas, the degrees of freedom (F) is given by:
- Translational motion: 3 degrees of freedom
- Rotational motion: 2 degrees of freedom
- Vibrational motion: Generally neglected at room temperature for diatomic gases.

Thus, the total degrees of freedom for diatomic gases is:
\[ F = 5 \]

### Step 3: Use the formula for molar specific heat at constant volume (C_V)
The formula for molar specific heat at constant volume for a gas is given by:
\[ C_V = \frac{F \cdot R}{2} \]
Where:
- \( R \) is the universal gas constant, which is given as \( R = 8.314 \, \text{J mol}^{-1} \text{K}^{-1} \).

### Step 4: Substitute the values into the formula
Substituting \( F = 5 \) and \( R = 8.314 \, \text{J mol}^{-1} \text{K}^{-1} \) into the equation:
\[ C_V = \frac{5 \cdot 8.314}{2} \]

### Step 5: Calculate \( C_V \)
Calculating the above expression:
\[ C_V = \frac{5 \cdot 8.314}{2} = \frac{41.57}{2} = 20.785 \, \text{J mol}^{-1} \text{K}^{-1} \]

### Step 6: Convert Joules to Calories
To convert the result from Joules to calories, we use the conversion factor \( 1 \, \text{cal} = 4.2 \, \text{J} \):
\[ C_V = \frac{20.785 \, \text{J mol}^{-1} \text{K}^{-1}}{4.2} \]

### Step 7: Perform the conversion
Calculating the above expression:
\[ C_V = \frac{20.785}{4.2} \approx 4.94 \, \text{cal mol}^{-1} \text{K}^{-1} \]

### Final Answer
The molar specific heat of oxygen gas at constant volume is approximately:
\[ C_V \approx 4.94 \, \text{cal mol}^{-1} \text{K}^{-1} \]

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