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 (\(C_V\)), we can follow these steps:

### Step 1: Identify the formula for molar specific heat at constant volume
The molar specific heat at constant volume for a gas is given by the formula:
\[ C_V = \frac{F}{2} R \]
where \(F\) is the degrees of freedom of the gas and \(R\) is the universal gas constant.

### Step 2: Determine the degrees of freedom for oxygen gas
Oxygen (\(O_2\)) is a diatomic gas. The degrees of freedom (\(F\)) for a diatomic gas is typically 5. This includes:
- 3 translational degrees of freedom (movement in x, y, and z directions)
- 2 rotational degrees of freedom (rotation about two axes, as the rotation about the axis connecting the two atoms does not contribute to energy)

### Step 3: Substitute the values into the formula
Now we can substitute \(F = 5\) and \(R = 8.314 \, \text{J mol}^{-1} \text{K}^{-1}\) into the formula:
\[ C_V = \frac{5}{2} \times 8.314 \]

### Step 4: Calculate \(C_V\)
Now, we perform the multiplication:
\[ C_V = \frac{5 \times 8.314}{2} = \frac{41.57}{2} = 20.785 \, \text{J mol}^{-1} \text{K}^{-1} \]

### Final Answer
Thus, the molar specific heat of oxygen gas at constant volume is:
\[ C_V = 20.785 \, \text{J mol}^{-1} \text{K}^{-1} \]

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