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

Explanation

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

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

### Step 2: Identify the degrees of freedom for a diatomic gas
For a diatomic gas, the degrees of freedom \( F \) 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 rotation about the axis connecting the two atoms does not count)

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

### Step 4: Calculate \( C_v \)
Calculating the above expression:
\[ C_v = \frac{5 \times 8.314}{2} = \frac{41.57}{2} = 20.785 \, \text{J mol}^{-1} \text{K}^{-1} \]

### Step 5: Convert \( C_v \) from Joules to Calories
To convert from Joules to Calories, we use the conversion factor \( 1 \, \text{cal} = 4.2 \, \text{J} \):
\[ C_v = \frac{20.785 \, \text{J mol}^{-1} \text{K}^{-1}}{4.2 \, \text{J/cal}} \]

### Step 6: Perform the conversion calculation
Calculating the above:
\[ C_v = \frac{20.785}{4.2} \approx 4.96 \, \text{cal mol}^{-1} \text{K}^{-1} \]

### Conclusion
Thus, the molar specific heat of a diatomic gas at constant volume is approximately:
\[ C_v \approx 4.96 \, \text{cal mol}^{-1} \text{K}^{-1} \]

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