Supposing the distance between the atoms of a diatomic gas to be constant, its specific heat at constant volume per mole (gram mole) is

Options

A. `(5)/(2)R`
B. `(3)/(2)R`
C. `R`
D. `(7)/(2)R`

Answer

To find the specific heat at constant volume (\(C_v\)) for a diatomic gas, we can follow these steps:

### Step 1: Understand the Degrees of Freedom
A diatomic gas consists of two atoms. The degrees of freedom for a molecule include translational, rotational, and vibrational motions.

### Step 2: Calculate Translational Degrees of Freedom
For a diatomic gas, each atom can move freely in three dimensions (x, y, z). Therefore, the translational degrees of freedom (\(F_t\)) is:
\[ F_t = 3 \]

### Step 3: Calculate Rotational Degrees of Freedom
In a diatomic molecule, the atoms can rotate about two axes (perpendicular to the bond between the two atoms). Thus, the rotational degrees of freedom (\(F_r\)) is:
\[ F_r = 2 \]

### Step 4: Consider Vibrational Degrees of Freedom
At room temperature, the vibrational degrees of freedom (\(F_v\)) for diatomic gases can be considered negligible. Therefore:
\[ F_v \approx 0 \]

### Step 5: Total Degrees of Freedom
Now, we can calculate the total degrees of freedom (\(F\)) for the diatomic gas:
\[ F = F_t + F_r + F_v = 3 + 2 + 0 = 5 \]

### Step 6: Use the Formula for Specific Heat at Constant Volume
The specific heat at constant volume per mole (\(C_v\)) is given by the formula:
\[ C_v = \frac{F \cdot R}{2} \]
where \(R\) is the universal gas constant.

### Step 7: Substitute the Total Degrees of Freedom into the Formula
Now, substituting \(F = 5\) into the formula:
\[ C_v = \frac{5 \cdot R}{2} \]

### Step 8: Final Answer
Thus, the specific heat at constant volume per mole for the diatomic gas is:
\[ C_v = \frac{5R}{2} \]

### Conclusion
The correct answer is \(C_v = \frac{5R}{2}\).

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