Which of the following statement given below are correct for monoatomic gas?

Options

A. ` C_v = (5)/(2)R `
B. ` C_P = (3)/(2)R `
C. ` C_p -C_v = 2R `
D. ` (C_P)/(C_V) = (5)/(3)`

Explanation

To determine which statements are correct for a monoatomic gas, we need to analyze the properties of monoatomic gases based on the kinetic theory of gases. Here’s a step-by-step solution:

### Step 1: Understand Degrees of Freedom
For a monoatomic gas, the molecules consist of single atoms. The degrees of freedom (f) for a monoatomic gas is given by the translational motion of the gas particles.

- **Degrees of Freedom (f)**: For monoatomic gases, f = 3 (only translational motion).

### Step 2: Calculate the Molar Heat Capacity at Constant Volume (C_V)
The molar heat capacity at constant volume (C_V) can be calculated using the formula:
\[ C_V = \frac{f}{2} R \]
Where R is the universal gas constant.

- Since f = 3 for a monoatomic gas:
\[ C_V = \frac{3}{2} R \]

### Step 3: Calculate the Molar Heat Capacity at Constant Pressure (C_P)
The relationship between C_P and C_V is given by:
\[ C_P = C_V + R \]
Substituting the value of C_V:
\[ C_P = \frac{3}{2} R + R = \frac{5}{2} R \]

### Step 4: Calculate the Ratio of C_P to C_V
The ratio of the heat capacities (γ) is given by:
\[ \gamma = \frac{C_P}{C_V} \]
Substituting the values we calculated:
\[ \gamma = \frac{\frac{5}{2} R}{\frac{3}{2} R} = \frac{5}{3} \]

### Conclusion
Based on the above calculations, we can summarize the properties of a monoatomic gas:
1. Degrees of freedom (f) = 3
2. Molar heat capacity at constant volume (C_V) = \(\frac{3}{2} R\)
3. Molar heat capacity at constant pressure (C_P) = \(\frac{5}{2} R\)
4. Ratio of heat capacities (γ) = \(\frac{5}{3}\)

### Correct Statements
Based on the analysis, the correct statements regarding a monoatomic gas are:
- The degrees of freedom is 3.
- \(C_V = \frac{3}{2} R\).
- \(C_P = \frac{5}{2} R\).
- The ratio \( \frac{C_P}{C_V} = \frac{5}{3} \).