Monatomic diatomic and triatomic gases whose initial volume and pressure are same, are compressed till their volume becomes half te initial volume.

Options

A. If the compression is adiabatic then monatomic gas wil have maximum final pressure.
B. If the compression is adiabatic then triatomic gas will have maximum final pressure.
C. if the compression is isothermal then their final pressure will be same.
D. If te compression is adiabatic then their final pressure will be minimum for diatomic gas.

Explanation

To solve the problem of comparing the final pressures of monatomic, diatomic, and triatomic gases when compressed to half their initial volume, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Initial Conditions**:
- All gases (monatomic, diatomic, and triatomic) start with the same initial volume (V₀) and initial pressure (P₀).

2. **Determine the Final Volume**:
- The gases are compressed to half of their initial volume, so the final volume (V_f) is:
\[ V_f = \frac{V₀}{2} \]

3. **Apply the Ideal Gas Law**:
- The ideal gas law states:
\[ PV = nRT \]
- Since the number of moles (n) and the temperature (T) are constant for each gas during the compression process, we can analyze the pressure change.

4. **Consider the Type of Compression**:
- We need to consider two types of processes: **adiabatic** and **isothermal**.

5. **For Isothermal Compression**:
- In an isothermal process, the temperature remains constant. Therefore, we can use the relationship:
\[ P₀V₀ = P_fV_f \]
- Substituting \(V_f = \frac{V₀}{2}\):
\[ P₀V₀ = P_f \left(\frac{V₀}{2}\right) \]
- Rearranging gives:
\[ P_f = 2P₀ \]
- Thus, the final pressure for all gases in an isothermal process is twice the initial pressure.

6. **For Adiabatic Compression**:
- For adiabatic processes, the relationship between pressure and volume is given by:
\[ PV^\gamma = \text{constant} \]
- Here, \(\gamma\) (gamma) is the heat capacity ratio (C_p/C_v):
- For monatomic gases, \(\gamma = \frac{5}{3}\)
- For diatomic gases, \(\gamma = \frac{7}{5}\)
- For triatomic gases, \(\gamma\) is typically around \(\frac{8}{6}\) or \(\frac{4}{3}\) depending on the specific gas.

7. **Calculate Final Pressure for Each Gas Type**:
- For monatomic gas:
\[ P_f = P₀ \left(\frac{V₀}{V_f}\right)^\gamma = P₀ \left(\frac{V₀}{\frac{V₀}{2}}\right)^{\frac{5}{3}} = P₀ \cdot 2^{\frac{5}{3}} \]
- For diatomic gas:
\[ P_f = P₀ \left(\frac{V₀}{\frac{V₀}{2}}\right)^{\frac{7}{5}} = P₀ \cdot 2^{\frac{7}{5}} \]
- For triatomic gas:
\[ P_f = P₀ \left(\frac{V₀}{\frac{V₀}{2}}\right)^{\frac{4}{3}} = P₀ \cdot 2^{\frac{4}{3}} \]

8. **Compare Final Pressures**:
- The final pressures will differ based on the value of \(\gamma\):
- Monatomic: \(P_f = P₀ \cdot 2^{\frac{5}{3}}\)
- Diatomic: \(P_f = P₀ \cdot 2^{\frac{7}{5}}\)
- Triatomic: \(P_f = P₀ \cdot 2^{\frac{4}{3}}\)

9. **Conclusion**:
- The monatomic gas will have the highest final pressure, followed by diatomic, and then triatomic gases.