Compressibility factor under critical state of a gas is

Options

A. `frac{5}{8]`
B. `frac{8}{3}`
C. `frac{3}{8R}`
D. `frac{3}{8}`

Answer

To determine the compressibility factor (Z) under the critical state of a gas, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Compressibility Factor (Z)**:
The compressibility factor (Z) is defined as:
\[ Z = \frac{P V_m}{R T} \]
where \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the universal gas constant, and \(T\) is the temperature.

2. **Use the Critical State Relationships**:
At the critical state, we have a relationship involving the critical pressure (\(P_c\)), critical volume (\(V_c\)), and critical temperature (\(T_c\)):
\[ \frac{R T_c}{P_c V_c} = \frac{8}{3} \]

3. **Rearranging the Equation**:
Rearranging the above equation gives:
\[ \frac{P_c V_c}{R T_c} = \frac{3}{8} \]

4. **Identifying the Compressibility Factor at Critical State**:
From the rearranged equation, we can see that:
\[ Z = \frac{P_c V_c}{R T_c} = \frac{3}{8} \]

5. **Conclusion**:
Therefore, the compressibility factor (Z) under the critical state of a gas is:
\[ Z = \frac{3}{8} \]
Hence, the correct option is option number 4.