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

### Step-by-Step Solution:

1. **Understanding the Compressibility Factor (Z)**:
The compressibility factor (Z) is defined as:
\[ Z = \frac{PV}{nRT} \]
where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the temperature.

2. **Using Critical Constants**:
Under critical conditions, we need to use the critical constants:
- \( P_c \) (critical pressure)
- \( V_c \) (critical volume)
- \( T_c \) (critical temperature)

3. **Substituting Critical Values**:
The critical constants can be expressed in terms of the parameters a and b (from the van der Waals equation):
- \( T_c = \frac{8a}{27bR} \)
- \( V_c = 3b \)
- \( P_c = \frac{8a}{27b^2} \)

4. **Substituting into the Compressibility Factor Formula**:
Now, substituting these values into the formula for Z:
\[ Z = \frac{P_c V_c}{R T_c} \]

5. **Calculating Z**:
Substitute \( P_c \), \( V_c \), and \( T_c \) into the equation:
\[ Z = \frac{\left(\frac{8a}{27b^2}\right) \cdot (3b)}{R \cdot \left(\frac{8a}{27bR}\right)} \]

6. **Simplifying the Expression**:
Simplifying the above expression:
\[ Z = \frac{\frac{24ab}{27b^2}}{\frac{8a}{27b}} = \frac{24ab \cdot 27b}{27b^2 \cdot 8a} \]
Cancelling out common terms:
\[ Z = \frac{24}{8} = 3 \]

7. **Final Result**:
Therefore, the compressibility factor under critical state of a gas is:
\[ Z = \frac{3}{8} \]