At critical temp. the value of z is :-

Options

A. `(8)/(3)`
B. `(3)/(8)`
C. `(1)/(8)`
D. `(1)/(27)`

Answer

To determine the value of the compressibility factor (Z) at the critical temperature (Tc), we can follow these steps:

### Step-by-Step Solution:

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

2. **Critical Constants**:
- At the critical point, we use the critical constants:
- \( P_c \) = critical pressure
- \( V_c \) = critical volume
- \( T_c \) = critical temperature
- The expression for Z at critical conditions becomes:
\[ Z_c = \frac{P_c V_c}{nRT_c} \]
- For simplicity, we can assume \( n = 1 \) mole.

3. **Substituting Critical Constants**:
- We know:
- \( P_c = \frac{a}{27b^2} \)
- \( V_c = 3b \)
- \( T_c = \frac{8a}{27Rb} \)
- Substitute these values into the Z equation:
\[ Z_c = \frac{P_c V_c}{RT_c} = \frac{\left(\frac{a}{27b^2}\right) (3b)}{R \left(\frac{8a}{27Rb}\right)} \]

4. **Simplifying the Expression**:
- Simplifying the numerator:
\[ \frac{a \cdot 3b}{27b^2} = \frac{3a}{27b} = \frac{a}{9b} \]
- Now substituting this back into the Z equation:
\[ Z_c = \frac{\frac{3a}{27b}}{\frac{8a}{27Rb}} = \frac{3a}{27b} \cdot \frac{27Rb}{8a} \]
- Canceling \( a \) and \( b \):
\[ Z_c = \frac{3R}{8} \]

5. **Final Value of Z at Critical Temperature**:
- Thus, the final value of the compressibility factor at the critical temperature is:
\[ Z_c = \frac{3}{8} \]

### Final Answer:
The value of Z at critical temperature is \( \frac{3}{8} \).