At critical temperature pressure and volume. The compressibility factor (Z) is-

Options

A. `(8)/(3)`
B. `(5)/(3)`
C. `(3)/(5)`
D. `(3)/(8)`

Answer

To find the compressibility factor (Z) at critical temperature, pressure, and volume, we can follow these steps:

### Step-by-Step Solution:

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

2. **Using Critical Parameters**:
At critical conditions, we can express Z in terms of critical pressure (\(P_c\)), critical volume (\(V_c\)), and critical temperature (\(T_c\)):
\[ Z = \frac{P_c V_c}{R T_c} \]

3. **Substituting Critical Values**:
We know the following relationships for critical parameters:
- \(T_c = \frac{8a}{27bR}\)
- \(V_c = 3b\)
- \(P_c = \frac{a}{27b^2}\)

Substituting these into the equation for Z:
\[ Z = \frac{P_c V_c}{R T_c} = \frac{\left(\frac{a}{27b^2}\right) \left(3b\right)}{R \left(\frac{8a}{27bR}\right)} \]

4. **Simplifying the Expression**:
Now we simplify the expression step by step:
- Substitute \(P_c\) and \(V_c\):
\[ Z = \frac{\frac{a \cdot 3b}{27b^2}}{R \cdot \frac{8a}{27bR}} \]
- This simplifies to:
\[ Z = \frac{3a}{27b} \cdot \frac{27bR}{8a} \]
- Canceling out common terms:
\[ Z = \frac{3}{8} \]

5. **Conclusion**:
Thus, the compressibility factor (Z) at critical temperature, pressure, and volume is:
\[ Z = \frac{3}{8} \]

### Final Answer:
The compressibility factor (Z) at critical temperature, pressure, and volume is \( \frac{3}{8} \).

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