For critical constant factor, compression factor Z is

Options

A. 1
B. `gt 1`
C. `lt 1`
D. 0

Explanation

To determine the compressibility factor \( Z \) for critical constants, we will follow these steps:

### Step 1: Understand the Definition of Compressibility Factor \( Z \)
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{RT} \]
where \( P \) is the pressure, \( V \) is the volume, \( R \) is the gas constant, and \( T \) is the temperature.

### Step 2: Identify Critical Constants
The critical constants are:
- \( P_C \): Critical Pressure
- \( V_C \): Critical Volume
- \( T_C \): Critical Temperature

### Step 3: Substitute Critical Constants into the Compressibility Factor Equation
At the critical point, we can substitute the critical constants into the equation for \( Z \):
\[ Z = \frac{P_C V_C}{RT_C} \]

### Step 4: Use Relationships Between Critical Constants
From the Van der Waals equation, we know:
- \( T_C = \frac{8A}{27Rb} \)
- \( P_C = \frac{A}{27b^2} \)
- \( V_C = 3b \)

### Step 5: Substitute \( P_C \), \( V_C \), and \( T_C \) into the Equation for \( Z \)
Substituting these values into the equation for \( Z \):
\[ Z = \frac{\left(\frac{A}{27b^2}\right)(3b)}{R\left(\frac{8A}{27Rb}\right)} \]

### Step 6: Simplify the Expression
Now, simplify the expression:
\[ Z = \frac{\frac{3A}{27b}}{\frac{8A}{27R}} = \frac{3A \cdot 27R}{27b \cdot 8A} = \frac{3R}{8b} \]

### Step 7: Analyze the Value of \( Z \)
Since \( b \) is a positive constant, the value of \( Z \) will be:
\[ Z = \frac{3}{8} < 1 \]

### Conclusion
Thus, the compressibility factor \( Z \) at the critical point is less than 1.

### Final Answer
The compressibility factor \( Z \) for critical constant factors is less than 1.

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