the value of compression factor ,Z for critical constants is

Options

A. `1/2`
B. `3/4`
C. `2/3`
D. `3/8`

Answer

To find the value of the compression factor \( Z \) for the critical constants, we will follow these steps:

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

### Step 2: Use the Critical Constants
At the critical point, the compression factor can be expressed using the critical pressure \( P_c \), critical volume \( V_c \), and critical temperature \( T_c \):
\[ Z_c = \frac{P_c V_c}{R T_c} \]

### Step 3: Substitute the Critical Constants
The critical constants can be expressed in terms of constants \( a \) and \( b \) (from van der Waals equation):
- Critical pressure \( P_c = \frac{a}{27b^2} \)
- Critical volume \( V_c = 3b \)
- Critical temperature \( T_c = \frac{8a}{27bR} \)

### Step 4: Substitute \( P_c \), \( V_c \), and \( T_c \) into the Equation for \( Z_c \)
Substituting these values into the equation for \( Z_c \):
\[ Z_c = \frac{P_c V_c}{R T_c} = \frac{\left(\frac{a}{27b^2}\right) (3b)}{R \left(\frac{8a}{27bR}\right)} \]

### Step 5: Simplify the Expression
Now simplify the expression:
\[ Z_c = \frac{\frac{3ab}{27b^2}}{\frac{8a}{27R}} = \frac{3ab \cdot 27R}{27b^2 \cdot 8a} \]
Cancel \( a \) and \( b \):
\[ Z_c = \frac{3R}{8b} \]

### Step 6: Final Result
Since \( b \) is a constant and does not affect the value of \( Z_c \) at the critical point, we can conclude:
\[ Z_c = \frac{3}{8} \]

Thus, the value of the compression factor \( Z \) for the critical constants is:
\[ \boxed{\frac{3}{8}} \]