Under critical states of a gas for one mole of a gas, compressibility factor is :

Options

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

Explanation

To find the compressibility factor (Z) under critical states for one mole of a gas, we can follow these steps:

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

### Step 2: Identify Critical Conditions
At critical conditions, we denote the critical pressure as \( P_c \), the critical volume as \( V_c \), and the critical temperature as \( T_c \). We need to find the compressibility factor at these critical conditions, denoted as \( Z_c \).

### Step 3: Use the Critical State Formulas
From the theory of gases, we have the following relationships for critical conditions:
- Critical pressure (\( P_c \)) can be expressed as:
\[ P_c = \frac{A}{27B^2} \]
- Critical volume (\( V_c \)) can be expressed as:
\[ V_c = 3B \]
- The relationship between critical pressure, critical temperature, and the constants A and B can be expressed as:
\[ P_c T_c = \frac{8A}{27R} \]

### Step 4: Substitute Values into the Compressibility Factor Equation
Now, we can substitute these values into the compressibility factor equation:
\[ Z_c = \frac{P_c V_c}{R T_c} \]

Substituting the expressions for \( P_c \) and \( V_c \):
\[ Z_c = \frac{\left(\frac{A}{27B^2}\right)(3B)}{R \left(\frac{8A}{27R}\right)} \]

### Step 5: Simplify the Expression
Now, simplify the expression:
1. Substitute \( P_c \) and \( V_c \):
\[ Z_c = \frac{\frac{3AB}{27B^2}}{\frac{8A}{27}} \]
2. Cancel out common terms:
\[ Z_c = \frac{3B}{27B^2} \cdot \frac{27}{8A} \]
3. This simplifies to:
\[ Z_c = \frac{3}{8} \]

### Conclusion
Thus, the compressibility factor for one mole of a gas at critical conditions is:
\[ Z_c = \frac{3}{8} \]

### Final Answer
The compressibility factor under critical states of a gas for one mole of gas is \( \frac{3}{8} \).

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