The van der Waals' equation of law of corresponding states for 1 mole of gas is :

Options

A. `(P_(r)+(3)/(V_(r)^(2)))(3V_(r)-1)=8T_(r)`
B. `(P_(r)-(3)/(V_(r)^(2)))(3V_(r)-1)=8T_(r)`
C. `(P_(r)+(3)/(V_(r)^(2)))(3V_(r)+1)=8piT_(r)`
D. `(P_(r)+(3)/(V_(r)^(2)))(3V_(r)+1)=8`

Explanation

To derive the van der Waals' equation of law of corresponding states for 1 mole of gas, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the van der Waals' Equation**:
The van der Waals' equation for one mole of gas is given by:
\[ P + \frac{a}{V^2}(V - b) = RT \]
where \( P \) is the pressure, \( V \) is the volume, \( a \) and \( b \) are van der Waals constants, \( R \) is the universal gas constant, and \( T \) is the temperature.

2. **Introduce Reduced Parameters**:
The reduced parameters are defined as follows:
- Reduced pressure: \( P_R = \frac{P}{P_C} \)
- Reduced volume: \( V_R = \frac{V_m}{V_C} \)
- Reduced temperature: \( T_R = \frac{T}{T_C} \)
where \( P_C \) and \( T_C \) are the critical pressure and critical temperature, respectively.

3. **Express Critical Constants in Terms of van der Waals Constants**:
The critical constants can be expressed in terms of the van der Waals constants:
- \( P_C = \frac{a}{27b^2} \)
- \( T_C = \frac{8a}{27Rb} \)

4. **Substitute the Reduced Parameters into the van der Waals' Equation**:
Substitute \( P \), \( V \), and \( T \) in terms of reduced parameters:
\[ P = P_R \cdot P_C = P_R \cdot \frac{a}{27b^2} \]
\[ V = V_R \cdot V_C \]
\[ T = T_R \cdot T_C = T_R \cdot \frac{8a}{27Rb} \]

5. **Rearranging the van der Waals' Equation**:
Substitute these into the original equation:
\[ P_R \cdot \frac{a}{27b^2} + \frac{a}{(V_R \cdot V_C)^2}(V_R \cdot V_C - b) = R \cdot T_R \cdot \frac{8a}{27Rb} \]

6. **Simplifying the Equation**:
After substituting and simplifying, we can rearrange the equation to isolate terms:
\[ P_R + \frac{3}{V_R^2}(3V_R - 1) = 8T_R \]

7. **Final Form**:
Thus, the van der Waals' equation in terms of reduced parameters is:
\[ P_R + \frac{3}{V_R^2}(3V_R - 1) = 8T_R \]

### Conclusion:
The derived equation is the law of corresponding states for one mole of gas, which shows that the behavior of all gases can be described by this equation when expressed in terms of reduced parameters.