At low pressure the van der Waals' equation is reduced to `[P +(a)/(V^(2)]]V =RT` The compressibility factor can be given as .

Options

A. `1-(a)/(RTV)`
B. `1-(RTV)/(a)`
C. `1 + (a)/(RTV)`
D. `1 + (RTV)/(a)`

Explanation

To solve the problem of finding the compressibility factor \( Z \) from the given van der Waals' equation at low pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Van der Waals' Equation**:
The van der Waals' equation at low pressure can be expressed as:
\[ \left(P + \frac{a}{V^2}\right)V = RT \]
where \( P \) is the pressure, \( V \) is the molar volume, \( R \) is the universal gas constant, \( T \) is the temperature, and \( a \) is a constant specific to the gas.

2. **Express the Compressibility Factor**:
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{nRT} \]
Assuming \( n = 1 \) (which means we are considering one mole of gas), we can simplify this to:
\[ Z = \frac{PV}{RT} \]

3. **Rearranging the Van der Waals' Equation**:
We can rearrange the van der Waals' equation:
\[ PV + \frac{a}{V} = RT \]
This can also be written as:
\[ PV = RT - \frac{a}{V} \]

4. **Substituting into the Compressibility Factor**:
Now, substituting \( PV \) from the rearranged equation into the compressibility factor:
\[ Z = \frac{PV}{RT} = \frac{RT - \frac{a}{V}}{RT} \]

5. **Simplifying the Expression**:
We can simplify this expression:
\[ Z = \frac{RT}{RT} - \frac{a}{VRT} = 1 - \frac{a}{VRT} \]

6. **Final Expression for the Compressibility Factor**:
Therefore, the compressibility factor \( Z \) can be expressed as:
\[ Z = 1 - \frac{a}{VRT} \]

### Conclusion:
The required value of the compressibility factor \( Z \) for the given van der Waals' equation at low pressure is:
\[ Z = 1 - \frac{a}{VRT} \]