At low pressures, the van der Waals equation is written as `[P+(a)/(V^(2))]V=RT`
The compressibility factor is then equal to

Options

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

Explanation

To find the compressibility factor (Z) using the van der Waals equation at low pressures, we can follow these steps:

### Step-by-Step Solution

1. **Write the van der Waals equation**:
The van der Waals equation is given as:
\[ \left(P + \frac{a}{V^2}\right)V = RT \]

2. **Rearrange the equation**:
We can rearrange the equation to isolate \(P\):
\[ P + \frac{a}{V^2} = \frac{RT}{V} \]
\[ P = \frac{RT}{V} - \frac{a}{V^2} \]

3. **Express the compressibility factor (Z)**:
The compressibility factor \(Z\) is defined as:
\[ Z = \frac{PV}{RT} \]

4. **Substitute the expression for P into the Z equation**:
Substitute the expression for \(P\) from step 2 into the equation for \(Z\):
\[ Z = \frac{\left(\frac{RT}{V} - \frac{a}{V^2}\right)V}{RT} \]

5. **Simplify the expression**:
Simplifying the equation gives:
\[ Z = \frac{RT}{RT} - \frac{a}{RTV} \]
\[ Z = 1 - \frac{a}{RTV} \]

### Final Result
Thus, the compressibility factor \(Z\) at low pressures is:
\[ Z = 1 - \frac{a}{RTV} \]