At high pressure, the compressibility factor for one mole of van der waals gas will be

Options

A. 1 + `frac{b}{RT}`
B. 1 - `frac{Pb}{RT}`
C. 1 + `frac{Pb}{RT}`
D. 1 + `frac{a}{VRT}`

Answer

To solve the question regarding the compressibility factor for one mole of a van der Waals gas at high pressure, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Van der Waals Equation**:
The Van der Waals equation for n moles of gas is given by:
\[ P + \frac{a n^2}{V^2} \left( V - nb \right) = nRT \]
where \( P \) is the pressure, \( V \) is the volume, \( T \) is the temperature, \( R \) is the universal gas constant, \( a \) and \( b \) are the Van der Waals constants.

2. **Simplify for One Mole**:
For one mole of gas (\( n = 1 \)), the equation becomes:
\[ P + \frac{a}{V^2} \left( V - b \right) = RT \]

3. **Consider High Pressure**:
At high pressure, the term \( P \) is very large compared to \( \frac{a}{V^2} \). Thus, we can neglect the \( \frac{a}{V^2} \) term:
\[ P \approx RT \left( \frac{V}{V - b} \right) \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ PV - Pb = RT \]
or
\[ PV = Pb + RT \]

5. **Divide by RT**:
Now, divide both sides of the equation by \( RT \):
\[ \frac{PV}{RT} = \frac{Pb}{RT} + 1 \]

6. **Define Compressibility Factor**:
The compressibility factor \( Z \) is defined as:
\[ Z = \frac{PV}{RT} \]
Therefore, substituting the previous result:
\[ Z = 1 + \frac{Pb}{RT} \]

7. **Final Result**:
Thus, the compressibility factor for one mole of a van der Waals gas at high pressure is:
\[ Z = 1 + \frac{Pb}{RT} \]

### Conclusion:
The correct answer is \( 1 + \frac{Pb}{RT} \), which corresponds to option three.