At very high pressure, the compressibility factor of one mole of a gas is given by :

Options

A. `1+(Pb)/(RT)`
B. `(Pb)/(RT)`
C. `1-(Pb)/(RT)`
D. `1-(b)/((VRT))`

Answer

To solve the problem regarding the compressibility factor of one mole of a gas at very high pressure, we will use the Van der Waals equation and derive the expression step by step.

### Step-by-Step Solution:

1. **Start with the Van der Waals Equation:**
The Van der Waals equation for one mole of a gas is given by:
\[ \left(P + \frac{a}{V^2}\right)(V - b) = RT \]
Here, \(P\) is the pressure, \(V\) is the volume, \(R\) is the gas constant, \(T\) is the temperature, and \(a\) and \(b\) are the Van der Waals constants.

2. **Substituting for One Mole of Gas:**
For one mole of gas, we can set \(n = 1\). Thus, the equation simplifies to:
\[ \left(P + \frac{a}{V^2}\right)(V - b) = RT \]

3. **Consider Very High Pressure:**
At very high pressure, the term \(\frac{a}{V^2}\) becomes negligible compared to \(P\). Therefore, we can simplify the equation to:
\[ P(V - b) \approx RT \]

4. **Rearranging the Equation:**
Rearranging the equation gives:
\[ PV - Pb = RT \]
This can be rewritten as:
\[ PV = RT + Pb \]

5. **Dividing by RT:**
To find the compressibility factor \(Z\), we divide the entire equation by \(RT\):
\[ \frac{PV}{RT} = 1 + \frac{Pb}{RT} \]

6. **Defining the Compressibility Factor:**
The compressibility factor \(Z\) is defined as:
\[ Z = \frac{PV}{RT} \]
Therefore, we can express \(Z\) as:
\[ Z = 1 + \frac{Pb}{RT} \]

### Final Expression:
Thus, at very high pressure, the compressibility factor \(Z\) for one mole of a gas is given by:
\[ Z = 1 + \frac{Pb}{RT} \]