At relatively high pressure, van der Waal's equation for one mole of gas reduces to

Options

A. PV = RT
B. PV = RT + a/V
C. PV = RT + Pb
D. `PV = RT - (a/ V^2)`

Answer

To solve the problem, we need to analyze the Van der Waals equation for one mole of gas and see how it simplifies at relatively high pressure.

### Step-by-Step Solution:

1. **Write the Van der Waals Equation**:
The Van der Waals equation for n moles of a gas is given by:
\[ \left( P + \frac{a n^2}{V^2} \right) (V - nb) = nRT \]
For one mole of gas (n = 1), this simplifies to:
\[ \left( P + \frac{a}{V^2} \right) (V - b) = RT \]

2. **Expand the Equation**:
Expanding the equation gives:
\[ PV - Pb + \frac{a}{V^2}V - \frac{ab}{V^2} = RT \]
This simplifies to:
\[ PV - Pb + \frac{a}{V} - \frac{ab}{V^2} = RT \]

3. **Consider High Pressure**:
At relatively high pressure, the volume (V) becomes small. Therefore, the term involving 'b' (the volume correction) becomes significant, while the term involving 'a' (the pressure correction) becomes negligible. Thus, we can ignore the \(\frac{a}{V^2}\) term.

4. **Neglect Pressure Correction**:
The equation simplifies to:
\[ PV - Pb = RT \]
Rearranging gives:
\[ PV = RT + Pb \]

5. **Final Form**:
Therefore, at relatively high pressure, the Van der Waals equation for one mole of gas reduces to:
\[ PV = RT + Pb \]

### Conclusion:
The correct answer is:
\[ PV = RT + Pb \]