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 under conditions of relatively high pressure.

### Step-by-Step Solution:

1. **Understanding the Van der Waals Equation**:
The Van der Waals equation for one mole of gas is given by:
\[ \left(P + \frac{a}{V^2}\right)(V - b) = RT \]
where \( P \) is the pressure, \( V \) is the volume, \( T \) is the temperature, \( R \) is the universal gas constant, \( a \) is a measure of the attraction between particles, and \( b \) is the volume occupied by the gas particles.

2. **High Pressure Condition**:
At relatively high pressures, the volume \( V \) of the gas becomes significantly smaller compared to the volume of the gas particles themselves. This means that \( V \) approaches \( b \) and the term \( (V - b) \) can be approximated as \( V \).

3. **Neglecting the Volume Correction**:
Under high pressure, the volume correction \( b \) becomes less significant, so we can simplify the equation to:
\[ P + \frac{a}{V^2} \approx \frac{RT}{V} \]

4. **Rearranging the Equation**:
Rearranging the simplified equation gives:
\[ P \approx \frac{RT}{V} - \frac{a}{V^2} \]

5. **Ideal Gas Behavior**:
As pressure increases, the term \( \frac{a}{V^2} \) becomes negligible compared to \( \frac{RT}{V} \). Therefore, we can further simplify the equation to resemble the ideal gas law:
\[ PV \approx RT \]
This shows that at high pressure, the behavior of the gas approaches that of an ideal gas.

### Final Result:
Thus, at relatively high pressure, Van der Waals equation for one mole of gas reduces to the ideal gas equation:
\[ PV = RT \]