At low pressure and high temperature the Van der Waals equation is finally reduced (simplified) to

Options

A. `(p + (a)/(V_m^2))(V_m - b) = RT`
B. `p(V_m - b) = RT`
C. `(p + a/(V_m^2))V_m = RT`
D. `pV_(m) = RT`

Answer

To simplify the Van der Waals equation at low pressure and high temperature, we start with the original Van der Waals equation for real gases:

1. **Write the Van der Waals equation**:
\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]
where \(P\) is the pressure, \(V\) is the volume, \(n\) is the number of moles, \(R\) is the universal gas constant, \(T\) is the temperature, \(a\) is a measure of the attraction between particles, and \(b\) is the volume occupied by the gas particles.

2. **Consider low pressure and high temperature**:
At low pressure, the volume \(V\) is large compared to the volume occupied by the gas particles (\(nb\)), and the interactions between gas molecules become negligible. Therefore, we can assume:
- \(V \gg nb\) (the volume of the gas is much larger than the volume occupied by the gas particles)
- \(P + \frac{a n^2}{V^2} \approx P\) (the correction term becomes small)

3. **Neglect the volume correction**:
Since \(nb\) becomes negligible at low pressure, we can simplify the equation:
\[ P + \frac{a n^2}{V^2} \approx P \]
Thus, we can rewrite the equation as:
\[ P \cdot V \approx nRT \]

4. **Final simplification**:
Rearranging gives us:
\[ PV = nRT \]
This is the ideal gas equation, which indicates that at low pressure and high temperature, real gases behave similarly to ideal gases.

5. **Conclusion**:
Therefore, at low pressure and high temperature, the Van der Waals equation reduces to the ideal gas equation:
\[ PV = nRT \]