Question

For van der Waals gases, at the critical point, \( \frac{dP}{dV_m} = 0 \) and

• A. \( \frac{d^2 P}{dV_m^2} = 0 \)
• B. \( \frac{d^2 P}{dV_m^2}<0 \)
• C. \( \frac{d^2 P}{dV_m^2}>0 \)
• D. \( \frac{d^2 P}{dV_m^2} \) diverges

Hint:

At the critical point for van der Waals gases, the second derivative of pressure with respect to molar volume is zero, indicating a point of inflection on the pressure-volume curve.

Answer 1

The Correct Option is A

Step 1: Van der Waals equation at the critical point.
For van der Waals gases, the equation of state is: \[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \] where \( P \) is pressure, \( V_m \) is the molar volume, \( a \) and \( b \) are constants, and \( T \) is the temperature.
At the critical point, the first and second derivatives of pressure with respect to volume become zero, indicating a point of inflection in the pressure-volume curve. This leads to \( \frac{d^2 P}{dV_m^2} = 0 \).
Step 2: Conclusion.
Thus, at the critical point, the second derivative of pressure with respect to molar volume equals zero.
Final Answer: \[ \boxed{\frac{d^2 P}{dV_m^2} = 0} \]