For a gas obeying the van der Waals' equation, at the critical point

Options

A. both `((delP)/(delV))_(T)`and `((del^(2)P)/(delV^(2)))_(T)`are zero
B. only `((del^(2)P)/(delV^(2)))_(T)` is not zero
C. `((delP)/(delV))_(T)` is zero but `((del^(2)P)/(delV^(2)))_(T)` is non-zero
D. `((delP)/(delV))_(T)` is non-zero but `((del^(2)P)/(delV^(2)))_(T)` is zero

Explanation

To solve the question regarding the behavior of a gas obeying the van der Waals equation at the critical point, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Van der Waals Equation**:
The van der Waals equation for real gases is given by:
\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]
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, and \( a \) and \( b \) are constants specific to the gas.

2. **Identify the Critical Point**:
At the critical point, the gas exhibits unique properties where the distinction between liquid and gas phases disappears. The critical point is characterized by specific values of pressure, volume, and temperature.

3. **Derivatives at the Critical Point**:
At the critical point, the first and second derivatives of pressure with respect to volume at constant temperature are both zero:
\[ \left(\frac{\partial P}{\partial V}\right)_T = 0 \]
\[ \left(\frac{\partial^2 P}{\partial V^2}\right)_T = 0 \]

4. **Analyze the Options**:
Given the options, we need to determine which statement correctly reflects the conditions at the critical point for a gas obeying the van der Waals equation. Since both derivatives are zero, any statement suggesting that these derivatives are non-zero is incorrect.

5. **Conclusion**:
The correct statement regarding a gas obeying the van der Waals equation at the critical point is that both the first and second derivatives of pressure with respect to volume at constant temperature are equal to zero.

### Final Answer:
The correct option is that both the first derivative and the second derivative of pressure with respect to volume at constant temperature are equal to zero at the critical point.