The van der Waals' equation for one mole may be expressed as
`V_(M)^(3)-(b+(RT)/(P))V_(m)^(2)+(aV_(m))/(P)-(ab)/(P)=0`
where `V_(m)` is the molar volume of the gas. Which of the followning is incorrect?

Options

A. For a temperature less than `T_(C),V` has three real roots
B. For a temperature less than `T_(C),V` has three imaginary roots
C. For a temperature equal to `T_(C)` all three roots of V are real and identical
D. On increasing the temp. `(TltT_(C))`, the three roots become closer to one another

Explanation

To solve the question regarding the van der Waals equation for one mole, we need to analyze the equation and the properties of its roots under different conditions. The van der Waals equation is given as:

\[ V_m^3 - \left(b + \frac{RT}{P}\right)V_m^2 + \frac{aV_m}{P} - \frac{ab}{P} = 0 \]

where:
- \( V_m \) is the molar volume of the gas,
- \( R \) is the universal gas constant,
- \( T \) is the temperature,
- \( P \) is the pressure,
- \( a \) and \( b \) are van der Waals constants specific to the gas.

### Step-by-Step Solution:

1. **Understand the Roots of the Equation**:
The cubic equation can have three roots (real or imaginary) depending on the values of \( T \), \( P \), \( a \), and \( b \). The nature of the roots changes with temperature.

2. **Critical Temperature Condition**:
At the critical temperature \( T_c \), the behavior of the roots changes. It is known that at \( T < T_c \), the equation can yield one real root and two imaginary roots. This is a crucial point in understanding the behavior of gases under different conditions.

3. **Analyze the Statements**:
The question asks to identify which statement regarding the roots of the equation is incorrect. We need to evaluate the statements related to the number of real and imaginary roots at different temperatures.

4. **Evaluate the Statements**:
- **Statement 1**: At \( T < T_c \), there is one real root and two imaginary roots. This is **true**.
- **Statement 2**: At \( T = T_c \), all roots are real and equal. This is also **true**.
- **Statement 3**: At \( T > T_c \), all roots are real. This is **true**.
- **Statement 4**: The statement claiming that at \( T < T_c \) all roots are real is **incorrect**.

5. **Conclusion**:
The incorrect statement is the one that suggests all roots are real at temperatures below the critical temperature \( T_c \).

### Final Answer:
The incorrect statement is that at \( T < T_c \), all roots of the van der Waals equation are real.