The value of critical temperature in terms of van der Waals' constants a and b is given by

Options

A. `T_(C) = (8a)/(27 Rb)`
B. `T_(C) = (27a)/(8 Rb)`
C. `T_(C) = (a)/(2 Rb)`
D. `T_(C) = (a)/(27 Rb)`

Answer

To find the critical temperature in terms of van der Waals' constants \( a \) and \( b \), we can follow these steps:

### Step 1: Understand Critical Temperature
Critical temperature (\( T_c \)) is defined as the temperature above which a gas cannot be liquefied by pressure alone. It represents a point where the properties of the gas and liquid phases become indistinguishable.

### Step 2: Recall the van der Waals Equation
The van der Waals equation for real gases is given by:
\[ \left( P + \frac{a}{V_m^2} \right) (V_m - b) = RT \]
where:
- \( P \) is the pressure,
- \( V_m \) is the molar volume,
- \( R \) is the universal gas constant,
- \( T \) is the temperature,
- \( a \) and \( b \) are van der Waals constants.

### Step 3: Identify Critical Constants
From the van der Waals equation, we can derive expressions for the critical constants. The critical volume (\( V_c \)), critical pressure (\( P_c \)), and critical temperature (\( T_c \)) can be expressed in terms of \( a \) and \( b \):
- \( V_c = 3b \)
- \( P_c = \frac{a}{27b^2} \)
- \( T_c = \frac{8a}{27Rb} \)

### Step 4: Derive Critical Temperature
Using the relationships derived, we can express the critical temperature as:
\[ T_c = \frac{8a}{27Rb} \]

### Conclusion
Thus, the value of the critical temperature in terms of van der Waals' constants \( a \) and \( b \) is:
\[ T_c = \frac{8a}{27Rb} \]