The van der Waal's gas constant, `a` is given by

Options

A. `3 V_c`
B. `27b^2 P_c`
C. `27 Rb T_c`
D. `3 Rb P_c`

Answer

To find the van der Waals gas constant \( a \), we can use the relationships between the critical constants \( T_c \), \( V_c \), and \( P_c \) for a van der Waals gas. The van der Waals equation is given by:

\[ (P + \frac{a}{V^2})(V - b) = RT \]

Where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( T \) is the temperature,
- \( R \) is the universal gas constant,
- \( a \) is a measure of the attraction between particles,
- \( b \) is the volume occupied by the particles.

### Step 1: Recall the critical constants relationships
The critical constants for a van der Waals gas are defined as:
- \( T_c = \frac{8a}{27bR} \)
- \( V_c = 3b \)
- \( P_c = \frac{a}{27b^2} \)

### Step 2: Rearranging the equations
From the equation for \( T_c \), we can express \( a \) in terms of \( T_c \), \( b \), and \( R \):
\[ a = \frac{27bRT_c}{8} \]

### Step 3: Express \( a \) in terms of \( P_c \)
From the equation for \( P_c \), we can express \( a \) in terms of \( P_c \) and \( b \):
\[ a = 27b^2P_c \]

### Step 4: Substitute \( b \) from \( V_c \)
Since \( V_c = 3b \), we can express \( b \) as:
\[ b = \frac{V_c}{3} \]

### Step 5: Substitute \( b \) into the equation for \( a \)
Substituting \( b \) into the equation for \( a \):
\[ a = 27\left(\frac{V_c}{3}\right)^2P_c \]
\[ a = 27\left(\frac{V_c^2}{9}\right)P_c \]
\[ a = 3V_c^2P_c \]

### Conclusion
Thus, the van der Waals gas constant \( a \) is given by:
\[ a = 3V_c^2P_c \]

### Final Answer
The correct expression for the van der Waals gas constant \( a \) is \( 3V_c^2P_c \).

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