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 correct expression for the van der Waals gas constant \( a \), we can follow these steps:

### Step 1: Understand the van der Waals constants
The van der Waals equation for real gases is given by:
\[ (P + \frac{a}{V_m^2})(V_m - b) = RT \]
where:
- \( P \) is the pressure,
- \( V_m \) is the molar volume,
- \( T \) is the temperature,
- \( R \) is the universal gas constant,
- \( a \) is the measure of the attractive forces between particles,
- \( b \) is the volume occupied by one mole of particles.

### Step 2: Identify the relationships for critical constants
We have the following relationships for critical constants:
1. Critical temperature \( T_c = \frac{8a}{27Rb} \)
2. Critical volume \( V_c = 3b \)
3. Critical pressure \( P_c = \frac{a}{27b^2} \)

### Step 3: Rearranging the critical temperature equation
From the critical temperature equation, we can express \( a \) in terms of \( T_c \), \( R \), and \( b \):
\[ a = \frac{27RbT_c}{8} \]

### Step 4: Rearranging the critical pressure equation
From the critical pressure equation, we can also express \( a \) in terms of \( P_c \) and \( b \):
\[ a = 27b^2P_c \]

### Step 5: Compare the options provided
Now, we need to check the options given in the question. We found two expressions for \( a \):
1. \( a = \frac{27RbT_c}{8} \)
2. \( a = 27b^2P_c \)

If one of these expressions matches with the options provided, we can conclude which option is correct.

### Conclusion
Based on the analysis, the expression \( a = 27b^2P_c \) is one of the derived expressions and is likely to be the correct answer if it matches one of the options provided.

### Final Answer
Thus, the correct expression for the van der Waals constant \( a \) is:
\[ a = 27b^2P_c \]

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