What is the value of `root(3)(a/(P_(c )b^(2)))` ?

Answer

To find the value of \( \sqrt[3]{\frac{A}{P_c b^2}} \), we can follow these steps:

### Step 1: Understand the critical pressure
The critical pressure \( P_c \) is the pressure required to liquefy a gas at its critical temperature.

### Step 2: Use the expression for critical pressure
The expression for critical pressure in terms of the Van der Waals constants \( A \) and \( B \) is given by:
\[ P_c = \frac{A}{27B^2} \]

### Step 3: Rearrange the expression
We need to rearrange the equation to express \( \frac{A}{P_c b^2} \):
\[ \frac{A}{P_c b^2} = \frac{A}{\left(\frac{A}{27B^2}\right) b^2} \]
This simplifies to:
\[ \frac{A}{P_c b^2} = \frac{A \cdot 27B^2}{A \cdot b^2} = \frac{27B^2}{b^2} \]

### Step 4: Simplify further
Since \( \frac{B^2}{b^2} = 1 \) (assuming \( B \) and \( b \) are the same constants), we have:
\[ \frac{A}{P_c b^2} = 27 \]

### Step 5: Take the cube root
Now, we can find the cube root:
\[ \sqrt[3]{\frac{A}{P_c b^2}} = \sqrt[3]{27} = 3 \]

### Final Answer
Thus, the value of \( \sqrt[3]{\frac{A}{P_c b^2}} \) is \( 3 \).

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