The virial equation for 1mole of a real gas is written as `:`
`PV=RT`
`[1+(A)/(V)+(B)/(V^(2))+(C)/(V^(3))+` ........ To higher power of `n]`
Where `A,B,` and `C` are known as virial coefficients . If Vander wall's equation is written in virial form, then what will be value of `B :`

Options

A. `a-(b)/(RT)`
B. `b^(3)`
C. `b-(a)/(RT)`
D. `b^(2)`

Answer

To find the value of the virial coefficient \( B \) when the Van der Waals equation is expressed in virial form, we can follow these steps:

### Step 1: Understand the Van der Waals Equation
The Van der Waals equation for one mole of a real gas is given by:
\[ \left( P + \frac{a}{V^2} \right)(V - b) = RT \]
where \( a \) and \( b \) are constants specific to each gas.

### Step 2: Rearranging the Van der Waals Equation
We can rearrange the Van der Waals equation to isolate \( P \):
\[ P = \frac{RT}{V - b} - \frac{a}{V^2} \]

### Step 3: Expressing in Terms of Virial Coefficients
To express this in the virial form \( PV = RT \left(1 + \frac{A}{V} + \frac{B}{V^2} + \frac{C}{V^3} + \ldots \right) \), we need to manipulate the equation further.

### Step 4: Simplifying the Expression
Using the Taylor expansion for \( \frac{1}{V - b} \):
\[ \frac{1}{V - b} = \frac{1}{V} \cdot \frac{1}{1 - \frac{b}{V}} \approx \frac{1}{V} \left( 1 + \frac{b}{V} + \frac{b^2}{V^2} + \ldots \right) \]
Substituting this back into the expression for \( P \):
\[ P = \frac{RT}{V} \left(1 + \frac{b}{V} + \frac{b^2}{V^2} + \ldots \right) - \frac{a}{V^2} \]

### Step 5: Combining Terms
Now, we can combine the terms:
\[ P = \frac{RT}{V} + \frac{RTb}{V^2} + \frac{RTb^2}{V^3} - \frac{a}{V^2} \]
\[ P = \frac{RT}{V} + \left( \frac{RTb - a}{V^2} \right) + \frac{RTb^2}{V^3} + \ldots \]

### Step 6: Identifying the Coefficients
From the above expression, we can identify the coefficients:
- The first virial coefficient \( A = 0 \) (since there is no term independent of \( V \)).
- The second virial coefficient \( B = RTb - a \).

### Conclusion
Thus, the value of \( B \) in the virial form of the Van der Waals equation is:
\[ B = RTb - a \]