At low pressure Vander Waal's equation for 3 moles of a real gas will have its simplified from

Options

A. `(PV)/(RT-(3a//V))=3`
B. `(PV)/(RT + Rb) =3`
C. `(PV)/(RT-3Pb)=1`
D. `(PV)/(RT - (9//V)) =3`

Answer

To find the simplified form of the Van der Waals equation for 3 moles of a real gas at low pressure, we can follow these steps:

### Step 1: Write the Van der Waals equation
The Van der Waals equation is given by:

\[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]

Where:
- \( P \) = pressure of the gas
- \( V \) = volume of the gas
- \( n \) = number of moles of the gas
- \( R \) = universal gas constant
- \( T \) = temperature
- \( a \) and \( b \) = Van der Waals constants specific to the gas

### Step 2: Consider low pressure conditions
At low pressure, the volume \( V \) of the gas is very large compared to the volume occupied by the gas molecules (which is represented by \( nb \)). Therefore, we can neglect the term \( nb \):

\[ V - nb \approx V \]

### Step 3: Substitute into the Van der Waals equation
Substituting this approximation back into the Van der Waals equation gives:

\[ \left( P + \frac{a n^2}{V^2} \right)V = nRT \]

### Step 4: Rearranging the equation
Expanding and rearranging the equation, we get:

\[ PV + \frac{a n^2}{V} = nRT \]

### Step 5: Isolate the pressure term
Rearranging further to isolate \( PV \):

\[ PV = nRT - \frac{a n^2}{V} \]

### Step 6: Divide through by \( RT \)
Now, we divide the entire equation by \( RT \):

\[ \frac{PV}{RT} = n - \frac{a n^2}{RT V} \]

### Step 7: Substitute \( n = 3 \)
For 3 moles of gas, substitute \( n = 3 \):

\[ \frac{PV}{RT} = 3 - \frac{3a}{RT V} \]

### Final Form
This simplifies to:

\[ \frac{PV}{RT} + \frac{3a}{RT V} = 3 \]

This is the simplified form of the Van der Waals equation for 3 moles of a real gas at low pressure.

### Conclusion
Thus, the simplified form of the Van der Waals equation for 3 moles of a real gas at low pressure is:

\[ \frac{PV}{RT} - \frac{3a}{V} = 3 \]