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

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

### Step-by-Step Solution:

1. **Write the Van der Waals Equation**:
The Van der Waals equation is given by:
\[ \left(P + \frac{a n^2}{V^2}\right)(V - nb) = nRT \]
For this problem, we will focus on the pressure correction term since we are at low pressure.

2. **Simplify for Low Pressure**:
At low pressure, we can ignore the volume correction term (the \(nb\) term). Therefore, the equation simplifies to:
\[ P + \frac{a n^2}{V^2} = \frac{nRT}{V} \]

3. **Substituting the Number of Moles**:
We are given that \(n = 3\). Substituting this into the equation gives:
\[ P + \frac{a (3)^2}{V^2} = \frac{3RT}{V} \]
This simplifies to:
\[ P + \frac{9a}{V^2} = \frac{3RT}{V} \]

4. **Rearranging the Equation**:
Now, we can rearrange the equation to isolate \(P\):
\[ P = \frac{3RT}{V} - \frac{9a}{V^2} \]

5. **Final Form**:
The final simplified form of the Van der Waals equation for 3 moles of a real gas at low pressure is:
\[ P = \frac{3RT}{V} - \frac{9a}{V^2} \]