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 solve the problem regarding 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:
\[ P + \frac{a n^2}{V^2} (V - nB) = nRT \]
where \( P \) is the pressure, \( n \) is the number of moles, \( V \) is the volume, \( R \) is the universal gas constant, \( T \) is the temperature, and \( a \) and \( b \) are Van der Waals constants.

2. **Consider Low Pressure**:
At low pressure, the volume correction term \( nB \) becomes negligible, so we can simplify the equation to:
\[ P + \frac{a n^2}{V^2} V = nRT \]

3. **Simplify the Equation**:
This simplifies to:
\[ P + \frac{a n^2}{V} = nRT \]

4. **Substitute the Number of Moles**:
Since we are given that \( n = 3 \) moles, we substitute \( n \) into the equation:
\[ P + \frac{a \cdot 3^2}{V} = 3RT \]
which simplifies to:
\[ P + \frac{9a}{V} = 3RT \]

5. **Rearranging the Equation**:
Rearranging the equation to isolate \( P \):
\[ P = 3RT - \frac{9a}{V} \]

6. **Express in the Required Form**:
To express this in the form \( \frac{PV}{RT} - \frac{3A}{V} = 3 \), we multiply the entire equation by \( V \):
\[ PV = 3RTV - 9a \]
Dividing through by \( RT \):
\[ \frac{PV}{RT} = 3V - \frac{9a}{RT} \]
Rearranging gives:
\[ \frac{PV}{RT} - \frac{9a}{RT} = 3 \]

7. **Final Form**:
Thus, the simplified form of the Van der Waals equation for 3 moles of a real gas at low pressure is:
\[ \frac{PV}{RT} - \frac{9a}{V} = 3 \]

### Final Answer:
The simplified form of the Van der Waals equation for 3 moles of a real gas at low pressure is:
\[ \frac{PV}{RT} - \frac{9a}{V} = 3 \]