The van der Waals equation for one mol of `CO_(2)` gas at low pressure will be

Options

A. (a) `(P + (a)/(V^(2))) V = RT`
B. (b) `P(V-b) = RT - (a)/(V^(2))`
C. (c) `P = (RT)/(V-b)`
D. (d) `P = ((RT)/(V-b) - (a)/(V^(2)))`

Explanation

To derive the van der Waals equation for one mole of \( CO_2 \) gas at low pressure, we can follow these steps:

### Step 1: Write the van der Waals equation
The van der Waals equation for one mole of a gas is given by:

\[ \left( P + \frac{a}{V^2} \right) (V - b) = RT \]

where:
- \( P \) is the pressure,
- \( V \) is the volume,
- \( T \) is the temperature,
- \( R \) is the universal gas constant,
- \( a \) and \( b \) are van der Waals constants specific to the gas.

### Step 2: Analyze the condition of low pressure
At low pressure, the volume of the gas becomes very large. This implies that:

\[ V \gg b \]

This means that the volume \( V \) is much greater than the volume excluded by the finite size of the gas molecules (represented by \( b \)).

### Step 3: Simplify the equation
Since \( V \) is much larger than \( b \), we can ignore \( b \) in the equation. Therefore, the equation simplifies to:

\[ P + \frac{a}{V^2} = \frac{RT}{V} \]

### Step 4: Rearranging the equation
Rearranging the simplified equation gives us:

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

This is the van der Waals equation for one mole of \( CO_2 \) gas at low pressure.

### Final Form
Thus, the final form of the van der Waals equation at low pressure for one mole of \( CO_2 \) is:

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