At low pressure, if `RT=2sqrt(a.p),` then the volume occupied by a real gas is :

Options

A. `(2RT)/(P)`
B. `(2P)/(RT)`
C. `(RT)/(2P)`
D. `(2RT)/(P)`

Explanation

To solve the problem, we need to find the volume occupied by a real gas at low pressure given the equation \( RT = 2\sqrt{a \cdot p} \).

### Step-by-Step Solution:

1. **Understanding the Given Equation**:
We start with the equation provided:
\[ RT = 2\sqrt{a \cdot p} \]
This equation relates the gas constant \( R \), temperature \( T \), the pressure \( p \), and a constant \( a \) which is specific to the gas.

2. **Using the Van der Waals Equation**:
For a real gas, we can use the Van der Waals equation:
\[ \left( p + \frac{a}{V^2} \right) (V - b) = RT \]
At low pressure, we can assume that the volume \( V \) is very large compared to \( b \), allowing us to neglect the term \( b \). Thus, we can simplify the equation to:
\[ p + \frac{a}{V^2} = \frac{RT}{V} \]

3. **Rearranging the Equation**:
Rearranging gives:
\[ pV + \frac{a}{V} = RT \]
This can be rewritten as:
\[ pV^2 + a = RTV \]

4. **Substituting for RT**:
Now, substitute \( RT \) from the first step:
\[ pV^2 + a = 2\sqrt{a \cdot p} \cdot V \]
Rearranging this gives us a quadratic equation in terms of \( V \):
\[ pV^2 - 2\sqrt{a \cdot p} \cdot V + a = 0 \]

5. **Using the Quadratic Formula**:
We can solve for \( V \) using the quadratic formula:
\[ V = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = p \), \( b = -2\sqrt{a \cdot p} \), and \( c = a \). Thus:
\[ V = \frac{2\sqrt{a \cdot p} \pm \sqrt{(2\sqrt{a \cdot p})^2 - 4 \cdot p \cdot a}}{2p} \]

6. **Simplifying the Expression**:
Simplifying the term under the square root:
\[ (2\sqrt{a \cdot p})^2 - 4pa = 4ap - 4ap = 0 \]
This means the discriminant is zero, indicating there is one solution:
\[ V = \frac{2\sqrt{a \cdot p}}{2p} = \frac{\sqrt{a \cdot p}}{p} \]

7. **Final Volume Expression**:
Therefore, the volume occupied by the real gas is:
\[ V = \frac{RT}{2p} \]

### Final Answer:
The volume occupied by the real gas at low pressure is:
\[ V = \frac{RT}{2p} \]