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

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 under the given condition \( RT = 2\sqrt{a \cdot P} \). We can start by using the Van der Waals equation for real gases, which is given by:

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

However, at low pressure, the excluded volume \( b \) becomes negligible, and we can simplify the equation to:

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

Rearranging gives:

\[ PV - a/V^2 = RT \]

Now, substituting the given condition \( RT = 2\sqrt{a \cdot P} \) into the equation:

\[ PV - \frac{a}{V^2} = 2\sqrt{a \cdot P} \]

Next, we can multiply through by \( V^2 \) to eliminate the fraction:

\[ PV^3 - a = 2\sqrt{a \cdot P} \cdot V^2 \]

Rearranging this gives:

\[ PV^3 - 2\sqrt{a \cdot P} \cdot V^2 - a = 0 \]

This is a cubic equation in terms of \( V \). To solve for \( V \), we can use the quadratic formula, but first, we need to express it in a standard form. Let's denote \( x = V \):

\[ P x^3 - 2\sqrt{a \cdot P} x^2 - a = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = P, b = -2\sqrt{a \cdot P}, c = -a \):

\[ x = \frac{2\sqrt{a \cdot P} \pm \sqrt{(2\sqrt{a \cdot P})^2 - 4P(-a)}}{2P} \]

Simplifying gives:

\[ x = \frac{2\sqrt{a \cdot P} \pm \sqrt{4aP + 4aP}}{2P} \]
\[ x = \frac{2\sqrt{a \cdot P} \pm \sqrt{8aP}}{2P} \]
\[ x = \frac{2\sqrt{a \cdot P} \pm 2\sqrt{2aP}}{2P} \]
\[ x = \frac{\sqrt{a \cdot P} \pm \sqrt{2aP}}{P} \]

Choosing the positive root (since volume cannot be negative), we have:

\[ V = \frac{\sqrt{a \cdot P} + \sqrt{2aP}}{P} \]

Now we can factor out \( \sqrt{aP} \):

\[ V = \frac{\sqrt{aP}(\sqrt{1} + \sqrt{2})}{P} \]

This simplifies to:

\[ V = \frac{(2\sqrt{aP})}{P} \]

Finally, substituting back to find the volume in terms of \( RT \):

Since \( RT = 2\sqrt{aP} \), we can write:

\[ V = \frac{2RT}{P} \]

Thus, the volume occupied by the real gas is:

\[ \boxed{\frac{2RT}{P}} \]