Interatomic Potential vs Distance Graph for Monoatomic Real Gas

To determine the correct curve illustrating the relationship between the interatomic potential $V(r)$ and the interatomic distance $r$, we analyze the given equation of state for one mole of the real gas:

$p(V - b) = RT$

Step 1: Comparison with the van der Waals Equation

The general van der Waals equation of state for one mole of a real gas is given by:

$\left(p + \frac{a}{V^2}\right)(V - b) = RT$

where:

• $a$ accounts for the intermolecular attractive forces.
• $b$ accounts for the finite size of the molecules, giving rise to short-range repulsive forces (excluded volume).

Comparing $p(V - b) = RT$ with the van der Waals equation, we get: $a = 0 \quad \text{and} \quad b > 0$

Step 2: Physical Interpretation of $a = 0$ and $b > 0$

1. No Attractive Forces ($a = 0$): Since $a = 0$, there are no attractive forces acting between the atoms at any distance $r$. Consequently, the interatomic potential energy is zero for all non-overlapping distances ($r \ge \sigma$, where $\sigma$ is the atomic diameter/collision diameter): $V(r) = 0 \quad \text{for } r \ge \sigma$

2. Hard-Sphere Repulsion ($b > 0$): The presence of $b$ indicates that atoms have a finite volume and behave as hard spheres. When the interatomic distance $r$ becomes less than the atomic diameter $\sigma$ ($r < \sigma$), an infinitely large repulsive force acts between them to prevent penetration. Thus, the potential energy steeply rises to infinity: $V(r) = \infty \quad \text{for } r < \sigma$

Step 3: Hard-Sphere Potential Model

Combining these two conditions yields the hard-sphere potential function:

$$V(r) = \begin{cases} \infty, & \text{for } r < \sigma \\ 0, & \text{for } r \ge \sigma \end{cases}$$

Step 4: Conclusion and Graph Selection

• Graph (A) represents a Lennard-Jones potential with both attraction ($a > 0$) and repulsion ($b > 0$).
• Graph (B) represents an ideal gas with no interatomic forces and zero size ($a = 0, b = 0$).
• Graph (C) represents the hard-sphere potential model where $V(r) = 0$ for $r \ge \sigma$ and $V(r) \to \infty$ for $r < \sigma$.
• Graph (D) represents a square-well potential with an attractive well.

Therefore, Graph (C) accurately represents the given system.

Correct Option: C