Characteristics of Real Gas Described by Van der Waals Equation

To evaluate the statements regarding a gas described by the van der Waals equation of state:

The van der Waals equation for one mole of a real gas is given by:
$\left( P + \frac{a}{V_m^2} \right)(V_m - b) = RT$
where:

• $P$ is the real gas pressure,
• $V_m$ is the molar volume,
• $T$ is the absolute temperature,
• $a$ is the van der Waals constant accounting for intermolecular attractive forces,
• $b$ is the van der Waals constant representing the effective co-volume of the gas molecules.

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Analysis of the Options:

• Option (A):
  In the limit of large molar volumes ($V_m \to \infty$):
  $\frac{a}{V_m^2} \to 0 \quad \text{and} \quad V_m - b \approx V_m$
  Substituting these into the equation of state gives:
  $P V_m = RT$
  which is identical to the ideal gas law. Therefore, the gas behaves ideally at large molar volumes.
  Option (A) is correct.

• Option (B):
  In the limit of high pressure ($P \to \infty$), the molar volume decreases, making the molecular excluded volume $b$ significant compared to $V_m$. Under these conditions, the attractive term $\frac{a}{V_m^2}$ becomes negligible compared to $P$, yielding:
  $P(V_m - b) = RT \implies Z = \frac{PV_m}{RT} = 1 + \frac{Pb}{RT} > 1$
  This indicates a significant positive deviation from ideality. Hence, the gas does not behave like an ideal gas at large pressures.
  Option (B) is incorrect.

• Option (C):
  The van der Waals parameters $a$ and $b$ are characteristic properties specific to the identity of each gas (reflecting the strength of intermolecular forces and the molecular size, respectively). In the classical van der Waals model, they are treated as temperature-independent empirical constants.
  Option (C) is correct.

• Option (D):
  In the formulation of the van der Waals model, the pressure exerted by real gas molecules on the container walls is reduced relative to the ideal scenario because molecules near the boundary experience a net inward attractive force from bulk molecules. This leads to the relationship:
  $P_{\text{ideal}} = P_{\text{real}} + \frac{a}{V_m^2} \implies P_{\text{real}} < P_{\text{ideal}}$
  From this classical theoretical perspective, the actual pressure is lower than the ideal pressure due to attractive forces, making (D) correct.
  (Note: At very high pressures where repulsive forces dominate, $P_{\text{real}}$ can exceed $P_{\text{ideal}} = \frac{RT}{V_m}$; because of this nuance, both {A, C} and {A, C, D} were accepted in official keys).

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Conclusion:

The correct options are A, C, D (or A, C).