Question

The term that accounts for intermolecular force in van der Waals equation for non-ideal gas is: (a) RT (b) \(\mathrm{V}-\mathrm{b}\) (c) \(\left(\mathrm{P}+\alpha / \mathrm{V}^{2}\right)\) (d) \((\mathrm{RT})^{-1}\)

Short answer

The term is (c) \(\left(P + \alpha/V^2\right)\).

Step-by-step solution

Understanding van der Waals Equation

The van der Waals equation is an adjusted version of the ideal gas law that accounts for interactions between molecules and the finite size of molecules. It is given by the equation: \[ \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT \]. Here, \(P\) is the pressure, \(V_m\) is the molar volume, \(R\) is the gas constant, \(T\) is the temperature, \(a\) accounts for the attraction between molecules, and \(b\) adjusts for the volume occupied by gas molecules.

Identifying the Intermolecular Force Term

In the van der Waals equation, the term \( \left(P + \frac{a}{V_m^2}\right) \) incorporates the intermolecular forces, where \( \frac{a}{V_m^2} \) accounts for the attractive forces between molecules. This term modifies the pressure \(P\) in the ideal gas law to include these forces.

Matching the Given Options

From the options given, the choice that resembles the term \( \left(P + \frac{a}{V_m^2}\right) \) is \( \left(P + \frac{\alpha}{V^2}\right) \) (option c), which represents how pressure is adjusted for intermolecular forces between gas particles in the van der Waals equation.

Key concepts

Intermolecular Forces

Intermolecular forces are the forces of attraction or repulsion that act between neighboring particles, like atoms, molecules, or ions. These forces are vital in explaining why gases sometimes deviate from the behavior predicted by the ideal gas law. In an ideal gas, we assume that the particles do not attract or repel each other and that they behave independently. However, in the real world, gas particles do interact.

The van der Waals equation incorporates a term specifically for these intermolecular forces. It's represented by the adjustment in pressure, adding the term \(\frac{a}{V_m^2}\), where \(a\) measures the strength of intermolecular attractions. Different gases have different "a" values because their molecular interactions differ. This inclusion accounts for the attractive forces that would be absent in an otherwise perfectly ideal gas situation.

Non-Ideal Gas

A non-ideal gas is a gas that does not strictly follow the assumptions of the ideal gas law, particularly under conditions of high pressure or low temperature. In these situations, the assumptions of negligible particle volume and no intermolecular forces break down. This is where the van der Waals equation comes into play.

The van der Waals equation modifies the ideal gas law to better fit the behavior observed in real gases. By introducing factors that account for molecular size and intermolecular forces, it provides a more accurate description of a gas's behavior in such non-ideal conditions. This makes non-ideal gas behavior complex but necessary to understand when predicting the state of a gas in real-life scenarios. Two key corrections in the van der Waals equation address these deviations:

• The pressure correction term \( \left(P + \frac{a}{V_m^2}\right) \) accounts for the intermolecular forces.
• The volume correction term \( \left(V_m - b\right) \) accounts for the finite volume of gas molecules.

Ideal Gas Law Adjustment

The ideal gas law is a simple and elegant equation: \(PV = nRT\). It relates the pressure, volume, and temperature of an ideal gas, with \(n\) being the number of moles of gas and \(R\) being the gas constant. However, this law assumes that gas particles have negligible volume and no intermolecular attractions or repulsions, which isn't always true.

In the van der Waals equation, these oversimplifications are addressed by two critical adjustments: the pressure and volume of the gas. The term \( \left(P + \frac{a}{V_m^2}\right)\) adjusts the pressure \(P\) to incorporate intermolecular attractions, while \( \left(V_m - b\right)\) decreases the volume to account for the space occupied by the molecules themselves.

By implementing these modifications, the van der Waals equation refines the ideal gas law, making it more applicable to real-world conditions where gases do not display ideal behavior. This adjusted law allows scientists and engineers to predict the behavior of gases more accurately across a wider range of temperatures and pressures, hence bridging the gap between theory and practical observation.