Problem 94 - Chemistry: The Central Science

Question

Briefly explain the significance of the constants \(a\) and \(b\) in the van der Waals equation.

Step-by-Step Solution

Answer

In the van der Waals equation, \(\left(P + \frac{an^2}{V^2}\right)(V-nb) = nRT\), the constant \(a\) accounts for the intermolecular forces between gas molecules. The larger the value of \(a\), the stronger these forces are, leading to deviations from the ideal behavior. The constant \(b\) represents the volume occupied by gas molecules, with the term (V-nb) accounting for the volume unavailable for free movement of molecules. Both constants \(a\) and \(b\) are responsible for making the van der Waals equation a more accurate description of real gases than the Ideal Gas Law.

Step 1: Write the Ideal Gas Law and the van der Waals equation

To better understand the significance of the constants a and b, first write down the Ideal Gas Law and the van der Waals equation: Ideal Gas Law: \(PV = nRT\) van der Waals equation: \(\left(P + \frac{an^2}{V^2}\right)(V-nb) = nRT\) where: P = pressure V = volume n = number of moles of the gas R = ideal gas constant T = temperature a = constant taking into account intermolecular forces b = constant taking into account the volume occupied by gas molecules

Step 2: Explain the significance of constant a

Constant a is a constant specific to each gas, accounting for the intermolecular forces (namely, the attractive forces) between gas molecules. The term \(\frac{an^2}{V^2}\) in the equation is added to the pressure, which is a correction factor for the attractive forces between molecules in a real gas. The larger the value of the constant a, the stronger the intermolecular forces are. The intermolecular forces make the gas deviate from the ideal behavior predicted by the Ideal Gas Law.

Step 3: Explain the significance of constant b

Constant b is also a constant specific to each gas, representing the volume occupied by gas molecules. In the term (V-nb) in the equation, nb represents the volume of the gas that is not available for other molecules to move freely due to the actual volume of the molecules. The larger the value of the constant b, the larger the volume of each gas particle is. This constant takes into account the fact that real gas molecules have a finite volume, while in the Ideal Gas Law, they are considered as point particles taking no volume. By considering the intermolecular forces (constant a) and the volume occupied by gas molecules (constant b), the van der Waals equation provides a more accurate description of real gases behavior than the Ideal Gas Law.