Problem 21 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

(a) State (i) the ideal gas equation (ii) the van der Waals equation of state. Explain how the additional terms in the van der Waals equation account for the actual behaviour of real gases. (b) Without performing any numerical calculations, show that, in the limit of high temperatures and low pressures, the van der Waals and ideal gas equations are identical. (Section 8.6)

Step-by-Step Solution

Answer

(a) Ideal gas: \( PV = nRT \); van der Waals: \( \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \). Van der Waals accounts for particle volume and attractions. (b) At high T, low P, the equations converge due to negligible \( a \) and \( b \) terms.

Step 1: Ideal Gas Equation

The ideal gas equation is a simple equation of state for an ideal gas. It is given by:\[ PV = nRT \]where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: van der Waals Equation

The van der Waals equation is a modified version of the ideal gas equation that accounts for the volume of gas particles and inter-particle attractions. This equation is:\[ \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \]Here, \( a \) is a measure of the attraction between particles, and \( b \) accounts for the finite volume occupied by gas particles.

Step 3: Explanation of Additional Terms

In the van der Waals equation, the term \( a \) reflects attractive forces between particles, which lower the effective pressure compared to an ideal gas. The term \( b \) accounts for the finite size of particles, reducing the available volume for motion, contrasting with the assumption of zero particle size in ideal gases.

Step 4: High Temperature and Low Pressure Limit

Consider the van der Waals equation in the limit of high temperatures and low pressures. As temperature increases and pressure decreases, the terms \( \frac{a n^2}{V^2} \) and \( nb \) become negligible. This is because:- \( \frac{a n^2}{V^2} \rightarrow 0 \) as\( V \rightarrow \infty \) (low pressure scenario).- \( nb \rightarrow 0 \) as \( n \) becomes small or \( V \) becomes large.Thus, the equation simplifies to the ideal gas law: \( PV = nRT \).

Key Concepts

Ideal Gas Law

The ideal gas law is a cornerstone of classical thermodynamics and provides an equation of state that describes the behavior of an ideal gas. The equation is represented as \( PV = nRT \), where:

• \( P \) stands for pressure,
• \( V \) is volume,
• \( n \) represents the number of moles of gas,
• \( R \) is the universal gas constant, and
• \( T \) is the temperature in Kelvin.

This simple relationship assumes that gas particles are point-like with no volume and that they do not interact with one another aside from elastic collisions. When using the ideal gas law, keep in mind that its accuracy is highest under conditions of low pressure and high temperature. These conditions help minimize the influence of gas particle volume and intermolecular forces, making the gas behave ideally.

Real Gases

Real gases, unlike their ideal counterparts, do not always follow the ideal gas law due to the presence of intermolecular forces and finite particle volumes. In real gases, attractive forces exist between particles, which can affect pressure and volume measurements. To account for these discrepancies in behavior, scientists and engineers often use the van der Waals equation and other equations of state that are adjusted for non-ideal behavior. To better understand real gases, you should consider how internal pressure changes under various conditions. At higher pressures, particles are closer together, increasing the effect of intermolecular attraction. Meanwhile, at lower temperatures, these attractions become significant enough to affect the overall behavior of the gas. Understanding these characteristics helps chemists and physicists predict the behavior of gases in more realistic scenarios.

Gas Particles

Gas particles are the individual atoms or molecules that make up a gas. In the ideal gas law, these particles are treated as hard spheres with no volume, colliding elastically and not attracting or repelling each other. This model is simplistic; however, it provides a useful approximation for many practical applications.In real gases, every particle exhibits a finite volume—an element well considered in the van der Waals equation with the term \( nb \). Inter-particle attractions, also crucial in real gases, are captured by the \( \frac{a n^2}{V^2} \) term in the equation. These particles engage in low-intensity adhesive interactions that affect how gases compress and expand. Upon analysis, having an appreciation for the true physical nature of gas particles leads to better models and predictions for various environmental and laboratory scenarios.

Equation of State

An equation of state is a mathematical equation describing the state of matter under a given set of physical conditions. Both the ideal gas law and the van der Waals equation provide equations of state but for different circumstances.

The ideal gas law applies to theoretical gases without inter-particle forces or volume, while the van der Waals equation modifies this by introducing corrections for these two important factors. The equation is \( \left( P + \frac{a n^2}{V^2} \right)(V - nb) = nRT \), where the additional terms \( a \) and \( b \) adjust for deviations from ideal behavior. Typically, equations of state are essential for engineers and scientists working with gases at a wide range of conditions, guiding activities such as the prediction of gas behavior and the development of gas-based technologies and processes.