# 16.3 Modification of the Ideal Gas Model to Discuss Behaviour of Non-Ideal Gases The simple gas laws work well under normal conditions, but they are based on an "ideal" gas model where molecules have no size and never interact with each other. In reality, this is not true. This section explores what happens when real gases are subjected to high pressure or very low temperatures. We will see why the simple laws break down and learn about the **Van der Waals equation**, a modification that accounts for the actual size of molecules and the forces between them. ## Behaviour of Ideal Gases **Ideal Gas Model Assumptions:** - The size of gas molecules is negligible. - There are no interactions between gas molecules. - Collisions between molecules and with container walls are perfectly elastic. **Ideal Gas Law:** Gases that obey the equation $PV = nRT$ are called ideal gases. **Variables:** - $P$: Pressure - $V$: Volume - $n$: Quantity of gas (moles) - $R$: Ideal gas constant - $T$: Absolute temperature **Macroscopic and Microscopic Connection:** The ideal gas law connects macroscopic properties ($P$, $V$, $T$, $n$, entropy) with microscopic quantities. For example, temperature $T = \frac{m}{3k}\langle v^2 \rangle$ is defined by the motion of individual particles. **Conditions for Ideal Behavior:** Gases approach the ideal state at low pressure and high temperatures. ## Statistical Mechanics **Definition:** A branch of physics that connects the microscopic details of a system (motion, energy, interaction of individual particles) with the macroscopic observables we measure (temperature, pressure, volume, entropy). **Role:** Provides a mathematical foundation for thermodynamics, which is otherwise a phenomenological theory. ## Behaviour of Non-Ideal Gases **Limitations of Ideal Gas Law:** The ideal gas law fails to explain gas behavior under specific conditions: - High pressure - High density - Low temperature (when molecules move slowly and interact) - Strong gravitational forces (e.g., in stars) ## Van der Waals Equation **Purpose:** A modification of the ideal gas law to account for deviations of real gases from ideal behavior, especially at wide ranges of temperature and pressure. **Van der Waals Assumptions for Real Gases:** 1. Gas molecules are hard spheres with definite volume. 2. Gas molecules interact with each other, especially at low temperatures. **Corrections Made to the Ideal Gas Law:** ### Volume Correction - **Problem:** Ideal gas assumes negligible molecule size. Real gas molecules have a finite size and occupy some volume. - **Effect:** The space available for molecule motion is less than the container volume $V$. - **Correction:** For one mole of gas, the effective volume is $(V - b)$, where $b$ is the volume occupied by one mole of gas molecules. For $n$ moles, it becomes $(V - nb)$. ### Pressure Correction - **Problem:** Ideal gas assumes no intermolecular interactions. Real gas molecules, especially at low temperatures, interact through attractive forces. - **Effect:** Molecules near the walls experience a net inward force from other molecules, reducing the impact force on the walls and thus the observed pressure. - **Correction:** The effective pressure is increased by an amount proportional to the square of the gas density (inversely proportional to $V^2$). For one mole, it is $\left(P + \frac{a}{V^2}\right)$. For $n$ moles, it becomes $\left(P + \frac{an^2}{V^2}\right)$. **Van der Waals Equation for 1 mole of a gas:** $$ \left(P + \frac{a}{V^2}\right)(V - b) = RT \quad (16.26) $$ **Van der Waals Equation for n moles of a gas:** $$ \left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT $$ - **Constants $a$ and $b$:** These are empirical constants specific to each gas, accounting for intermolecular attractive forces ($a$) and molecular volume ($b$). **Approaching Ideal Behavior:** At high temperatures or very low pressures, the volume $V$ becomes very large. In this situation, the terms $\frac{an^2}{V^2}$ (or $\frac{a}{V^2}$) and $nb$ (or $b$) become negligible, and the Van der Waals equation reduces to the ideal gas equation $PV = nRT$. | Law/Equation | Formula (for n moles) | | :----------- | :--------------------- | | Ideal Gas Law | $PV = nRT$ | | Van der Waals Equation | $\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT$ | The Van der Waals equation is a fundamental step in understanding the deviations of real gases from ideal behavior, providing a more accurate description of gas properties across a wider range of conditions.

16.3 Modification Of The Ideal Gas Model To Discuss Behaviour Of Non Ideal Gases | Statistical Mechanics And Thermodynamics | Physics 12