# Lecture 15: Gases: Kinetic Molecular Theory and Real Gases

## General Chemistry I

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## Learning Objectives

By the end of this lecture, students will be able to:

1. State the postulates of the kinetic molecular theory (KMT) of gases
2. Relate KMT to the observed gas laws (Boyle's, Charles's, Avogadro's, Dalton's)
3. Calculate root-mean-square speed and relate molecular speed to temperature and molar mass
4. Describe the Maxwell-Boltzmann distribution of molecular speeds
5. Explain effusion and diffusion and apply Graham's law
6. Explain deviations from ideal gas behavior and apply the van der Waals equation

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## Lecture Content

### I. Kinetic Molecular Theory (KMT) -- Postulates

The kinetic molecular theory provides a microscopic model that explains the macroscopic behavior of gases through five postulates. First, gases consist of a large number of particles (atoms or molecules) in constant, random, straight-line motion. Second, the volume of the individual gas particles is negligible compared to the total volume of the container, so the particles are effectively treated as point masses. Third, gas particles exert no attractive or repulsive forces on each other -- intermolecular forces are assumed to be zero. Fourth, collisions between gas particles and with the walls of the container are perfectly elastic, meaning total kinetic energy is conserved with no energy lost to friction or deformation. Fifth, the average kinetic energy of gas particles is directly proportional to the absolute temperature in Kelvin: KE_avg = (3/2) kT (where k is the Boltzmann constant), or equivalently KE_avg = (3/2) RT/N_A per molecule.

### II. KMT Explains the Gas Laws

The kinetic molecular theory elegantly accounts for each of the empirical gas laws. Boyle's law (P inversely proportional to V at constant T and n) is explained by the fact that decreasing the volume forces particles to hit the walls more frequently, increasing the pressure. Charles's law (V directly proportional to T at constant P and n) follows because higher temperature means faster-moving molecules that strike the walls harder and more often, requiring the container to expand to maintain constant pressure. Avogadro's law (V directly proportional to n at constant T and P) makes sense because more molecules produce more wall collisions, so the container must expand to keep the pressure constant. Dalton's law (P_total = sum of partial pressures) holds because gas molecules do not interact with each other, so each gas contributes independently to the total pressure.

### III. Molecular Speeds

Not all molecules in a gas sample move at the same speed -- there is a distribution of speeds at any given temperature. The root-mean-square (rms) speed is given by u_rms = sqrt(3RT/M), where R = 8.314 J/(mol*K), M is the molar mass in kg/mol, and T is in Kelvin. The rms speed is proportional to the square root of temperature and inversely proportional to the square root of molar mass. At the same temperature, lighter molecules move faster (H2 is faster than O2), and at higher temperatures all molecules move faster.

Three characteristic speeds describe the distribution. The most probable speed (u_mp), the speed at the peak of the distribution, equals sqrt(2RT/M). The average speed (u_avg), the mean of all molecular speeds, equals sqrt(8RT/(pi*M)). The rms speed equals sqrt(3RT/M). These three are always related by u_mp < u_avg < u_rms.

### IV. Maxwell-Boltzmann Distribution

The Maxwell-Boltzmann distribution is a probability function showing the fraction of molecules moving at each speed. At low temperature, the distribution is narrow and peaks at a relatively low speed. At high temperature, the distribution broadens and shifts to higher speeds, with the peak becoming lower. Heavier molecules produce narrower distributions peaked at lower speeds compared to lighter molecules at the same temperature. The total area under the curve is always 1 (representing 100% of the molecules). No molecules have exactly zero speed, and a small fraction occupies the high-speed tail of the distribution.

<image>A graph showing Maxwell-Boltzmann speed distributions. The x-axis is "Molecular speed (m/s)" and the y-axis is "Fraction of molecules." Three curves are plotted: Curve 1 (blue, labeled "T = 300 K, N2"): tall, narrow peak at a lower speed. Curve 2 (red, labeled "T = 1000 K, N2"): shorter, broader peak shifted to higher speed. Curve 3 (green, labeled "T = 300 K, He"): shorter, very broad peak at much higher speed than N2 at the same temperature (lighter molecule). Vertical dashed lines mark u_mp, u_avg, and u_rms for Curve 1, showing their relative positions. An annotation explains: "Higher T or lower M shifts the distribution to higher speeds and broadens it."</image>

### V. Effusion and Diffusion

Effusion is the escape of gas molecules through a tiny hole (smaller than the mean free path) into a vacuum. Diffusion is the mixing of gas molecules through another gas as a result of random motion.

#### Graham's Law of Effusion

Graham's law states that the rate of effusion is inversely proportional to the square root of the molar mass: rate_1 / rate_2 = sqrt(M_2 / M_1). Lighter gases effuse faster -- hydrogen effuses 4 times faster than oxygen because sqrt(32/2) = 4. Although Graham's law applies rigorously to effusion, it can also be used as a rough approximation for diffusion, though diffusion is complicated by intermolecular collisions that slow the net movement.

#### Applications

Graham's law has important practical applications. It is used in the separation of isotopes, most famously in uranium enrichment, where gaseous UF6 containing the lighter U-235 effuses slightly faster than UF6 containing U-238. It can also be used to determine the molar mass of an unknown gas by comparing its effusion rate to that of a known gas.

### VI. Mean Free Path

The mean free path is the average distance a molecule travels between collisions. It depends on the molecular diameter, the number density (N/V), temperature, and pressure. Under standard conditions (1 atm, 25 C), the mean free path for N2 is approximately 70 nm, roughly 200 molecular diameters. Lower pressure or higher temperature increases the mean free path because molecules are farther apart.

### VII. Real Gases: Deviations from Ideal Behavior

Real gases deviate from ideal behavior under two conditions. At high pressure, molecules are forced close together, and their finite volume is no longer negligible compared to the container volume. At low temperature, molecules move slowly enough that intermolecular attractive forces become significant, reducing the effective pressure. Two corrections must therefore be applied to the ideal gas model: one for the actual volume occupied by the molecules (which reduces the free space available) and one for intermolecular attractions (which pull molecules away from the walls and reduce the measured pressure).

### VIII. The van der Waals Equation

The van der Waals equation modifies the ideal gas law to account for these two effects: (P + a(n/V)^2)(V - nb) = nRT. The constant a corrects for intermolecular attractions (units: L^2*atm/mol^2), with larger a values indicating stronger intermolecular forces. Polar molecules and those capable of hydrogen bonding have large a values (for example, H2O has a = 5.46), while small nonpolar molecules have small a values (He has a = 0.034). The constant b corrects for molecular volume (units: L/mol), with larger b values corresponding to physically larger molecules. When both a and b are zero, the equation reduces to PV = nRT. The van der Waals equation is one of several equations of state for real gases; others include the Redlich-Kwong and Peng-Robinson equations.

<image>A comparison plot of PV/nRT (compressibility factor, Z) versus pressure for an ideal gas and several real gases. The x-axis is "Pressure (atm)" from 0 to 1000. The y-axis is "Z = PV/nRT" from 0 to 2.0. A horizontal dashed line at Z = 1.0 represents ideal gas behavior. Curves for H2, N2, CH4, and CO2 are shown. At moderate pressures, all real gases dip below Z = 1 (intermolecular attractions dominate, reducing V). At very high pressures, all curves rise above Z = 1 (molecular volume dominates, increasing V). H2, having the weakest intermolecular forces, shows the smallest dip and rises above 1 sooner. CO2, with stronger intermolecular forces, shows a more pronounced dip. Annotations explain each region.</image>

### IX. Compressibility Factor (Z)

The compressibility factor Z = PV/(nRT) provides a convenient measure of how much a real gas deviates from ideal behavior. For a perfect ideal gas, Z equals 1 at all conditions. When Z is less than 1, intermolecular attractions dominate, making the effective volume smaller than the ideal prediction. When Z is greater than 1, molecular volume effects dominate, making the effective volume larger. As pressure approaches zero, Z approaches 1 for all gases -- every gas becomes ideal at sufficiently low pressure. At a particular temperature called the Boyle temperature, Z remains approximately 1 over a wide range of pressures for a given gas.

### X. Conditions Favoring Ideal Behavior

Gases behave most ideally at high temperature, where the kinetic energy of molecules far exceeds the energy of intermolecular interactions, and at low pressure, where molecules are far apart and their volume is a negligible fraction of the total. Nonpolar, small molecules such as He and H2 behave most ideally, while polar, large molecules such as H2O, NH3, and SO2 show the greatest deviations from ideal behavior.