5 Kinetic Molecular Theory

[Heberto Calderon]

Theexperimental observations about the behavior of gases discussed so far can beexplained with a simple theoretical model known as the kinetic molecular theory.This theory is based on the following postulates, or assumptions. Gases are composed of a large number of particles that behave like hard, spherical objects in a state of constant, random motion. These particles move in a straight line until they collide with another particle or the walls of the container. These particles are much smaller than the distance between particles. Most of the volume of a gas is therefore empty space. There is no force of attraction between gas particles or between the particles and the walls of the container. Collisions between gas particles or collisions with the walls of the container are perfectly elastic. None of the energy of a gas particle is lost when it collides with another particle or with the walls of the container. The average kinetic energy of a collection of gas particles depends on the temperature of the gas and nothing else.The assumptions behind the kinetic molecular theory can be illustrated with theapparatus shown in the figure below, which consists of a glass plate surrounded by wallsmounted on top of three vibrating motors. A handful of steel ball bearings are placed ontop of the glass plate to represent the gas particles.

When the motors are turned on, the glass plate vibrates, which makes the ball bearingsmove in a constant, random fashion (postulate 1). Each ball moves in a straight line untilit collides with another ball or with the walls of the container (postulate 2). Althoughcollisions are frequent, the average distance between the ball bearings is much largerthan the diameter of the balls (postulate 3). There is no force of attraction between theindividual ball bearings or between the ball bearings and the walls of the container(postulate 4).

The collisions that occur in this apparatus are very different from those that occurwhen a rubber ball is dropped on the floor. Collisions between the rubber ball and thefloor are inelastic, as shown in the figure below. A portion of the energy of theball is lost each time it hits the floor, until it eventually rolls to a stop. In thisapparatus, the collisions are perfectly elastic. The balls have just as muchenergy after a collision as before (postulate 5).

At any time, some of the ball bearings on this apparatus are moving faster than others,but the system can be described by an average kinetic energy. When we increasethe "temperature" of the system by increasing the voltage to the motors, we findthat the average kinetic energy of the ball bearings increases (postulate 6).

The pressure of a gas results from collisions between the gas particles and the wallsof the container. Each time a gas particle hits the wall, it exerts a force on the wall.An increase in the number of gas particles in the container increases the frequency ofcollisions with the walls and therefore the pressure of the gas.

The last postulate of the kinetic molecular theory states that the average kineticenergy of a gas particle depends only on the temperature of the gas. Thus, the averagekinetic energy of the gas particles increases as the gas becomes warmer. Because the massof these particles is constant, their kinetic energy can only increase if the averagevelocity of the particles increases. The faster these particles are moving when they hitthe wall, the greater the force they exert on the wall. Since the force per collisionbecomes larger as the temperature increases, the pressure of the gas must increase aswell.

Gases can be compressed because most of the volume of a gas is empty space. If wecompress a gas without changing its temperature, the average kinetic energy of the gasparticles stays the same. There is no change in the speed with which the particles move,but the container is smaller. Thus, the particles travel from one end of the container tothe other in a shorter period of time. This means that they hit the walls more often. Anyincrease in the frequency of collisions with the walls must lead to an increase in thepressure of the gas. Thus, the pressure of a gas becomes larger as the volume of the gasbecomes smaller.

The average kinetic energy of the particles in a gas is proportional to the temperatureof the gas. Because the mass of these particles is constant, the particles must movefaster as the gas becomes warmer. If they move faster, the particles will exert a greaterforce on the container each time they hit the walls, which leads to an increase in thepressure of the gas. If the walls of the container are flexible, it will expand until thepressure of the gas once more balances the pressure of the atmosphere. The volume of thegas therefore becomes larger as the temperature of the gas increases.

As the number of gas particles increases, the frequency of collisions with the walls ofthe container must increase. This, in turn, leads to an increase in the pressure of thegas. Flexible containers, such as a balloon, will expand until the pressure of the gasinside the balloon once again balances the pressure of the gas outside. Thus, the volumeof the gas is proportional to the number of gas particles.

Imagine what would happen if six ball bearings of a different size were added to the molecular dynamicssimulator. The total pressure would increase because there would be morecollisions with the walls of the container. But the pressure due to the collisions betweenthe original ball bearings and the walls of the container would remain the same. There isso much empty space in the container that each type of ball bearing hits the walls of thecontainer as often in the mixture as it did when there was only one kind of ball bearingon the glass plate. The total number of collisions with the wall in this mixture istherefore equal to the sum of the collisions that would occur when each size of ballbearing is present by itself. In other words, the total pressure of a mixture of gases isequal to the sum of the partial pressures of the individual gases.

In 1829 Thomas Graham used an apparatus similar to the one shown in thefigure below to study the diffusionof gases the rate at which twogases mix. This apparatus consists of a glass tube sealed at one end with plaster that hasholes large enough to allow a gas to enter or leave the tube. When the tube is filled withH2 gas, the level of water in the tube slowly rises because the H2molecules inside the tube escape through the holes in the plaster more rapidly than themolecules in air can enter the tube. By studying the rate at which the water level in thisapparatus changed, Graham was able to obtain data on the rate at which different gasesmixed with air.

To understand the importance of this discovery we have to remember that equal volumesof different gases contain the same number of particles. As a result, the number of molesof gas per liter at a given temperature and pressure is constant, which means that thedensity of a gas is directly proportional to its molecular weight. Graham's law ofdiffusion can therefore also be written as follows.

Similar results were obtained when Graham studied the rate of effusionof a gas, which is the rate at which the gas escapes through a pinhole into a vacuum. Therate of effusion of a gas is also inversely proportional to the square root of either thedensity or the molecular weight of the gas.

Graham's law of effusion can be demonstrated with the apparatus in thefigure below. A thick-walled filter flask is evacuated with a vacuum pump. A syringe isfilled with 25 mL of gas and the time required for the gas to escape through the syringeneedle into the evacuated filter flask is measured with a stop watch.

As we can see when data obtained in this experiment are graphed in the figure below,the time required for 25-mL samples of different gases to escape into a vacuum isproportional to the square root of the molecular weight of the gas. The rate atwhich the gases effuse is therefore inversely proportional to the square root of themolecular weight. Graham's observations about the rate at which gases diffuse (mix) oreffuse (escape through a pinhole) suggest that relatively light gas particles such as H2molecules or He atoms move faster than relatively heavy gas particles such as CO2or SO2 molecules.

The kinetic molecular theory can be used to explain the results Graham obtained when hestudied the diffusion and effusion of gases. The key to this explanation is the lastpostulate of the kinetic theory, which assumes that the temperature of a system isproportional to the average kinetic energy of its particles and nothing else. In otherwords, the temperature of a system increases if and only if there is an increase in theaverage kinetic energy of its particles.

The basic version of the model describes an ideal gas. It treats the collisions as perfectly elastic and as the only interaction between the particles, which are additionally assumed to be much smaller than their average distance apart.

The theory's introduction allowed many principal concepts of thermodynamics to be established. It explains the macroscopic properties of gases, such as volume, pressure, and temperature, as well as transport properties such as viscosity, thermal conductivity and mass diffusivity. Due to the time reversibility of microscopic dynamics (microscopic reversibility), the kinetic theory is also connected to the principle of detailed balance, in terms of the fluctuation-dissipation theorem (for Brownian motion) and the Onsager reciprocal relations.

In about 50 BCE, the Roman philosopher Lucretius proposed that apparently static macroscopic bodies were composed on a small scale of rapidly moving atoms all bouncing off each other.[1] This Epicurean atomistic point of view was rarely considered in the subsequent centuries, when Aristotlean ideas were dominant.

3a8082e126