The ideal gas equation is an empirical relationship which describes how an ideal gas behaves under a given set of conditions, but it does not explain why gases behave as they do. To try to explain the behavior of an ideal gas a model which will enable us to image how gases behave at the atomic level is needed. Such a model has been developed and is known at the Kinetic-molecular theory for ideal gases. The model evolved over a period of 100 years and is the product of the work of many early scientists. The kinetic molecular model is based on the following postulates;

  1. Gases consist of tiny (submicroscopic) molecules which are in continuous, random motion.

  2. The distance between molecules is large compared with the size of the molecules themselves. The volume occupied by a gas consists mostly of empty space.

  3. Gas molecules have no attraction for one another.

  4. Gas molecules move in straight lines in all directions, colliding frequently with one another and with the walls of the container.

  5. No energy is lost by the collision of a gas molecule with another gas molecule or with the walls of the container. All collisions are perfectly elastic.

  6. The average kinetic energy for molecules is the same for all gases at the same temperature, and its value is directly proportional to the absolute temperature.

I have prepared a computer generated model to represent these postulates. So lets study the computer generated model and see how it behaves.

You can see a region (a box) which constrains the particles of gas. Although the particles are not particularly small for an ideal gas we do see they are in continuous, straight line motion. Again because of the computer, and to allow easy viewing the particles are not widely spaced. Notice the particles are not attracted to each other. The gas particles move in a straight line between collisions. If we color one of the particles differently than the others we can see the straight line motion easier. The collisions are also easier to identify.

If you are observant you may see that the colored particle has different speeds, sometimes fast, and sometimes slow. In this computer model the number of different speeds is limited. An important point must be made to identify the difference between this computer simulation and reality. For an ideal gas there is a much broader range of speeds of the particles. In fact a plot of molecular speed and fraction of gas particles with a particular speed is shown on the overhead. (Show overhead of molecular speed distribution). Notice that at a given temperature there are fractions of gas particles with very high speeds, and fractions of particles traveling very slowly. As the temperature is changed the speed distribution changes. If we increase the temperature the curve shifts to higher speeds. The kinetic energy of a collection of gas particles is given by

where m is the mass of a molecule of the gas and u is its root-mean-square velocity. The root-mean-square velocity is not exactly equal to the average velocity, but they are close. The rms velocity is the speed of a molecule that has the average molecular kinetic energy. This relationship will be important in later discussions of some interesting properties of gases. Right now it is important to recognize that a collection of gas particles has a range of speeds, not a single speed.

When a collision occurs energy can be transferred. So if the white gas particle is moving slowly and it collides with a fast moving particle, transfer of energy can result in the white particle moving faster, and the other particle slower after the collision. Total energy of the collision is conserved--elastic collision.

Now the usefulness of a model depends on its ability to reproduce experimental observations as well as make predictions which can be verified by further experiment.

Lets begin by verifing some of the experiments we performed earlier. Boyle's Law related pressure to volume of an ideal gas at constant mol and temperature. So if set the temperature at 300 K and 4 mol of gas we can observe how the pressure is effected by a decrease in volume. Initially the pressure is 2.59 atm.

Lets observe what happens as the number of moles of gas are changed at constant temperature and volume.

Now lets consider how changing the temperature of a gas effects the pressure at constant volume and constant moles. To help recognize how we use the Kinetic-molecular model, watch carefully what happens as the temperature is lowered.

Now lets consider how changing the temperature of a gas effects the volume at constant pressure and constant moles. We will change to the 'V' mode to calculate volume. To help recognize how we use the Kinetic-molecular model to explain Charles' Law, watch carefully what happens as the temperature is lowered.


Diffusion and effusion

The last postulate says the average kinetic energy for molecules is the same for all gases at the same temperature. It should be noted that this says if you have two samples of gas, one with a large mass and one with a small mass, the average speed of the sample with the large mass is slower than the average speed of the light gas. For example, the average speed for a collection of H2 molecules at 25 degrees C is 1766 m.sec-1, (2.55 x 104 while for Cl2 it is 286 m.sec-1 (4.13 x 103

The last postulate in the kinetic-molecular theory indicates there is a direct relationship between kinetic energy and temperature. We will not derive this relationship, but simply state it as;

where R is the ideal gas constant (8.314 J.K-1.mol-1), T is the absolute temperature and N is Avogadro's number. If this equation is solved for velocity, u, we have;

At a given temperature the velocity of a gas is inversely proportional to its molar mass. This relationship helps us understand two phenomena, effusion and diffusion. Effusion is the rate of escape of a gas particle through a tiny hole, as when a gas escapes from a balloon, and it depends on the molar mass of the gas. Diffusion refers to the spread of a gas through another gas, for example when perfume diffuses through a room.

For effusion the rate is directly proportional to the velocity of the gas. The higher the velocity of the gas the faster the gas effuses. Grahams law of effusion says the effusion rate of gases are inversely proportional to the molar masses.

cancelling the constants

which is the same as


The explanation for rate of diffusion is more complicated because of the effect of collisions with other molecules (these collision are absent in effusion) but the net result is the same, the rate of diffusion is identical to the rate of effusion.

The difference in the rate of diffusion due to the difference in mass is used to separate the relatively low abundant uranium isotope 235U (0.7 %) from the more abundant isotope 238U (99.3 %). The uranium is converted to the volatile UF6 which, while a liquid at room temperature, boils at 56 degrees C. The sample of gas is allowed to diffuse through a porous barrier from one chamber to the next. As a result in the small difference in mass of the volatile compounds the lighter isotope concentrates after passing through thousands of chambers.

Real Gases

Although ideal gas equation is a very useful description of gases, many gases, most gases do not obey this relationship. Gases which do not obey the ideal gas equation are call real gases.

To see how real gases behave we will consider plotting the quotient versus P for the gas. For 1 mol of an ideal gas a plot of this quotient as the pressure increase behaves as we see in the figure below. According to Boyle's law increasing the pressure decreases the volume. For an ideal gas the plot is a constant.

However, with real gases we observe deviation from this plot. Consider carbon dioxide and observe how this quotient changes as we increase the pressure.