Lesson: Chapter - 12

The Kinetic Theory of Gases and the Ideal Gas Law

We said earlier that temperature is a measure of the kinetic energy of the molecules in a material, but we didn’t elaborate on that remark. Because individual molecules are so small, and because there are so many molecules in most substances, it would be impossible to study their behavior individually. However, if we know the basic rules that govern the behavior of individual molecules, we can make statistical calculations that tell us roughly how a collection of millions of molecules would behave. This, essentially, is what thermal physics is: the study of the macroscopic effects of the microscopic molecules that make up the world of everyday things.

The kinetic theory of gases makes the transition between the microscopic world of molecules and the macroscopic world of quantities like temperature and pressure. It starts out with a few basic postulates regarding molecular behavior, and infers how this behavior manifests itself on a macroscopic level. One of the most important results of the kinetic theory is the derivation of the ideal gas law, which not only is very useful and important, it’s also almost certain to be tested on
Physics.

The Kinetic Theory of Gases

We can summarize the kinetic theory of gases with four basic postulates:

1. Gases are made up of molecules: We can treat molecules as point masses that are perfect spheres. Molecules in a gas are very far apart, so that the space between each individual molecule is many orders of magnitude greater than the diameter of the molecule.
2. Molecules are in constant random motion: There is no general pattern governing either the magnitude or direction of the velocity of the molecules in a gas. At any given time, molecules are moving in many different directions at many different speeds.
3. The movement of molecules is governed by Newton’s Laws: In accordance with Newton’s First Law, each molecule moves in a straight line at a steady velocity, not interacting with any of the other molecules except in a collision. In a collision, molecules exert equal and opposite forces on one another.
4. Molecular collisions are perfectly elastic: Molecules do not lose any kinetic energy when they collide with one another.

The kinetic theory projects a picture of gases as tiny balls that bounce off one another whenever they come into contact. This is, of course, only an approximation, but it turns out to be a remarkably accurate approximation for how gases behave in the real world.

These assumptions allow us to build definitions of temperature and pressure that are based on the mass movement of molecules.

Temperature

The kinetic theory explains why temperature should be a measure of the average kinetic energy of molecules. According to the kinetic theory, any given molecule has a certain mass, m; a certain velocity, v; and a kinetic energy of ¹/ ₂ mv². As we said, molecules in any system move at a wide variety of different velocities, but the average of these velocities reflects the total amount of energy in that system.

We know from experience that substances are solids at lower temperatures and liquids and gases at higher temperatures. This accords with our definition of temperature as average kinetic energy: since the molecules in gases and liquids have more freedom of movement, they have a higher average velocity.

Pressure

In physics, pressure, P, is the measure of the force exerted over a certain area. We generally say something exerts a lot of pressure on an object if it exerts a great amount of force on that object, and if that force is exerted over a small area. Mathematically:

P = F/A

Pressure is measured in units of pascals (Pa), where 1 Pa = 1 N/m².

Pressure comes into play whenever force is exerted on a certain area, but it plays a particularly important role with regard to gases. The kinetic theory tells us that gas molecules obey Newton’s Laws: they travel with a constant velocity until they collide, exerting a force on the object with which they collide. If we imagine gas molecules in a closed container, the molecules will collide with the walls of the container with some frequency, each time exerting a small force on the walls of the container. The more frequently these molecules collide with the walls of the container, the greater the net force and hence the greater the pressure they exert on the walls of the container.

Balloons provide an example of how pressure works. By forcing more and more air into an enclosed space, a great deal of pressure builds up inside the balloon. In the meantime, the rubber walls of the balloon stretch out more and more, becoming increasingly weak. The balloon will pop when the force of pressure exerted on the rubber walls is greater than the walls can withstand.

The Ideal Gas Law

The ideal gas law relates temperature, volume, and pressure, so that we can calculate any one of these quantities in terms of the others. This law stands in relation to gases in the same way that Newton’s Second Law stands in relation to dynamics: if you master this, you’ve mastered all the math you’re going to need to know. Ready for it? Here it is:

Video Lesson - Ideal Gas Equation

PV = nRT

Effectively, this equation tells us that temperature, T, is directly proportional to volume, V, and pressure, P. In metric units, volume is measured in m³, where 1m³ = 10⁶cm².

The n stands for the number of moles of gas molecules. One mole (mol) is just a big number—6.023 × 10²³ to be precise—that, conveniently, is the number of hydrogen atoms in a gram of hydrogen. Because we deal with a huge number of gas molecules at any given time, it is usually a lot easier to count them in moles rather than counting them individually.

The R in the law is a constant of proportionality called the universal gas constant, set at 8.31 J/mol · K. This constant effectively relates temperature to kinetic energy. If we think of RT as the kinetic energy of an average molecule, then nRT is the total kinetic energy of all the gas molecules put together.

Deriving the Ideal Gas Law

Imagine a gas in a cylinder of base A, with one moving wall. The pressure of the gas exerts a force of F = PA on the moving wall of the cylinder. This force is sufficient to move the cylinder’s wall back a distance L, meaning that the volume of the cylinder increases by ?V = AL. In terms of A, this equation reads A = ?/L. If we now substitute in ?/L for A in the equation F = PA, we get F = PL, or

P?V = FL

If you recall in the chapter on work, energy, and power, we defined work as force multiplied by displacement. By pushing the movable wall of the container a distance L by exerting a force F, the gas molecules have done an amount of work equal to FL, which in turn is equal to P?V.

P?V = FL

The work done by a gas signifies a change in energy: as the gas increases in energy, it does a certain amount of work on the cylinder. If a change in the value of PV signifies a change in energy, then PV itself should signify the total energy of the gas. In other words, both PV and nRT are expressions for the total kinetic energy of the molecules of a gas.

Boyle’s Law and Charles’s Law

Physics will not expect you to plug a series of numbers into the ideal gas law equation. The value of n is usually constant, and the value of R is always constant. In most problems, either T, P, or V will also be held constant, so that you will only need to consider how changes in one of those values affects another of those values. There are a couple of simplifications of the ideal gas law that deal with just these situations.

Boyle’s Law

Boyle’s Law deals with gases at a constant temperature. It tells us that an increase in pressure is accompanied by a decrease in volume, and vice versa: P_iV_i = P_fV_f. Aerosol canisters contain compressed (i.e., low-volume) gases, which is why they are marked with high-pressure warning labels. When you spray a substance out of an aerosol container, the substance expands and the pressure upon it decreases.

Video Lesson - Boyle's Law

Charles’s Law

Charles’s Law deals with gases at a constant pressure. In such cases, volume and temperature are directly proportional: V_i/T_i = V_fT_f. This is how hot-air balloons work: the balloon expands when the air inside of it is heated.

Video Lesson -

Gases in a Closed Container

You may also encounter problems that deal with “gases in a closed container,” which is another way of saying that the volume remains constant. For such problems, pressure and temperature are directly proportional: V_i/T_i = V_fT_f. This relationship, however, apparently does not deserve a name.

Example 1

A gas in a cylinder is kept at a constant temperature while a piston compresses it to half its original volume. What is the effect of this compression on the pressure the gas exerts on the walls of the cylinder?

Questions like this come up all the time on Physics. Answering it is a simple matter of applying Boyle’s Law, or remembering that pressure and volume are inversely proportional in the ideal gas law. If volume is halved, pressure is doubled.

Example 2

A gas in a closed container is heated from 0ºC to 273°C. How does this affect the pressure of the gas on the walls of the container?

First, we have to remember that in the ideal gas law, temperature is measured in Kelvins. In those terms, the temperature goes from 273 K to 546 K; in other words, the temperature doubles. Because we are dealing with a closed container, we know the volume remains constant. Because pressure and temperature are directly proportional, we know that if the temperature is doubled, then the pressure is doubled as well. This is why it’s a really bad idea to heat an aerosol canister.

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