The Ideal Gas Law

We begin by identifying three states of matter. Even if we don’t know the actual substructure of matter, its nature as a collection of atoms, we can organize matter into three basic forms:

1) under a gentle force the matter does not change its shape or size – we call this a solid;

2) under a gentle force the matter changes its shape but not its size – we call this a liquid; and

3) under a gentle force the matter changes both its shape and size – we call this a gas (or vapor).

We want to consider the third of those forms.

Imagine a gas contained inside a rigid container. Following the lead of Leukippos and his student, Demokritos, we subdivide the gas with infinitely thin barriers that cut through it and through each other. The principle of conservation of mass prevents us from using an infinite set of cuts: an infinite set of infinitesimals adds up to an indefinite finite number, so an infinite cutting of the gas would produce an indefinite mass of gas, which would violate the conservation law. We thus infer that the gas must consist of a finite, albeit extremely large, number of particles.

Further, we assert that in the absence of chemical transformations the number of particles in a container does not change. We augment the law of conservation of mass with a law that mandates the conservation of chemical species. We assume that the energy needed to transmute a particle of one chemical species into one or more particles of another chemical species far exceeds the energies available to each particle in the gas.

At a given temperature and pressure, equal volumes of gas
contain the same number of particles, regardless of the chemical composition of
the particles. Thus we have Avogadro’s law, a rule discovered by Lorenzo Romano
Amedeo Carlo Bernadette Avogadro di Quaregna e Cerreto (1776 Aug 09 – 1856 Jul
09), who published his result in 1811. Later studies enabled chemists and
physicists to determine the value of what they called Avogadro’s number:
6.0221415 x 10^{23} particles per mole (a mass in grams equal to the
gas’s atomic weight: 2 grams for Hydrogen, 4 grams for Helium, 44 grams for
Carbon Dioxide, etc.).

For the gas to have indeterminate size and shape the particles must have only faint interactions among themselves. We thus consider a dilute gas, one in which the total volume occupied by the particles is very much less than the volume of the container. Under that circumstance the gas consists of particles flying freely inside the container.

By assumption the container does not move, so we infer that the gas it contains manifests no net linear momentum. But the gas consists of mass-bearing particles in motion, so we infer that the linear momenta of those particles must continuously add up to a net zero. With the particles all moving independently from each other, they must have randomly distributed linear momenta, so the gas contains the scrambled kinetic energy that we identify as heat. If the gas contains N particles, it contains a quantity of energy E=Nθ, in which θ represents the average kinetic energy per particle.

We have two kinds of variables in that description. The number of particles represents an extensive variable, a number that represents some aspect of the full extent of the system. The average kinetic energy per particle represents an intensive variable, a number that represents the intensity of some property of the system around a given point within the system.

We now have a means to understand gas pressure in Newtonian terms. When one of the particles in the gas strikes a wall of its container, it rebounds and thereby reverses the component of its linear momentum oriented perpendicular to the wall. Newton’s third law necessitates that the wall obtain an equal and opposite change in its momentum. When we average such changes over all of the particle impacts on a given area of the container’s wall and a given interval of time, we get the equivalent of a uniform pressure exerted against the wall.

Let’s be more specific about how we contain the gas. The rigid container in which we have put the gas consists of a vertical cylindrical tube standing on a flat surface to which we have sealed it and a cylindrical piston that can slide freely within the tube without letting any of the gas escape from the tube. The weight of the piston divided by the surface area of the piston’s lower face equals the pressure that the piston exerts upon the gas. If the piston does not move, then, by Newton’s third law of motion, the gas must exert the same pressure upon the piston.

Start with no gas in the tube and the piston resting on the flat surface at the bottom of the tube. When we put gas into the tube at a constant pressure, it exerts a force that lifts the piston to make room for the gas to occupy. The gas thus does work on the piston as the piston moves over a distance Δh to fill a volume equal to the product of Δh and A, the area of the piston’s face. The force that moves the piston equals the product of the gas’s pressure and the area of the piston’s face, so we calculate the amount of work done to fill the volume V=AΔh as

(Eq’n 1)

That work represents a kind of potential energy, which we can locate in the piston or in the gas, depending upon which of those things we use to convert the energy back into work.

If we keep the amount of gas in the cylinder constant and we allow it to expand by pushing the piston, we get

(Eq’n 2)

if we also do nothing to change the amount of energy in the gas. In this example, in order to keep the energy in the gas unchanged, we allow heat to seep into the gas. Equation 2 expresses what we call Boyle’s law.

If we allow heat to seep into or out of the gas without letting the volume of the gas change, we get a different law. To deduce that law consider the following:

Imagine that we put the particles of the gas into the cylinder with no kinetic energy at all. They lie as a kind of liquid on the bottom of the tube, occupying a minuscule volume. If we now add heat to the particles, their average kinetic energy increases. As the particles pound the bottom of the piston, they exert a force that lifts the piston in the tube. The rising of the piston causes the density of the particles to diminish, thereby reducing the rate at which the particles hit the piston. That reduction in the rate of impacts reduces the force that the particles exert in direct proportion to the change in the volume occupied by the gas in accordance with Equation 2.

In that last statement we have tacitly assumed that the average energy of the gas particles doesn’t change as the piston moves. But as they make the piston move against its weight, the particles do work upon the piston and thereby lose energy. To maintain the average energy of the particles in accordance with our assumption, the gas must absorb additional heat.

If we keep the pressure in our system constant, as making the gas support the weight of the piston does, then the volume of the gas stands in direct proportion to the amount of heat we put into it. To prove and verify that statement we note that the average velocity of the particles stands in direct proportion to the square root of each particle’s average kinetic energy. The force that the gas exerts upon the lower face of the piston stands in direct proportion to the average velocity of the particles and to rate at which the particles hit the piston. That latter rate stands in direct proportion to the particles’ average velocity, because the faster a particle moves the sooner it goes from striking the piston to traversing the volume of the container to striking the piston again. Thus the force that the gas exerts on the piston stands in direct proportion to the square of the average velocity of the particles and, thus, in direct proportion to the heat content of the gas. In the case in which we keep the pressure holding the gas in its container constant we know, in accordance with Equation 2, that the volume of the gas stands in direct proportion to the amount of heat in the gas. Of course, we have tacitly assumed that none of the heat that we put into the gas goes into overcoming any forces binding the particles to each other.

Because the amount of heat in the gas stands in direct proportion to the average kinetic energy carried by the particles (so long as the number of particles doesn’t change) and we correlate the average kinetic energy of the particles with the temperature that we assign to the gas, we can say that at a given pressure the volume of a gas stands in direct proportion to the absolute temperature of the gas. Thus we have a statement of Charles’s law, which originated in work done by Jacques Alexandre César Charles (1746 Nov 12 – 1823 Apr 07) in the 1780's. Originally unpublished, Charles’s work received credit in 1802 from Joseph Louis Gay-Lussac (1778 Dec 06 – 1850 May 09), who rediscovered it in his own work (some people refer to Charles’s law as Gay-Lussac’s law).

We can exploit Charles’s law with our piston-in-a-cylinder arrangement to measure the heat content of the gas. We need only calibrate the cylinder in increments that represent the volume contained in the cylinder below each mark and assign to each mark a number that represents the heat contained in the gas. If we ensure that the number of particles in the gas remains unchanged, then that number can also represent the average kinetic energy per particle in the gas. Of course, we don’t actually measure the energy per particle directly, but rather measure a number related to it, the number that we call the temperature.

In order to assign a temperature to the gas, we must calibrate our temperature scale. We need only pick two points and we can choose those arbitrarily, so we make the temperature of melting water ice equal zero degrees Celsius and make the temperature of boiling water at standard sea level atmospheric pressure (101,325 newtons per square meter or 14.7 pounds per square inch) equal one hundred degrees Celsius. We can then mark the container to show the temperature of the gas by the position of the piston. If we measure the volume of the gas at two different temperatures, such as the freezing temperature and boiling temperature defined above, we can determine the real zero of temperature, the absolute zero. We get

V_{100}-V_{0}=V_{0}(0.3660992)=V_{0}/2.7315,

from which we calculate the absolute zero as lying 273.15 centigrade degrees below the temperature at which water freezes.

Alternatively, we have Amontons law, which tells us that, if we keep the volume of a gas unchanged, the pressure of the gas stands in direct proportion to the gas’s heat content. Between 1700 and 1702 Guillaume Amontons (1663 Aug 31 – 1705 Oct 11) discovered that at constant volume the pressure in a gas stands in direct proportional to the absolute temperature of the gas. Again Equations 1 and 2 tell us that increasing the energy contained in the gas while keeping the volume of the gas constant increases the pressure.

Now we combine Avogadro’s law, Amontons law, Boyle’s law, and Charles’s law. We thus get

(Eq’n 3)

in which the Boltzmann constant k=1.38066x10^{-23} joule per particle
per degree Kelvin. Of course, that law only describes ideal gases. For more real
gases, those with stronger interactions among their particles, either due to
stronger inter-particle forces or due to closer packing of the particles
together, we need to use modified versions of the law, such as that devised by
Johannes Diderik van der Waals (1837 Nov 23 – 1923 Mar 08) in 1873.

(Eqn 4)

In that equation the lower-case ay represents an energy characteristic of the forces between pairs of particles in the gas and the lower-case bee represents the volume occupied by the particles themselves. But pursuing such modifications to the ideal gas law takes us beyond the scope of this essay, so I will stop here.

gabh