Lecture Notes from CHM 1341
12 June 1996

The Ideal Gas Law

Gases are one of the four common phases of matter. Yes, there are four: solid, liquid, gas, and PLASMA. Plasmas are fluids all right but they occur at such high temperatures that the outer ("valence") electrons are stripped off, leaving them as ions (charged atoms) in a sea of electrical currents. You encounter plasmas daily in fluorescent lights. You see them briefly during lightning strokes. And they are of use in technology for etching surfaces.

While plasmas are spectacular, we'll dwell here on the other three.

Solids are molecules cheek-by-jowl at their closest proximity, locked in fixed crystalline position, incompressible, and, of course, unable to flow. All materials become solid if one but reduces the temperature sufficiently or increases the pressure upon them sufficiently. Solid substances are mistakenly referred to as the state in which molecules no longer move; actually they do oscillate in place about their center point fixed by the intermolecular forces holding the solid together.

At sufficiently high temperature (-259°C for H but +3030°C for Os), that molecular vibration breaks free of its fixed centers and the molecules, while still adjacent, may now flow over one another in the liquid state. Liquids remain incompressible, but the overcoming of the lock on position means that liquids are fluid, filling their container to the level of their volume.

Both solids and liquids have interfaces (surfaces) which divide them from their surroundings. Across that interface, at any temperature above absolute zero, there is a finite vapor pressure of the substance. That vapor only makes itself terribly obvious when the temperature rises to the point that the vapor pressure competes with "atmospheric pressure," at which point, the liquid boils or the solid "sublimes." A common example of a subliming solid is "dry ice," solid carbon dioxide, which skips its liquid phase at ordinary pressures. The temperatures which liberate elemental vapor at one atmospheric pressure can range from -269°C for He to +5600°C for Rh.

Molecules don't change size when their condensed phases vaporize, so their gases contain molecular mass of finite volume, but that condensed volume is scattered at STP (Standard Temperature, 273°K, and Pressure, 1 atm) over a vastly larger volume of the gas. The increase will be in excess of 1,000 fold; so the normal assumption (and one hallmark of an "ideal gas") is that the molecules may as well be considered individually volumeless, point particles. In addition, the force fields which held them together in their condensed phases are still operational, but the forces fall effectively to nothing beyond a few molecular diameters. The intermolecular spacing in the gas is so great (over 50x solid spacing) that another assumption (and hallmark of an "ideal gas") is that they no longer interact with one another. Neither assumption is strictly true, yet the gas laws which result from them give results which are about 99% accurate. "Close enough for government work."

So if gas molecules don't see one another, it matters not to them even what kind of molecules share the volume of the container with them; that's Avogadro's Law. So an "ideal gas" can be of ANY substance.

In fact, the ONLY things which matter to an ideal gas are the temperature (which influences their average flight speed) and the walls of the vessel, against which they bash themselves with great frequency and (another hidden ideal gas assumption) great elasticity. That is, on every collision, gas molecules bounce back from the wall undiminished in AVERAGE energy...so they keep it up forever. Those bashing speeds vary with temperature and molecular weight, but it is safe to say that gas molecules are running around at about the speed of sound, 760 mph in air at STP. On these collisions, the forward momentum of the molecules is reversed as they richochet. Changing momentum means an equal-and-opposite (Newton's 2nd) force, obvious on the gas molecule but operative on the wall as pressure! The force of the rebounds per unit area of vessel surface is the very definition of pressure. And from this definition, we can infer some ideal gas relations.

It is true that on each collision with the wall, the molecule can exchange energy and either increase or decrease what it had, but since the walls are all at the same temperature, the average energy of colliding molecules continues to average to the same value. In fact, if that collisional energy exchange DIDN'T take place, you could never heat or cool gases by changing the temperature of their vessel walls!

For example, doubling the volume alone will halve the gas density (mass/unit volume) which halves the collisions per unit area which halves the pressure! Simple. So pressure and volume are INVERSELY related to one another: change either one by a factor F, say, and the other will change by the factor 1/F. In other words, PV = constant; that's Boyle's Law, and it holds as long as you don't mess around with the temperature at the same time. That constancy of temperature ensures constancy of molecular velocity.

But suppose you kept the volume fixed instead and chose to change the temperature. Those molecular velocities change...but by how much? Without proof, let us note that the kinetic energy (motion energy) of molecules, ½ m v², is directly proportional to the absolute temperature T measured in Kelvin. In other words, quadruple the T and you double the molecular velocities. What effect does that have? First, since the molecules are moving twice as fast, they hit the walls twice as often; that alone would double the pressure. But wait...there's more. Second, molecules going twice as fast have twice as much momentum to transmit on each collision; that too doubles the pressure. So...quadrupling the temperature doubles the pressure TWICE; a 4x T change has brought about a 4x P change. So pressure increases DIRECTLY with temperature (in a fixed volume). Or P=(constant)*T.

Let's do another Gedanken experiment. (Einstein loved those. Remember gedanken is German for thought.) Suppose we double the temperature at fixed volume. The pressure doubles. Then suppose we let the volume change to recover that original pressure (while retaining the doubled temperature). Since that requires the pressure to now decrease be the factor of 2, the volume must (by Boyle's Law) increase by that same factor of 2! We've done one experiment in two stages; what's wonderful about it is that the same result (doubled T, doubled V, original P) would have been obtained if we did the experiment all in one go simply by heating the gas to double its temperature while keeping the pressure constant; e.g., the volume would change DIRECTLY with temperature (at fixed pressure). That's Charles's Law: V=constant*T (if P fixed).

So Boyle knows PV is fixed at fixed T. And Charles knows V/T is fixed at fixed P. And Avogadro knows P/n is fixed at fixed V and T. n? The number of moles of any ideal gas or mixture of such gases. Remember that ideal gases act completely independently of one another; so putting two different substances as ideal gases in a vessel means that the total pressure is the sum of the independent substance pressures.

The relationship which brings all of these observations together, codifies the behavior of all ideal gases, is

where that new symbol R is a constant, specifically the Gas Constant, which has the values

            L atm             J
R = 0.0821 ------- = 8.314 -------
            mol K           mol K

where the volume unit L = Liter = (dm)³ = 1000 cm³  
and the energy unit J = Joule = km m²/ s²  .

Note that if nRT are fixed, Boyle is happy. Likewise if P and nR are fixed, Charles is happy. Even Avogadro can buy into this if V and RT are fixed.

So PV = nRT is worth knowing.

Chris ParrUniversity of Texas at Dallas
Programs in Chemistry, Room BE3.506
P.O. Box 830688  M/S BE2.6 (for snailmail)
Richardson, TX  75083-0688
   Voice: (214) 883-2485
     Fax: (214) 883-2925
     BBS: (214) 883-2168 (HST) or -2932 (V.32bis)
Internet: parr@utdallas.edu  (sends Chris e-mail.)