Classical Kinetic Theory of Gases: A Crash Review

The classical kinetic theory of gases is usually covered in a basic chemistry class. The key assumption is:

In the absence of significant molecular interactions,

The average kinetic energy of a molecule is independent of the kind of molecule.
The average kinetic energy of a molecule is directly proportional to the absolute temperature.

This is not intended to be a rigorous presentation.

Notes:

For a gas, "standard conditions" (when a substance is a gas under such conditions) is defined as a temperature of 0°C and a pressure of 1 atm(osphere). This is a historical artifact.
The volume of a mole of ideal gas molecules at "standard conditions" is approximately 22.4 liters.

What is the "average kinetic energy"?

We use the root-mean-square average [RMS average] to compute the average (RMS) kinetic energy. Cf. the expression on the right. For hydrogen [H₂] gas at 0°C, the average speed of a hydrogen molecule is about 1.84·10³ meters/second [m/s]. Part of the derivation of the Maxwell-Boltzmann probability distribution for the speed of the molecules defines absolute temperature so that this is the expression for the average kinetic energy of a molecule of gas:

3kT

T is absolute temperature, and k is Boltzmann's constant. I would assume that this expression fails when the temperature is low enough to cause the Maxwell-Boltzmann probability distribution to require quantum corrections...but I don't know this.

What are "significant molecular interactions"?

Anything that would measurably alter the quantity of the gas. A non-exhaustive list would include:

The absolute temperature is high enough to ionize some of the molecules. [That is a plasma, not a gas!].
Phase changes: sublimation (to or from a solid), evaporation (from liquid), or condensation (to liquid). Evaporation and condensation don't work above the critical temperature.
Chemical reactions. Even if the chemical species reach equilibrium, the reaction will respond to changes in the physical parameters (temperature, pressure, and volume), thus altering the composition of the gas. Even if the reaction doesn't alter the molecule count, it may well affect the temperature.

There are at least two interactions that do not affect the basic assumptions of classical kinetic gas theory, but do affect more detailed equations.

Molecules don't overlap.
Molecules that are far part tend to polarize each other, and then attract each other (very weakly).

What is the probability distribution for the speed of the molecules?

Historically, formally deriving this algebraically is a test of Quantum Mechanics. There are at least two possible approaches to this; both are not that clean. It was empirically extrapolated by James Clerk Maxwell (1831-1879) in 1860, about five to six decades before Quantum Mechanics was beginning to be formulated.

At sufficiently intermediate temperatures (well above the boiling point of liquid helium, but below thermal ionization of the gas), the following differential equation is a good approximation (i.e., not the accuracy-limiting factor):

dN = N·4π(m/[2πkT])^3/2e^-mv²/[2kT] v²dv
Integrate the right-hand side on v from the lower bound of the classical velocity to the upper bound of the classical velocity in question, to get the expected number of molecules with a speed in that range. In the above:

m: mass of the molecule
v: classical velocity
N: number of molecules
k: Boltzmann's constant. Value depends on the units of measurement for velocity and absolute temperature.
T: absolute temperature.

We call this the Maxwell-Boltzmann probability distribution for the speed of a molecule. The above formulation of the equation has the advantage of highlighting the ratio m/(kT)...which is the only occurence of k, m, and T. If we set a := m/(kT), we can simplify the above differential equation to
dN = N·[2/π]^1/2(α)^3/2e^-αv²/2 v²dv

Unless we are concerned about very small Lorentz corrections, it is not worth the effort to find the General-Relativistic version of the above equation -- in practice, the gas will ionize from high temperatures before the corrections become significant. Gravity-field emulation techniques (if necessary) should suffice.

However, the algebraic morass from which the formula comes does have significant corrections to a key part at low temperatures. The corrections depend on the distinction between bosons and fermions...and seriously affect the form of the equation.

What is the "perfect gas law" i.e. "ideal gas law"?

The perfect gas law equation is:
PV = nRT
In the above,

P: pressure
V: volume
n: moles of gas molecules
R: molar gas constant. The value depends on the units of measurement for pressure, volume, and absolute temperature.
T: absolute temperature.

Historically, a formal test of the Maxwell-Boltzmann speed distribution law is the derivation of the perfect gas law from it. This is another algebraic morass. This derivation ignores both volume exclusion and the normally-weak electromagnetic attraction between molecules. The induced deviations are most extreme when a gas is close to changing phase.

The finite volume of the molecules tends to increase the volume above an ideal gas' volume.
Molecules tend to attract each other slightly when far away (this is a weak electromagnetic effect). This tends to make the pressure less than an ideal gas' pressure.

Errors tend to be minimal far away from both the boiling point and critical pressure of a gas. For gases similar to the Earth's atmosphere, the errors are generally less than 1% for room temperature or higher, and pressures below 10 atm.

NOTE: The derivation of the ideal gas law does assume that gravity is insignificant: that is, it is legitimate to represent the center-of-mass trajectories of the gas molecules between collisions as straight lines. Two obvious violations of this are strong gravity fields (neutron star or stronger), and protostars. In both cases, the ideal gas law works on scales small enough to not see the curvature in the center-of-mass trajectories.

What about Boyle's, Avogadro's, and Charles' laws?

These are all partial versions of the perfect gas law. Using variable names as defined above:

1662: (Robert) Boyle's (1627-1691) law: PV = constant when T, quantity of gas constant.
1787: (Jacques Alexander) Charles (1746-1823) reported that different gases expanded by the same fraction for the same rise in temperature.
1801: Dalton conjectures the law of partial pressures: molecular type does not matter when figuring the overall pressure of a gas...so the partial pressure for a given gas compound is the percentage (by molecule count) of that compound in the gas. Add the partial pressures to get the total pressure.
1802: (Joseph Louis) Gay-Lussac (1778-1850) empirically determines an absolute temperature scale. He refines Charles' law to: V/T = constant when pressure, gas quantity constant. T must be an absolute temperature. He found that the Celsius scale could be converted to an absolute temperature scale [now called Kelvin] by adding about 273 degrees to the Celsius scale.
1805: Gay-Lussac started experimenting with gases that chemically reacted. He finds that they reacted in ratios defined by small integers. For instance, 2000 ml of hydrogen + 1000 ml of oxygen completely reacted to form water. Also, 1000 ml of carbon monoxide reacted with 500 ml of oxygen to form 1000 ml of carbon dioxide.
Obviously, some of these chemical names are translated into our nomeclature. As a conceptual tool, reaction equations (and chemical formulae as we know them) don't exist yet. In particular, at this stage what is known about carbon monoxide and carbon dioxide is that they are both gases, and they both contain only carbon and oxygen.
1811: Amadeo Avogadro (1776-1856) proposes a hypothesis (Avogadro's law): equal volumes of dilute gases contain equal numbers of molecules. Avogadro's law's practical applications escape notice.
1858: Stanislao Cannizarro (1826-1910) showed how to apply Avogadro's law systematically. This immediately solves the questions of both correct atomic weight ratios and correct formulae.
For instance, before 1858 it was unclear whether the correct chemical formula for water was H₂O or HO. In our nomeclature, water is H₂O...and HO is a hydroxide radical without its electron, imminently about to resume its normal formula HO^-.
At this point, the perfect gas law PV = nRT can instantly be obtained (aside from how to measure the quantity of the gas n) as an algebraic extrapolation of Boyle's, Charles', and Avogadro's laws.
1860: James Clerk Maxwell (1831-1879) empirically extrapolates the Maxwell-Boltzmann probability distribution. The algebraic derivation of the perfect gas law, from the Maxwell-Boltzmann probability distribution, is a formal test of the Maxwell-Boltzmann probability distribution. Starting in 1858, the perfect gas law is capable of falsifying probability distributions for the speed of molecules.
1920: Otto Stern attempts an empirical verification of the Maxwell-Boltzmann probability distribution. By heating a tungsten wire, he constructed a beam of silver atoms emitted at about 1200°C By using a system of slits and a rotating drum, it was possible to measure how many emitted silver atoms were travelling in various speed ranges. By 1947, empirical verifications can measure agreement with the Maxwell-Boltzmann probability distribution to within 1% or so.

When do evaporation and condensation become impossible?

Thomas Andrews [1813-1885] discovered that for a pure gas, the distinction between the liquid and gas phases vanishes at the "critical temperature". This is formally called the "continuity of the gas and liquid phases". Qualitatively, there is still a difference (gases are obviously compressible, liquids aren't very compressible).

Above the critical temperature: the gas can be compressed into a liquid without a phase change.
Below the critical temperature: increasing the pressure eventually causes the gas to saturate, followed by condensation into a liquid.

Ideal gas law from average kinetic energy

Consider a single gas molecule with mass m, and assume that its collisions with a wall are both practically instantaneous, and perfectly elastic. To further simplify things, let's consider a cubical container of volume V, whose sides are aligned with the x-y-z cartesian coordinate system. [The sides being of length ³√V.]

The molecule has velocity v := (v_x,v_y,v_z). As the collisions are perfectly elastic, each collision with a wall only changes the sign of the corresponding coordinate in the velocity. The magnitude of the impulse at a collision with a y-z plane wall is then 2m|v_x|.

For an ideal gas, the distance in the x-coordinate traveled between two collisions with the same y-z wall is

2 ³√V

|v_x|

We thus get an average magnitude of orthogonal force on each y-z wall of

2m|v_x|

2 ³√V

|v_x|

i.e.

m|v_x|²

³√V

After conducting similar calculations for v_y and the x-z walls, and v_z and the x-y walls, we add the magnitudes of force on all six walls:

2m|v_x|²+2m|v_y|²+2m|v_z|²

³√V

But this reduces (by definition of Euclidean norm of a vector) to an expression in the kinetic energy of the gas molecule.

2m|v|²

³√V

When we have enough particles to average, we can replace m|v|²/2 with the average kinetic energy. In particular, for one mole of ideal gas we have the following as the magnitude of force on all six walls:

6RT

³√V

Since each wall has an area of (³√V)², and the molecules of an ideal gas have zero length, the pressure on a single wall of this cube from a mole of ideal gas is

RT

V

And thus we get PV=RT for one mole of ideal gas. With n as the number of moles, this recovers the ideal gas law PV=nRT.

Heat capacity of a noble gas at constant volume

This depends on the definition used in the Maxwell-Boltzmann probability distribution for the speed of the molecules of the (noble) gas. The average RMS kinetic energy of

3kT

per molecule immediately translates to a total kinetic energy of

3RT

per mole, where k, T, and R are defined as before. Informally, we computed the average RMS kinetic energy per molecule from the total kinetic energy per mole, so "obviously" we can reverse the procedure. We might suspect that R/k is the conversion factor from "per molecule" to "per mole"...in which case it should be Avogadro's constant. Skimming the fundamental constants list at the

U.S. NIST, we find that this is indeed the case empirically.

We differentiate by T to find the heat capacity in deg^-1mol^-1:

(d/dT)([3/2]RT) deg^-1mol^-1 = [3/2]R deg^-1mol^-1 Thus, the heat capacity C_V of an ideal noble gas is (3/2)R deg^-1mol^-1. Numerically (NIST data): 1.5^.[8.314 472 ± 0.000 015] J^.mol^-1K^-1, which is approximately 12.471708 ± 0.000 023] J^.mol^-1K^-1.

Real noble gases approximate this very well. This does not apply to more complicated gases, because they have additional methods of storing heat besides kinetic energy.

Heat capacity of a noble gas at constant pressure

This depends on the perfect gas equation (and has all of that equation's foibles). The general technique is to notice that when the volume of a gas increases at constant pressure [say, in an effectively friction-free gas-tight piston], the work done by an increase in volume dV is P dV. If we want to increase the volume of a gas in such conditions by increasing the temperature, the relevant quantity is dV/dT. We can implicitly differentiate the gas law by T, holding n, R, and P constant: (d/dT)(PV) = (d/dT)(nRT) i.e. P(dV/dT) = nR i.e. P dV = nR dT . This heat energy is in addition to that required to heat the gas at constant volume, so the heat capacity C_P at constant pressure is R deg^-1mol^-1 greater than the heat capacity C_V at constant volume. That is, C_P for an ideal noble gas is (5/2)R deg^-1mol^-1. Numerically (NIST data): 2.5^.[8.314 472 ± 0.000 015] J^.mol^-1K^-1, which is approximately 20.786180 ± 0.000 038] J^.mol^-1K^-1.

Real noble gases approximate this very well. Again, this does not apply to more complicated gases -- for the same reason.

Van der Waals' Equation

Purdue on Van der Waal's equation
In 1873, the Dutch physicist Johannes van der Waals extrapolated how to implement first-order corrections to the ideal gas law, that accounted for molecular volume and the small attractive forces responsible for the gas-liquid phase distinction at low temperatures. These extrapolations require empirical constants: [P+(an²/V)][V-nb]=nRT In addition to the prior notation, we now have two empirical Van der Waals constants:

a: a proportionality constant measuring the weak electromagnetic attractive forces between the gas molecules.
b: molar volume of the gas molecules.

When going from a sufficiently chemically pure gas, to a realistic mixture: b (molar volume) isn't too difficult -- but a poses some interesting questions.

Estimating Van der Waals constants from critical temperature and pressure

Phase change diagrams for a Van der Waals gas
NOTE: this page uses different notation : their N is my n, and their k is my R. (Confusing! I need the Boltzmann constant...).

For a sufficiently chemically pure gas: Let T_c be the critical temperature, P_c be the critical pressure, and V_c be the critical volume.

["Critical temperature" is what we've seen before. "Critical pressure" is the limit of the vapor pressure of the liquid as the liquid's temperature approaches the critical temperature from below. "Critical volume" is (similarly) the limit of the volume of the liquid as the liquid's temperature approaches the critical temperature from below. Another way of thinking about it is as the liquid is brought to its critical temperature from below, eventually the vapor (immediately after vaporization) will approach the liquid in density. The critical state then can be thought of as occuring when the vapor density equals the liquid density.]

Then, for a perfect Van der Waals gas, both Linus Pauling and the above link report that:

T_c = 8a/(27bR) P_c = a/(27b²) V_c = 3bn

Linus Pauling reports that, in practice, the critical temperature and critical pressure are much closer to the theoretical values than the critical volume, and that the pragmatic approximation to critical volume is usually close to 2.25bn. That means it is possible to estimate a and b from the critical temperature and critical pressure (a minor algebra exercise).

EXERCISE: The natural temperature, pressure and volume units for a Van der Waals gas are the critical temperature, critical pressure, and critical volume. That is: for a mole of gas [n=1], when the units are changed to T' := T/T_c, P' := P/P_c, and V' := V/V_c, the Van der Waals equation algebraically reduces to:

[P'+3/V'²][3V'-1] = 8T' The ' marks are to indicate changed coordinates, not first derivatives.

EXERCISE: In SI units, a is m³·K/mol. b is m³/mol, i.e. molar volume.

Most chemically pure gases have the same physical properties at the critical state when the units are normalized this way. This is called the Law of Corresponding States. It also suggests that any proposed improvements on the Van der Waals equation also have this property.

The Van der Waals equation as a Laurent series in V, i.e. power series in 1/V: the Boyle temperature

Phase change calculator
The Van der Waals gas equation can be rewritten as the leading coefficients of a Laurent series in V, when the number of moles n is set to 1: PV/(RT) = 1 + [b-a/(RT)]/V + b²/V² + ... We define a Boyle temperature T_Boyle as a temperature at which the "linear" term of the Laurent series is 0. T_Boyle is not useful for an ideal gas [no correction terms]. For a Van der Waals gas, an algebraic exercise indicates that T_Boyle = a/(bR).

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Classical Kinetic Theory of Gases: A Crash Review

Classical Kinetic Theory Survival 101

Noble gases i.e. monoatomic gases: Classical Kinetic Theory Survival 201

When volume and molecular interactions matter: Van der Waals Equation

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