[The below derivation of the ideal gas law by Rudolf Clausius with commentary is excerpted from Chapter 40 in my book, Block by Block – The Historical and Theoretical Foundations of Thermodynamics]
The math behind the kinetic theory of gases
So let’s now look at how Clausius started this journey. I’ll walk you through the essence of his logic by modifying his text with some of Feynman’s own explanations.[1]
Clausius started by assuming the gas to be ideal, meaning that only a small space relative to the whole is filled by molecules and that intermolecular forces are insignificant. He also assumed the molecules to all have the same velocity even though he realized that this was not actually true, writing “There is no doubt that actually the greatest possible variety exists amongst the velocities.”[2] This assumption served to greatly simplify the mathematics and so enabled Clausius to reach an answer and test his hypothesis. (Note: Velocity and speed are used somewhat interchangeably in the literature. Going forward I will use velocity to mean speed unless otherwise stated.)
Clausius’ first goal was to determine the relationship between the translational motion of the molecules and the pressure of the gas. Pressure was a well-understood property to start with, better understood than temperature as it readily lent itself to a mechanical calculation. The force acting against the wall is caused by molecules hitting the wall. The important variables to consider are the mass and velocity of the molecules and then the rate at which they hit the wall.
For a single molecule and a single hit, the force generated against the wall is the change in momentum of the molecule. If the collision is perfectly elastic and the molecule is aimed directly at the wall, the change in momentum of the molecule is
Before: mv1
After: mv2
But we know that for elastic collisions v2 equals –v1 and so the change in momentum of the molecule is simply 2mv1. From Newton’s 2nd Law of Motion,
Force = rate of change of momentum
= d(mv)/dt
= (change in momentum of single molecule) x
(rate of collision of molecules against wall)
= 2mv x (rate of collision of molecules against wall)
To calculate the collision rate, we need to know how many molecules simultaneously strike the wall at each moment in time. Assume the molecules travel an infinitesimal distance (dx) for an infinitesimal time (dt). The molecules striking the wall dt into the future will be those dx distance away from the wall. In other words, a range of molecules with different velocities will strike at the same moment in time if their relative distance from the wall is proportional to their relative velocity.[3] The total number of molecules thus striking the wall is equal to the density of the gas (N/V) times that volume of the gas that will strike the wall, which is the area of the wall (A) times the “striking” distance, which as discussed above is vx. Note the “x” subscript on the velocity. We’re only interested in the velocity of the molecule in the direction of the wall. Since the molecules move equally in all three x, y and z directions—the system as a whole is stationary—the square of the mean velocities in each direction are equal to each other and thus equal to the square of the mean velocity. But we need to calculate the total number of molecules moving in just one of these three directions, specially the direction aimed directly at the wall. This number is one-third of the total, and actually one-half of this number since the other half is heading directly away from the wall. So let’s see what we have:
Force = (2mvx) (1/2) [(N/V) x (Avx)]
= A (N/V) mvx2
= A (N/V) 1/3 mv2
where “v” now represents the mean velocity. Re-arranging some gives us
Pressure = Force/Area = (N/V) x (2/3) x (1/2 mv2)
PV = 2/3 N (1/2 mv2)
for which the last term, ½ mv2, is naturally the kinetic energy of the gas molecule moving at the mean velocity.[4] Clausius arrived at this answer from a more circuitous pathway involving angles of collisions.
* * * * *
This result can be carried one step further. Comparing PV = (2/3)N(½mv²) with the ideal gas law PV = nRT = NkT, wherein n = number of moles, R = gas constant, N = number of individual particles, and k = Boltzmann constant, the two right-hand sides must be equal:
(2/3)N⟨½mv²⟩ = NkT
⟨½mv²⟩ = (3/2)kT
This is the result that ties temperature directly to translational kinetic energy. It is also the relation used throughout this book whenever internal energy, heat capacity, or temperature itself is discussed at the atomic level.
* * * * *
Once Clausius arrived at this equation, he compared it against the available experimental data available in the form of the gas laws of Boyle (PV = k at constant T), Gay-Lussac (P = kT at constant V, V = kT at constant P) and Clapeyron who tied the two together in the Ideal Gas Law,
PV = T x constant
and concluded that the kinetic energy, or vis viva as he referred to it, of the molecules is proportional to absolute temperature. He then took this one step further by calculating the densities (V/Nm) at a given pressure for a range of ideal gases, which enabled determination of their mean velocities:
Oxygen = 461 m/s
Nitrogen = 492 m/s
Hydrogen = 1844 m/s
These values represent the translational speed of the molecules. Clausius next sought to quantify the relative proportion of total energy (CvT) caused by translational motion (1/2 mv2), referencing Rankine’s view that Cv represents the “true specific heat.” The unstated assumptions behind this logic was that U is a function of T only and that Cv is a constant that quantifies the total energy associated with thermal motion—translational, rotational, vibrational—in a gas. This was one of the early versions of the equipartition theory, a concept we’ll address in more detail later.
Translational Energy = N ½ mv2 = 3/2 PV
Total Energy = CvT = Cv PV/R
Translational Energy / Total Energy = 3/2 R/Cv = 3/2 (γ – 1)
where
R = Cp – Cv
and
γ = Cp/Cv
for which the quantity γ is a useful number to have in certain calculations involving gases.[5]
At the end of his paper, seeking to gain insight into the structure of atoms, Clausius inserted some heat capacity data obtained by Regnault for air and arrived at
Translational Energy / Total Energy = 0.63
and thus concluded that translational motion alone does not account for the total energy of motion contained in an ideal gas, in this case air. As he wrote, “We must conclude, therefore, that besides the translatory motion of the molecules as such, the constituents of these molecules perform other motions, whose vis viva also forms a part of the contained quantity of heat.”[6]
References
[1] Feynman, Richard Phillips, Robert B. Leighton, Matthew L. Sands, and Richard Phillips Feynman. 1989a. The Feynman Lectures on Physics. Volume I. Mainly Mechanics, Radiation, and Heat. Vol. 1. The Feynman Lectures on Physics 1. Redwood City, Calif.: Addison-Wesley. Chapter 39, The Kinetic Theory of Gases.
[2] Clausius, R. 2003. “The Nature of the Motion Which We Call Heat.” In The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary, edited by Stephen G. Brush and Nancy S. Hall, 111–34. History of Modern Physical Sciences 1. London : River Edge, NJ: Imperial College Press ; Distributed by World Scientific Pub. pp. 126-127
[3] Note that this calculation is based on continuous hitting of the wall by waves of molecules. The calculation is independent of time. Regardless of the time chosen, the calculation tells you how many molecules are striking the walls at any given moment in time. The change in momentum is thus continuously applied.
[4] Note that the force balance involves change in momentum, not kinetic energy. Kinetic energy falls out of the mathematics from the time element: total momentum change equals change per atom x number of atoms striking at one moment in time; the additional velocity component comes in with this last variable.
[5] γ = Cp/Cv is a useful ratio because 1) it is independent of units used, 2) it tells you something about the complexity of the molecules, and 3) it appears in calculations involving adiabatic processes, including sound. See (Brush, 2003a) p. 13.
[6] Clausius, p. 134.
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