I believe I’ve already found an answer (below) to this long-standing question of mine. Let me walk you through the derivation.
First, let’s start with the following derivation for an ideal gas found in Chapter 40 – Clausius: the kinetic theory of gases in my book Block by Block. The Historical and Theoretical Foundations of Thermodynamics (2020).
Assumptions:
- fixed volume, isolated system of gas molecules
- gas is ideal – intermolecular forces are insignificant
- the force acting against the wall is caused by molecules hitting the wall, with the important variables being 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) in an infinitesimal time (dt). The molecules striking the wall during the time interval dt into the future will be those within the distance dx from the wall. Since vx = dx/dt, then this “striking distance” is equal to vx. The total number of molecules thus striking the wall during dt 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 vx.
Note the “x” subscript on the velocity. We’re only interested in the velocity of the molecules 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.
Comparing this result to the Ideal Gas Law,
PV = T x constant
indicates that the kinetic energy the molecules is proportional to absolute temperature. From this, one can calculate the relative speeds of molecules at room temperature (298K):
Oxygen = 482 m/s
Nitrogen = 515 m/s
Hydrogen = 1928 m/s
These values represent the translational speed of the molecules.
Now, with this calculation behind us, let’s return to the question motivating this page.
What happen to the above mathematics when a molecule strikes an inward moving piston? It comes away with a higher velocity. The faster the speed of the piston, the faster the molecule moves after impact. So given this, why isn’t the temperature rise during adiabatic compression dependent on the speed of the moving piston? We know that it isn’t based on the classical thermodynamic derivation (available in any introductory textbook) of the equation quantifying these process for a reversible ideal gas
T2/T1 = (V1/V2) γ – 1 = (P2/P1) (γ – 1)/γ
and PV γ = constant
where γ = Cp/Cv (heat capacity at constant pressure and volume, respectively)
But we don’t know why it isn’t.
So this is an interesting problem, right? The speed of the piston does influence the rebound speed of the striking molecules and yet doesn’t affect the final answer. Why?
The kinetic theory of gases and the classical thermodynamic equations quantifying adiabatic volume change for an ideal gas were developed during the mid-1800s. One would think that shortly thereafter, someone would have used the understanding of the former to bring a fundamental understanding to the latter. Such was not the case. I had a difficult time finding anyone who had resolved this; hence, the reason why this problem ended up on my MUQ list. But with further research I eventually found someone who did answer the question: Karoly Tettamanti, Polytechnical University, Budapest, in 1962. Wrote Tettamanti, “[it was a] surprising conclusion that a direct deduction of the functions of the adiabatic state, and of the equation of work, from a kinetic theory of gases, and based on the transfer of impulse, is not to be found in the literature of the past hundred years.” [1]
While his derivation is too lengthy to share here, Tettamanti showed that the temperature and pressure changes during reversible adiabatic compression of an ideal gas are dependent solely on the change in volume. While his work demonstrates that the speed of the piston plays no role, he didn’t draw attention to this fact. But if one pulls apart his derivation, one sees why this is so. While the inward moving piston increases the rebound speed of the striking molecules, it decreases the amount of time to arrive at a fixed final volume, i.e., a fixed value of V1/V2. So the molecules gain more energy but spend less time gaining that energy such that the total energy gained (due to the work of the piston) is the same, regardless of piston speed.
[See the complete derivation in Tettamanti’s paper. You can also ask me for a copy, if you’d like. I”ll gladly share it.]
Of note, I separately asked Grok3 to address this question and it came up with the same answer. I was concerned that it was making this answer up and so asked it to provide me a mathematical proof based on the kinetic theory of gases. And it did! This was an amazing experience for me to witness. Wrote Grok, “The mathematical proof using kinetic theory confirms that, while a faster piston imparts more energy per molecular collision, the total energy transferred to the gas is independent of piston speed because the collision frequency and compression time adjust inversely.” Fascinating!
References
[1] Tettamanti, K. “A KINETIC EXPLANATION OF ADIABATIC COMPRESSION”, Periodica Polytechnica Chemical Engineering, 6(3), pp. 139–148, 1962.