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While reading The kinetic Theory of gases in Feynman Lectures , i was hoping for an explanation concerning the factor (2) that shows that only half of the atoms are headed towards a piston, while the other half is heading on other directions. Actually , also in Class , the professor , gave the same explanation , but i didn't quiet understand why , i imagine some (say 7 balls) thrown at the different time on a while with different velocities , i cannot imagine how the half of them are going directly towards the wall while half of them are going on other ways after hitting the wall. another question pop up to my mind ,when we say half of them , we are talking about number of molecules or atoms inside an enclosure , then , how come the half can be interpreted if the number of molecules is odd? I may wish for a profound explanation if it is possible , besides , a demonstration , i think probabilities will be needed and some approximations ,but truly this question is making me a bit nervous . Thank you . enter image description here

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  • $\begingroup$ The argument to get from $P=2nm v_{x}^2$ to $P=nm\langle v_{x}^2\rangle$ was stated nicely by @Jeffrey J Weimer. To get from $P=nm\langle v_{x}^2\rangle$ to $P= nm\langle v^2\rangle/3$, where $\langle v^2\rangle=\langle v_{x}^2\rangle+\langle v_{y}^2\rangle+ \langle v_{z}^2\rangle$ you have to account for all $3$ directions by dividing by $3$. In the end, you're dividing by $6$.This can be re-written as $P=\frac{2}{3} nm\langle v^2\rangle/2$ where $m\langle v^2\rangle/2$ looks like the kinetic energy. $\endgroup$ Commented Apr 24, 2019 at 21:06

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The half is taken statistically.

The gas law theory is based on statistical behaviour of a large set of molecules.

22.4 L of air contains at normal conditions about 6.023e+23 molecules.

For a large set of molecules, the probability of a positive/negative vertical component of the velocity of an atom is 0.5/0.5.

Half of atoms move upwards, half downwards, regardless of horizontal velocity components.

The same for any other direction line.

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Nothing profound is required to understand the physical system that is tied to the equations.

First, distinguish between unsteady state and steady state. The initial action of "throwing the balls at the wall" is unsteady state. Nothing should be said about this state with regard to pressure because the concentration of particles (number per volume) in this case is not uniform throughout the container. To appreciate this, realize that gas flows due to a concentration gradient, "throwing balls at the wall" is the equivalent of having gas flow only in one direction, and flow in one direction indicates an unsteady state gradient.

Using the last statement from above, recognize that a steady state conditions means, we will have no net flux of gas at the plane of the wall. The only way to have no net flux is to have equal amounts of gas moving to and from the plane at any instant in time. Amounts are normalized as concentration using numbers of particles per unit volume. The amounts move with a velocity to and from the wall. Concentration (number per volume) times velocity (distance per time) is flux (number per area per time). To have zero net flux at the wall, we either have zero concentration or zero net velocity. The former is nonsense. This means, net velocity is zero. Therefore, the number of particles that hit the wall at any instant must be equal to the number of particles that leave the wall at that same instant.

The apparent dilemma of half a particle is resolved by recognizing that half of a particle is striking the wall while half of some other particle is leaving the wall. Also, the flux expression deals with finite areas. The analysis for an infinitesimal area where perhaps only one particle or even a partial particle strikes is not amenable to this approach. As area goes to the limit of zero, the concept of pressure as a macroscopic measure of the system vanishes with it.

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Feynman's words might've meant the following; suppose that all particles in the volume $nv_xtA$ were moving towards the piston (let us say that the piston is on the right). If that was the case, wouldn't there be an empty region of space as all molecules in that volume moved towards the right? If you say that the neighbouring particles on the left (left of the volume under consideration) fill in this space, then who fills their space? If we follow this reasoning then it seems that the gas molecules have a net rightwards motion. We know this isn't true. A box filled with gas doesn't have extra pressure on any one face of the box.

Thus, to prevent this 'imbalance' or to maintain the randomness of motion of gas molecules, we can say that only half the molecules in that volume will hit the piston while the other half move towards the left. This causes no voids to be formed at any point in the box.

This is just an intuitive understanding as to why statistics gives us the half.

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