Pressure exerted by an ideal gas according to kinetic theory of gases

Question

In my textbook and Wikipedia, I have observed that force exerted on a wall of the container by one molecule is taken into account. Such that $F=\frac{mu} {\Delta t}$ where ${\Delta t}=\frac{2l}{u}$. But this change in time is the time required for a molecule to move from one wall to the opposite. In a gas container, each gas molecule doesn't get to move this freely. Then why do we assume ${\Delta t}=\frac{2l}{u}$? Is it that the molecules remain in random motion and tends to maintain constant density all over the place for which the statistical value of $\Delta t$ turns out to be the same?

Another small question, were polyatomic molecules also considered as one sphere each in the kinetic theory of gas? Or was it each atom resembled a sphere but not a molecule?

Answer 1

Your answer to the first question is correct.

We assume individual atoms travel the whole way across. This is not true. They collide some (although much less often than you’d think; try creating two jets of air from fans and notice they seem to have no effect on each other. Like point one at your face, and then bring in another going sideways across that first stream, but not hitting you. No change is noticed, as if they don’t interact at all.)

But if the collisions are perfectly elastic and no energy is lost during collision, the average of the entire situation is the same as each particle having to make the whole journey.

The answer to the second question is that we model each molecule as a sphere; we don’t try to make shapes of atoms connected as spheres. More accurately, we don’t consider shape here. That was just for illustration. The only sense in which we pretend it’s a sphere is that the colliding particle is returned along the impact line, but that is it. We assume a point and mass and elastic collision. But to answer, the unit of analysis is a molecule not an atom.

Answer 2

Addition to Al Browns answer.

The change in momentum would be given by $$F\Delta t=2mv_\text{rms}$$ or $$F= \frac{2mv_\text{rms}}{\Delta t}$$ where $v_{\text{rms}}$ is the root-mean-square velocity, usually given by $$v_{rms} = \sqrt{\overline{v}^2} = \sqrt{\dfrac{3k_BT}{m}}$$ where $\bar v$ is the average speed.

The time is given by $$\Delta t=\frac{2l}{v_\text{rms}}$$ where $l$ is the distance between each side of the container.

were polyatomic molecules also considered as one sphere each in the kinetic theory of gas? Or was it each atom resembled a sphere but not a molecule?

Yes. Each molecule is modelled as a hard sphere, and not a point particle, and all collisions are assumed to be elastic.