# # of gas particles hitting one face of a cubic container

1. Oct 10, 2016

### FranzDiCoccio

1. The problem statement, all variables and given/known data

1 mol of gas at temperature $T$ is contained in a cubic container of side $L$.
Estimate the number of collisions per second between the atoms in the gas and one of the walls of the cubic container.

My book gives this formula for that quantity
$$\frac{N_A}{6L}\sqrt{\frac{3 k T}{m}}$$

Where $N_A$ is Avogadro's number, $k$ is the Boltzmann's constant and $m$ is the atomic mass.

I tried two different lines of reasoning, and came up with the same answer, which is different from that of the book
$$\frac{N_A}{2L}\sqrt{\frac{ k T}{m}}$$

2. Relevant equations

a) Root mean square velocity

$$v_{rms}=\sqrt{\frac{3 k T}{m}}$$

b) average velocity along one direction:

$$\bar{v}_x=\frac{v_{rms}}{\sqrt{3}}=\sqrt{\frac{ k T}{m}}$$

3. The attempt at a solution

A) Via the momentum transferred to a wall in a time $\Delta t$.
The average number of atoms hitting a wall in a time $\Delta t$ is
$$\bar{N} = \frac{\bar{F} \Delta t}{2 m \bar{v}_x}=\frac{p L^2 \Delta t}{2 m \bar{v}_x}=\frac{p V \Delta t}{2 L m \bar{v}_x}=\frac{N_A k T \Delta t}{2 L m \bar{v}_x}=\Delta t \frac{N_A}{2L}\sqrt{\frac{kT}{m}}$$

B) Via the density of atoms.
The average velocity along direction x is $\bar{v}_x$. On average, half of the particles goes in the direction of the wall I'm interested in.
The average number of atoms hitting a wall in a time $\Delta t$ is half of the number of particles contained in a volume $\bar{V}=L^2 \Delta t \bar{v}_x$.
Thus
$$\bar{N} =\frac{1}{2} \bar{V} \frac{N_A}{V}=\frac{1}{2} L^2 \Delta t \bar{v}_x \frac{N_A}{L^3}= \frac{N_A}{2L}\sqrt{\frac{kT}{m}}$$

C) My book says "on average $N_A/3$ atoms hit the same face in the time $\Delta t$", and takes
$$\Delta t = \frac{2 L}{v_{rms}}$$.

The statement about 1/3 of the particles sounds wrong to me.
Can someone help with this? Shouldn't it take into account my formula b) above? Isn't the square velocity along one direction 1/3 of the square modulus of the velocity?

Thanks a lot
Franz

Last edited: Oct 10, 2016
2. Oct 10, 2016

Both answers are reasonably good=I think yours is closer to being right, and I think the book's is somewhat inaccurate. Both are in the same ballpark. I think I can offer an improvement, but it is only exact for collisionless gases: The effusion rate out of a small aperture $A$ is given by $dN/dt=(n \, \overline{v}/4)A$ where $n$ is the number of particles per unit volume. $\overline{v}=((8/\pi)kT/m)^{1/2}$ (mean speed) for a Maxwellian distribution. This gives $dN/dt=(N_A/4L)(8/ \pi)^{1/2} (kT/m)^{1/2}$. (Note: I have previously derived the effusion rate formula $( R=n \, \overline{v}/4)$from first principles for a collisionless gas (by performing a double integral in real space as well as velocity space to find the number of particles that have the necessary velocity to reach the aperture in time $\Delta t$, etc.), and it is exact. A little arithmetic gives my answer is approximately $dN/dt=(1/2.5)(kT/m)^{1/2}$. The book's answer you quoted has $\sqrt{3}/6=1/3.4$ (approximately), compared to your $(1/2)$ and my $(1/2.5)$.

Last edited: Oct 10, 2016
3. Oct 11, 2016

### FranzDiCoccio

Hi Charles,

and thanks a lot for your insight.

So what you're saying is "since both answers rely on a large number of approximations, they are equally good".
I think I see what you mean.

But what I thought is that the average number of collisions with a wall should be connected with the pressure. Specifically the pressure is related to the net force (or rate of momentum transfer) normal to the wall. So I think one should use the velocity along one direction $\bar{v}_x$ and not $v_{rms}$.
What I'm saying that, even at this approximate level, the book's solution does not seem to be consistent with the equation of state for an ideal gas...
There would be a coefficient $\sqrt 3$ that should not be there.
Right now I have no time to check that (school is starting in half an hour), I'll do that later.

As for your alternative, it is really interesting, and gives more insight, but it's definitely too difficult compared to the level of the book.

Thanks a lot again!
Franz

4. Oct 11, 2016

I didn't want to be too critical of the book, but I think a good textbook should provide the answer that I provided you. I think you could do an alternative derivation of the collision rate that gives the same answer I got by simply considering the x component of the velocity, but I'll need to give that one some thought. Meanwhile, the pressure is known to satisfy $PV=NkT=(2/3)N(1/2)mv_{rms}^2$.This makes $(1/2)mv_{x \, rms}^2=(1/2)kT$.

5. Oct 11, 2016

### FranzDiCoccio

Hi Charles,
your last sentence is what I'm referring to. I'm not sure the book's answer is consistent with that relation.

By the way, the book is for Italian high school students. That means 4th year, 17-18 year old people.
The book contains a standard derivation of kinetic theory, and what I'm discussing here is an exercise.
I found the derivation I'm referring to in the "cheatsheet" that the editor gives to the teachers.
It's just a few lines but, as I say, I'm not very convinced.

Thanks again, as soon as I have some time I'll also try to rethink the derivation to keep your suggestion into account.
Franz

6. Oct 11, 2016

Your $v_x$ proof for the exact answer would proceed as follows: You have a velocity distribution $g(v_x)=A(N_A/L^3)exp^{-v_x^2/(2kT/m)}$. At a distance x, particles will make it to the wall (assumes collisionless) in time $\Delta t$ if $v_x>x/(\Delta t)$ . Thereby $\Delta N$ in time $\Delta t$ is $\Delta N=L^2 \, \int\limits_{x=0}^{+\infty} \int\limits_{v_x=x/\Delta t}^{+\infty} g(v_x) \, dv_x \,dx$. If you know how to evaluate the double integral, I think you should be able to get the same answer with this integral that I previously obtained. I'm also going to evaluate it myself and see what I get. editing.. $A=\sqrt{m/(2 \pi kT)}$ and the order of integration can be changed on the double integral to get $\$ $I=\int\limits_{v_x=0}^{+\infty} \int\limits_{x=0}^{v_x \Delta t} g(v_x) \, d x \, d v_x$. The $dx$ integral just gives $v_x \Delta t$. The integral $\int\limits_{v_x=0}^{+\infty} v_x \, g(v_x) \, dv_x$ is straightforward. This results in $\Delta N=(N_A/L) (\Delta t) \sqrt{kT/(2 \pi m)}$ $\$ which is in agreement with with my first result. Please let me know if you have further questions about evaluating this last integral.

Last edited: Oct 11, 2016
7. Oct 11, 2016

### FranzDiCoccio

Ok, I see what you mean. Integrals are not a problem for me, but they are for the book's audience, for at least one more year or so.
Integrals are almost the last topic in 5th year mathematics, and even then most students struggle with the easy ones.
At this point the students in the Physics course have just met the concept of Maxwell distribution and its properties, but in general they are not given its explicit form, and anyway they won't make much of it, mathematically.

The exercise is meant to be solved without integrals. I do realize that, at that level, the approximation is so rough that one can't really say that one choice is better than another. Yet the book's derivation sounds wrong to me, even assuming that one really has a bunch of particles with velocity $v_{rms}$ and random directions.

Thanks again, this is useful.