How Is Gas Pressure Derived in the Kinetic Theory of Gases?

[DorelXD]

Hello! It's been around two months since I started to learn about the knetic theory of ideal gases. But I haven't managed to completely understand it. What I don't understand is how we derive the formula for the presure that the gas molecules exerts on a wall's surface.

Here's what I've managed to understand; first we begin by studying a single molecule. The change in momentun when a molecule hits a wall is (in modulus): 2mv_x . So, the force exerted by a single molecule is: f=\frac{2mv_x}{\Delta t}. The total force exerted by all the molecules that collide with the wall within a time interval /Delta t is: F=\Sigma f=\Sigma \frac{2mv_x}{\Delta t}

Different molecules have different velocities. Ley n=\frac{N}{V} be the number of particles per unit of volume. n_1 molecules have a x-compomnent u_1, n_2 molecues have a x-component u_2, and so on. It's obvious that:

n=n_1+n_2+n_3+...n_i

Now, for a molecule to collide with a wall, it must get to the wall within the time interval \Delta t. So it must be within a distance v_x\Delta t. Given the fact that the area of the wall is A, for a molecule to colide with a wall it must be found in the volume determined by the area A and the distance v_x\Delta t. This volume is: v_x\Delta tA. The number of mlecules found in this volume is:n_iv_x\Delta tA, but only half of them will hit the wall, because of some fancy mathematics which I'll hopefully understand in a few years: \frac{1}{2}n_iu_i\Delta tA.

The force exerted on the wall by a certain category of molecules that have a certain velocity will be, \frac{n_i}{2}u_i\Delta tA times the force exerted by one molecule, \frac{2mu_i}{\Delta t}: n_iAmu_i^2;

The total force exerted by all the molecules will be:

F=\Sigma f=\Sigma f(u_1)+ \Sigma f(u_2)+...\Sigma f(u_i)=Am(n_1u_1^2+n_2u_2^2+...+n_iu_i^2)

Now, the average squared velocity on the x direction, \overline{u^2} is: \frac{n_1u_1^2+n_2u_2^2+...+n_iu_i^2}{n}=\overline{u^2}, so replacing the parentheses
we obtain: F=nAm\overline{u^2}

So far, so good. I understand that we need to replace the sum of that velocities with the average velocity because we can't find the value of each velocity, but I don't get what are we doing next. Everything I wrote before, I understood from a physics book. Now, after that, the book replaces n with \frac{N}{V}. I know that we defined to be like that, but It doesn't seem right to replace it. I mean, in our expresion for the force we did the sum for the molecules that hit the wall. Why all that molecules are contained in one unit of volume? I don't understand.

I'm sorry for my English and I hope you guys will help me to finally understand this theory. If you have another approach, I'm willing to listen. If you find anything wrong in what I said, or don't understand, please let me know. I really hope that you'll help me understand.

 

[voko]

The force exerted on the wall by a certain category of molecules that have a certain velocity will be, \frac{n_i}{2}u_i\Delta tA times the force exerted by one molecule, \frac{2mu_i}{\Delta t}: n_iAmu_i^2;

The total force exerted by all the molecules will be:

F=\Sigma f=\Sigma f(u_1)+ \Sigma f(u_2)+...\Sigma f(u_i)=\frac{N}{V}Am(n_1u_1^2+n_2u_2^2+...+n_iu_i^2)

This must be $$ F=\Sigma f=\Sigma f(u_1)+ \Sigma f(u_2)+...\Sigma f(u_i)=Am(n_1u_1^2+n_2u_2^2+...+n_iu_i^2) = nAm\overline{u^2} $$

Now, after that, the book replaces n with \frac{N}{V}. I know that we defined to be like that, but It doesn't seem right to replace it. I mean, in our expresion for the force we did the sum for the molecules that hit the wall. Why all that molecules are contained in one unit of volume? I don't understand.

The computation indeed took into account only the molecules that hit the wall within an arbitrary span of time. The it averaged the speeds of the molecules, and the result was that the force depends on the total number of molecules in a unit volume and the average speed of the molecules in the unit volume. I find it quite intuitive, but you seem to have a problem with that - can you describe what exactly you dislike?

 

[DorelXD]

This must be $$ F=\Sigma f=\Sigma f(u_1)+ \Sigma f(u_2)+...\Sigma f(u_i)=Am(n_1u_1^2+n_2u_2^2+...+n_iu_i^2) = nAm\overline{u^2} $$



The computation indeed took into account only the molecules that hit the wall within an arbitrary span of time. The it averaged the speeds of the molecules, and the result was that the force depends on the total number of molecules in a unit volume and the average speed of the molecules in the unit volume. I find it quite intuitive, but you seem to have a problem with that - can you describe what exactly you dislike?

First of all, thank you for your answer!

This must be

Indeed, my bad. I've just modified it.

I find it quite intuitive, but you seem to have a problem with that - can you describe what exactly you dislike?

I dislike the part where we replace n with frac{N}{V}. I don't know exactly why but it doesn't seem right. The span of time is indeed arbirtrary. I don't get it. I don't know how to put it exactly.

 

[voko]

I dislike the part where we replace n with frac{N}{V}. I don't know exactly why but it doesn't seem right. The span of time is indeed arbirtrary. I don't get it. I don't know how to put it exactly.

Without your being explicit about your concerns, it will be difficult to dispel them. Especially this one, where $n = N/V$ by definition.