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): [itex]2mv_x[/itex] . So, the force exerted by a single molecule is: [itex]f=\frac{2mv_x}{\Delta t}[/itex]. The total force exerted by all the molecules that collide with the wall within a time interval /Delta t is: [itex] F=\Sigma f=\Sigma \frac{2mv_x}{\Delta t}[/itex]

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

[itex]n=n_1+n_2+n_3+...n_i[/itex]

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

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

The total force exerted by all the molecules will be:

[tex]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)[/tex]

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

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 [itex]n[/itex] with [itex]\frac{N}{V}[/itex]. I know that we defined to be like that, but It dosen'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, [itex]\frac{n_i}{2}u_i\Delta tA[/itex] times the force exerted by one molecule, [itex] \frac{2mu_i}{\Delta t}[/itex]: [itex]n_iAmu_i^2[/itex];

The total force exerted by all the molecules will be:

[tex]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)[/tex]

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 [itex]n[/itex] with [itex]\frac{N}{V}[/itex]. I know that we defined to be like that, but It dosen'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 [itex]n[/itex] with [itex]frac{N}{V}[/itex]. I don't know exactly why but it dosen'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 [itex]n[/itex] with [itex]frac{N}{V}[/itex]. I don't know exactly why but it dosen'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.

 

[Nickie]

Hello! Thank you for sharing your struggles with the Kinetic Theory. It is a complex theory and it is understandable that you may need some clarification. I will try my best to explain it in a simpler way.

The Kinetic Theory is based on the idea that gases are composed of tiny particles (molecules) that are in constant motion. When these molecules collide with each other or with the walls of their container, they exert a force. This force is what we call pressure.

To understand how this pressure is derived, we first need to understand the concept of momentum. Momentum is the product of an object's mass and its velocity. When a molecule collides with a wall, its momentum changes. This change in momentum is what causes the molecule to exert a force on the wall.

Now, let's look at the formula for pressure that you mentioned: F=nAm\overline{u^2}. This formula tells us that the pressure exerted by a gas is directly proportional to the number of molecules (n), the area of the wall (A), and the average squared velocity (\overline{u^2}). The average squared velocity is important because it takes into account the fact that molecules in a gas have different velocities. By taking the average of their squared velocities, we are able to get a more accurate measure of the force they exert.

To address your confusion about replacing n with \frac{N}{V}, let's first define these variables. n is the number of molecules per unit volume (density) and N is the total number of molecules in the gas. V is the volume of the gas. Now, in order to calculate the total force exerted by all the molecules, we need to take into account all the molecules in the gas, not just the ones that hit the wall. That is why we use N instead of n in the formula.

I hope this explanation helps you better understand the Kinetic Theory. If you have any further questions or need more clarification, please do not hesitate to ask. As scientists, it is important for us to have a deep understanding of the theories we study, so never be afraid to ask questions and seek clarification. Best of luck in your studies!