# Relationship between pressure and volume using kinetic molecular theory

### How does kinetic molecular theory explain gas laws? | Socratic State the postulates of the kinetic-molecular theory; Use this theory's postulates to the relationships between molecular masses, speeds, and kinetic energies with (b) When volume decreases, gas pressure increases due to increased. The assumptions behind the kinetic molecular theory can be illustrated with Thus, the pressure of a gas becomes larger as the volume of the gas becomes smaller. This relationship eventually became known as Graham's law of diffusion. In order to tackle this question a general understanding of the the Kinetic Molecular Theory is required. The theory is based on assumptions of the.

A portion of the energy of the ball is lost each time it hits the floor, until it eventually rolls to a stop. In this apparatus, the collisions are perfectly elastic. The balls have just as much energy after a collision as before postulate 5.

Any object in motion has a kinetic energy that is defined as one-half of the product of its mass times its velocity squared.

Kinetic molecular theory of gases - Physical Processes - MCAT - Khan Academy

When we increase the "temperature" of the system by increasing the voltage to the motors, we find that the average kinetic energy of the ball bearings increases postulate 6. The kinetic molecular theory can be used to explain each of the experimentally determined gas laws. The Link Between P and n The pressure of a gas results from collisions between the gas particles and the walls of the container.

Each time a gas particle hits the wall, it exerts a force on the wall.

## How does kinetic molecular theory explain gas laws?

An increase in the number of gas particles in the container increases the frequency of collisions with the walls and therefore the pressure of the gas. Amontons' Law P T The last postulate of the kinetic molecular theory states that the average kinetic energy of a gas particle depends only on the temperature of the gas. Thus, the average kinetic energy of the gas particles increases as the gas becomes warmer.

Because the mass of these particles is constant, their kinetic energy can only increase if the average velocity of the particles increases. The faster these particles are moving when they hit the wall, the greater the force they exert on the wall. Since the force per collision becomes larger as the temperature increases, the pressure of the gas must increase as well.

If we compress a gas without changing its temperature, the average kinetic energy of the gas particles stays the same. There is no change in the speed with which the particles move, but the container is smaller. Thus, the particles travel from one end of the container to the other in a shorter period of time. This means that they hit the walls more often. Any increase in the frequency of collisions with the walls must lead to an increase in the pressure of the gas.

Thus, the pressure of a gas becomes larger as the volume of the gas becomes smaller. Charles' Law V T The average kinetic energy of the particles in a gas is proportional to the temperature of the gas.

Because the mass of these particles is constant, the particles must move faster as the gas becomes warmer. If they move faster, the particles will exert a greater force on the container each time they hit the walls, which leads to an increase in the pressure of the gas. If the walls of the container are flexible, it will expand until the pressure of the gas once more balances the pressure of the atmosphere.

The volume of the gas therefore becomes larger as the temperature of the gas increases. Avogadro's Hypothesis V N As the number of gas particles increases, the frequency of collisions with the walls of the container must increase. This, in turn, leads to an increase in the pressure of the gas. If the walls are cooler than the gas, they will get warmer, returning less kinetic energy to the gas, and causing it to cool until thermal equilibrium is reached.

## Basics of Kinetic Molecular Theory

We will have more to say about molecular velocities and kinetic energies farther on. Molecular Speed Although the molecules in a sample of gas have an average kinetic energy and therefore an average speed the individual molecules move at various speeds.

Some are moving fast, others relatively slowly Figure 5. At higher temperatures at greater fraction of the molecules are moving at higher speeds Figure 3. What is the speed velocity of a molecule possessing average kinetic energy? What is the difference between the average speed and root mean square speed of this gas?

Boyle's law is easily explained by the kinetic molecular theory. The pressure of a gas depends on the number of times per second that the molecules strike the surface of the container.

If we compress the gas to a smaller volume, the same number of molecules are now acting against a smaller surface area, so the number striking per unit of area, and thus the pressure, is now greater. Kinetic explanation of Charles' law: Kinetic molecular theory states that an increase in temperature raises the average kinetic energy of the molecules.

If the molecules are moving more rapidly but the pressure remains the same, then the molecules must stay farther apart, so that the increase in the rate at which molecules collide with the surface of the container is compensated for by a corresponding increase in the area of this surface as the gas expands.

Kinetic explanation of Avogadro's law: They have certain speeds and if you knew those speeds and you knew the distribution of speeds and the positions in here, you could figure out these macroscopic properties.

What I wanna basically do in this video is try to figure out what is the relationship if we know the microscopic properties, how could we predict the macroscopic properties, like if I knew the speed of all these molecules, how could I figure out what pressure would be in there or vice versa, if I knew the temperature of the gas, could I say what the average speeds are of these molecules in this gas.

That's what we're gonna do, but first, we have to make a few assumptions. One assumption is that these molecules don't really interact and if they do interact, it would only be because of the collision and if there is a collision between these molecules, we have to assume it's elastic and kinetic energy will be conserved, the momentum will be conserved.

Similarly, if one of these molecules strikes the wall of the container and has a collision there, that should also be elastic. There should be no kinetic energy lost. So then let's get to it. Let me just clean that up, get rid of that and start over up here. I need to figure out how to relate a microscopic quantity to a macroscopic quantity. Let's just start with speed. Let's say you got a particle in here, a molecule moving this way with some speed. I'll call it v x since I'm drawing it in the x direction and it collides with this wall, well, that's gonna impart a force on this wall and if you get a lot of these doing that in here, you'll get a pressure on this wall, but it's gonna be an elastic collision, so this particle is gonna bounce backwards with the same velocity and let's try to figure out what force that would exert 'cause if I can figure out the force on the wall, I can figure out the pressure 'cause pressure's just force per area.

So Force, yeah, it equals m a, but it also equals delta P, the change in momentum over the change in time. So this is an alternate way to write Newton's Second Law. What would be the change in momentum? So I'm gonna try to find the force on this wall, the change in momentum, momentum is m v, and if the mass doesn't change, then change in momentum is just m delta v where the V here is speed. So mass times, sorry, excuse me, velocity. Mass times change in velocity. So what would be the change in velocity for this collision right here, struck the wall and bounced back with the same speed?

Some people wanna say zero 'cause it comes in with the same speed that it goes out with, but V is velocity and so, the change in velocity is actually two times V because it came in with V and left with negative V. So technically, the change would be negative two V, but I'm gonna ignore negatives 'cause I just want the size of this force on this wall.

So m times two times v x over delta t but I don't want delta t in here. I want an equation of state that just has pressure and volume and speeds and stuff like that. So how can I get rid of delta t? Well I know the distance in here, let's just call the side lengths here L.

### The Kinetic Molecular Theory

Well, the time it takes between collisions, so there's an impulse here and delta v right when this collision takes place and then this particle travels over here to the left, bounces off at this wall then comes back over to here, again, hits it. How long is it between those impacts? Well, the time it would take to travel to the left and back, I know speed is distance per time, so the time, the delta t, is just gonna be the distance per speed and the distance is not just L 'cause it's gonna travel to this wall and then back.

I wanna know the force on this wall over here. I need to figure out how long is it between collisions with this wall, so it's gonna be two times L over the velocity in the x direction.

That's where I can substitute in over here and I get that F is gonna be m times two v x over delta t now is two L over v x but since I'm dividing by v x on the bottom, I moved that up top and look, I've already got one here, so I'm just gonna square it. I can cancel off the twos and I get that the force on the wall by this particle is mass times its velocity in the x direction squared divided by L. I should say this particle doesn't have to just be going in the x direction. It might have some total velocity this way where the x component is just a part of it, but it I just took the x component of the particle speed, whatever particle it is that had some velocity, the x component, I'd get the force contribution to the pressure on this wall over here.

So this is the force on this wall over here by one particle, but I wanna know the force from all the particles 'cause I wanna get the total pressure, so how could I do that?

Well, if I want the total force, I just need to add up the contributions from all the particles. So let's say there were other particles, well they're gonna have the same mass m. I'm assuming they got the same gas throughout. All molecules have the same mass and the L will be the same for all of them. So, the only difference in contribution will be that some may have a certain component of velocity in the x direction. I'll call this v x one squared plus there may be some other particle that has a different component, two, and there may be some particle that has a different component, three. You just have to add all these up. So I have x v two, the two references particle two squared plus v x three, the x component of particle three's, velocity squared plus, I'd keep going to N many times.