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First, let’s talk a little more about molecular speeds. Just to jog your memory, last week we determined the average translational kinetic energy of a gas molecule and the average kinetic translational kinetic energy per mole of gas. Those equations were, respectively,

and

We can use those equations to find what is called the root-mean-square speed, or the rms speed:

This value follows directly from our equations for the translational kinetic energy. As you can see, at any given temperature T, gas molecules of different masses can have the same kinetic energy but different rms speeds. For example, nitrogen molecules have a molar mass of 28 g/mol. Oxygen, by contrast, has a molar mass of 32 g/mol. From our rms speed equation, we find that nitrogen molecules in the air move more quickly than the oxygen molecules. This also explains why the most abundant element in the universe, hydrogen, is almost completely absent from Earth’s atmosphere. Hydrogen has a molar mass of only 2 g/mol. According to our rms speed equation, this means that hydrogen molecules move very, very fast. Faster, in fact, than the escape velocity of Earth. So hydrogen molecules escape into space, while heavier molecules like nitrogen and oxygen are trapped. Science!

The last topic in this section on the kinetic-molecular model of ideal gases has to do with collisions between molecules. So far we’ve been thinking about gas molecules hitting the wall of a container. However, gas molecules frequently hit each other. It’s true! So, you know…let’s think about that.

For this thought exercise we can’t consider gas molecules as point particles. If they were points, they would never collide. We need to think of these molecules in more realistic terms. Let’s assume that molecules are rigid spheres with a radius r. There are N number of these spherical molecules in some volume V. Let’s assume, for now, that only one molecule is moving. When this moving molecule collides with another molecule, the distance between the tow centers is 2r.

Now, let’s draw a cylinder with a radius 2r with its access in the direction of the velocity of the moving molecule. Let’s say that the moving molecule collides with a stationary molecule, whose center is inside our cylinder. During a very short interval of time that we’ll call dt, a molecule traveling with a speed v with travel a distance v dt. During this very short time interval, the moving molecule collides with any molecule in a cylinder with a volume of radius 2r and length v dt, or

There are a certain number of molecules per volume (N/V), so the number of molecules in our cylinder, denoted by dN, is

To find the number of collisions per unit of time, we get the differentials on the same side of the equation:

This equation assumes that only one molecule is moving, which isn’t very realistic. It turns out, though, that all we have to do is multiply the above equation by the square root of two to model more frequent collisions:

From this equation, we can find the average time between collisions. Just flip the above expression! This is called the mean free time:

We can use this mean free time to find the average distance traveled between collisions, called the mean free path. All we have to do is multiply the velocity by the mean free time:

We denote the mean free path by the Greek letter lambda.

Notice that the mean free path is inversely proportional to the number molecules per volume as well as inversely proportional to the cross-sectional area of a molecule. That means the more molecules there are and the bigger those molecules are, the shorter the mean distance between collisions. It doesn’t, however, depend at all on the velocity of the molecules.

The mean free path can also be related back to the ideal gas equation. Remember, the ideal gas equation is pV = NkT. So the mean free path written in terms of the ideal gas equation ends up being

Think about what this means. If the temperature is increased at a constant pressure, the gas expands and the average distance between the molecules increases, thus the mean free path increases. If the pressure is increased at a constant temperature, the gas is squeezed together and the mean free path decreases. Just like we would expect.

We’ve done it! We’ve completed the section on the kinetic-molecular model of ideal gases! Good job, everyone. Next week we’ll start discussing something called specific heat.

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Regardless, there is strain between Mystery and Katya. Mystery deals with it by bringing home various other women, but it is Katya who is blameworthy for hitting on other men in the house. After Mystery gives Herbal permission to go ahead and sleep with Katya (literally, permission is the term Style uses. Ick.), Mystery decides he really is in love with Katya after all and has the mental health episode which happened back in Step 1. But in this retelling we get what Style seems to think is a large and horrifying reveal, that is, Mystery gets a prescription for pills that are labeled as being for the treatment of schizophrenia. I think we’re supposed to be horrified by this, and I am a little horrified that we’re supposed to be horrified. It’s like the Lovecraft tale in which the horrifying reveal is that the woman who was murdered and buried in the basement and now her hair is crawling around murdering people was really “a negress.” Mental health should not be a horrifying reveal any more than race or ethnicity should be. Besides which, I have a vague recollection from a gen ed psych 101 class of the professor remarking that schizophrenia is the most common diagnosis given to people at a first diagnosing who don’t have symptoms that seem to be clearly anything else, but even if I were convinced Mystery was schizophrenic, and given that he is not under regular care, and if he were I probably shouldn’t know the details, who cares exactly? Based on Style’s descriptions of Mystery, I think he is a horrible person, but I also think he needs help. Ideally from people who actually care about other people, which would not so much be Style.

The by-now mandatory name-dropping subplot is about Courtney Love, who has moved into the mansion and whom everyone admires because she doesn’t give a fuck. Her habits of living without anything regarding hygiene, to the point that she doesn’t even twist open bread bags, she just rips them apart at the top, is seen as just a symptom of this so very admirable trait of not giving a fuck. Style meets a musician in one of her bands and wishes he could meet a “girl” like that at the clubs he frequents. I suppose he could try to meet women rather than girls and targets and victims, and possibly go places other than clubs and stop seeing sex as the only possible end goal for interactions between men and women, but those things might require some revision to his pronouncements on the natures of men and women, and then he might start thinking of people as people with varying personalities that don’t map neatly onto gendered categories, and that might make for interesting writing with sympathetic characters. We can’t have that.

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Let’s think about the absolute value of the velocity in the x-direction from last time. That value is not actually the same for all molecules, as you may have guessed. However, we could separate all the different velocities and determine their effect on the pressure. But in doing so, we’re basically replacing the square of the x-velocity in the pressure equation with the square of the average x-velocity. This average value is related to the speeds of the molecules. We know that

We can find the average of this relationship over all molecules in a similar way:

It turns out, though, that in our model there is no difference between the x-, y-, and z-directions. So the square of the velocity is can be represented as three times the velocity in the x-direction. Or, if we move the terms around a little bit:

So, instead of using the square of the velocity in our equation for pressure, we can use this. The equation becomes

Notice that term in the brackets. Look familiar? Of course it does! That’s the average translational kinetic energy! This term, multiplied by the total number of molecules N, is the total random kinetic energy of the the molecules’ translational motion. That means that we can write the above equation in an even simpler way:

That’s not all! Remember the ideal gas equation? You know, pV = nRT? If we set these two pV equations equal to each other and solve for the kinetic energy, we get

And just like that, we have the equation for the average translational kinetic energy for a given number of moles of an ideal gas. What this equation shows us that the kinetic energy is directly proportional to the temperature T.

We’re still not done, because we can look at this whole kinetic energy situation in a slightly different way. If the last equation shows the total translational kinetic energy per mole, we can do some mathematical manipulation to get an equation that shows the translational kinetic energy per molecule. Let’s get to work.

To find the average translational kinetic energy of a single molecule, we need to divide the total translational kinetic energy by the number of molecules N. In math, that looks like this:

Remember that we can find the total number of molecules N by multiplying the number of moles n by Avagadro’s number:

We can work this into our equation and get

Stay with me because this is about to get much simpler. The ratio R/N_A is often referred to as the Boltzmann constant k. It’s equal to 1.38 x 10^-23 J/molecule · K. That means that we can write this equation like this:

This is the average translational kinetic energy for a single molecule. Like the per mole situation, the translational kinetic energy is directly proportional to the temperature. Neat, right?

One last thing: We can use the Boltzmann constant k to write the ideal gas law in an alternative form. We know that

and

We can use these to right the ideal gas equation in a different way:

That’s all for now. More on the kinetic-molecular model of ideal gases next week!

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