This lecture continues the topic of thermodynamics, exploring in greater detail what heat is, and how it is generated and measured. The Boltzmann Constant is introduced. The microscopic meaning of temperature is explained. The First Law of Thermodynamics is presented.
So, what is it you have to remember from last time? You know, what are the main ideas I covered? And in the end we agreed to use the absolute Kelvin scale for temperature. And the way to find the Kelvin scale, you take the gas, any gas that you like, like hydrogen or helium, at low concentration, and put that inside a piston and cylinder. Remember now, this is in Kelvin. So, somewhere here will be the boiling point of water, somewhere the freezing point of water; that may be the boiling point of water.
But they will always be straight lines if the concentration is sufficiently low. In other words, it appears that pressure times volume is some constant.
Say c , times this temperature.
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And you can use that to measure temperature because if you know two points on a straight line then you know that you can find the slope and then you can calibrate the thermometer, then for any other value of P times V that you get, you can come down and read your temperature.
I can use one; you can use another one. People in another planet who have never heard of water—they can use a different gas. But all gases seem to have the property that pressure times volume is linearly proportional to this new temperature scale, measured with this new origin at absolute zero.
The next thing I mentioned was, people used to think of the theory of heat as a new theory.
You know, we got mechanics and all that stuff—levers and pulleys and all that. Then we defined specific heat, law of conservation of this caloric fluid that allows you to do some problems in calorimetry.
You mix so much of this with so much of that, where will they end up? That kind of problem.
So, that promoted heat to a new and independent entity, different from all other things we have studied. But something suggests that it is not completely alien or a new concept, because there seems to be a conservation of law for this heat, because the heat lost by the cold water was the heat gained by the hot water.
Heat gained by the cold water was the heat lost by the hot water. So, you have a conservation law.
Secondly, we know another way to produce heat. Take two automobiles; slam them. The question is what happened to the kinetic energy of the two cars?
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That is really gone. That was our legal way out of the whole issue. But you realize now this caloric fluid can be produced from nowhere, because there was no caloric fluid before, but slamming the two cars produce this extra heat. So, how do you do the conversion ratio? You know, how many calories can you get if you sacrifice one joule of mechanical energy?
So, Joule did the experiment. Not with cars. He had this gadget with him, which is a little shaft with some paddles and a pulley on the top, and you let the weight go down. Something will be missing. Keep track of the missing amount. So many joules—but meanwhile you find this water has become hot.
You find then how many calories should have gone in, because we know the specific of water, we know the rise in temperature, we know how many calories were produced. And then, you compare the two and you find that 4. So, that is the conversion ratio of calories to joules. One joule, 4. If the car was made of just one material, it had a specific heat, then it would go up by a certain temperature you can actually predict.
So today, I want to go a little deeper into the question of where is the energy actually stored in the car, and what is heat. We just said car heats up and the loss of joules divided by 4. Now, we can answer in detail exactly what is heat. When we say something is hotter, what do we mean on a microscopic level? So, how does that come into play?
PHYS 200: Fundamentals of Physics I
For that, I want you to take the simple example where the temperature enters. That is in the relation PV equal to some constant times temperature. We want to ask what is this proportionality constant. Suppose you were doing this. In the old days, this is what people did.
What did we think should be on the right-hand side? What is going to control this particular constant for the given experiment?
Do you know what it might be proportional to? Professor Ramamurti Shankar : Amount of gas.
By what metric? Professor Ramamurti Shankar : Right. Suppose you were not aware of particles. Professor Ramamurti Shankar : The mass. Now, if you guys ever said moles, I was going to shoot you down.
We are trying to deduce that. So, put yourself back in whatever stone ages we were in. Mass would be a reasonable argument, right? We know that if you have some amount of gas producing the pressure, and you put twice as much stuff, you would think it will produce twice as much pressure. Same reason why you think the expansion of a rod is proportionally change in temperature times the starting length.
So, put the mass that you had there. Then, you should put a constant still.
This is actually correct. But here is what people found. Somebody else puts in helium gas. So, each gas has a different constant. In other words, one gram of hydrogen and one gram of helium do not have the same pressure.
In fact, one gram of helium has to be divided by 4 to find its effect on pressure. So, you have to think about why is it that the mass directly is not involved. Mass has to be divided by a number, and the number is a nice, round number.
So, if things, like, carbon, as atoms, weigh 12 times as much as things called hydrogen, then if you took some amount of carbon, you divide it by a number, like, 12 to count the number of carbon atoms.
Okay, so hydrogen you want to count the number of hydrogen atoms. So then, what really you want here is not the mass, but the number of atoms of a given kind.
We are certainly free to write either a mass or the number of atoms, because the two are proportional.
But the beauty of writing it this way, you write it in this fashion, by this new constant k , k is independent of the gas. You can write it in terms of mass. TThen once you divide it by the number you can put a single constant in front. Or if you want a universal constant, what you should really be counting is the number of atoms or molecules. So, this is called the Boltzmann Constant.
The Boltzmann Constant has a value of 1. So, this is a universal constant. Take some random group gas. The number you will put in there is some number like 10 23 or 10 So, whenever a huge number is involved, what you try to do is to measure the huge number as a simple multiple off another huge number, which will be our units for measuring large numbers.
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For example, when you want to buy eggs, you measure in dozens. When you want to buy paper, you might want to measure it in thousands or five hundreds or whatever unit they sell them in.
When you want to find intergalactic distances, you may use a light year. You use units so that in that unit, the quantity of interest to us is some number that you can count in your hands. So, a mole is like a dozen. We wanted to buy 6 times 10 23 eggs, you will say get me one mole of eggs.