In lecture, we're going to continue our study of the physical properties of gases, building on the material and concept development study number 14. This will also build upon the material from lecture one, in which we studied the relationship between pressure of the gas and the volume of gas for fixed samples of gas at fixed temperatures. You'll recall that what we observed experimentally was something called Boyle's Law. Boyle's Law told us that the pressure is inversely related to the volume of the gas for a fixed sample of gas at a fixed temperature. If we vary the sample of the gas, we vary the proportionality. That's most commonly written in the form that pressure times volume is a constant, as we've written here. But we know that, that constant depends upon what the sample of the gas actually is. What though, if we were to change the temperature of the gas? That's actually the subject for today's lecture. To study the variation of the properties of the gases we change the temperature. We're going to use the same apparatus that we used before. This is just a cylinder, fitted with a piston. But in this particular case, what we're going to do is now apply a constant pressure to the piston, to the ex, to the end of the piston here. And in doing so, we're going to hold the pressure of the gas itself constant. The pressure of the gas has to equal the applied pressure outside, otherwise the piston will move. So we fix the pressure of the gas and we're going to vary the temperature of the gas and measure that variation with a thermometer on the end of the apparatus. We could vary the temperature of the gas by for example heating the cylinder or by immersing the entire cylinder into an ice bath and taking various temperature measurements. What we'll discover is that the volume of the gas trapped inside here varies as we vary the temperature. We'll collect that volume versus temperature data and this particular set of data looks something like this. For one particular trapped sample, and one particular pressure we get, oh, a half a dozen or so data points and if we draw a line through those data points what we discover is that beautifully, they form an almost perfect straight line, it would seem. Implying that there's a simple relationship between the volume and the temperature. Notice that this is not a proportionality. It doesn't pass through zero. It passes through some non zero intercept down here. What happens if we vary the sample of the gas, or measure, or vary the pressure of the gas? Let's actually sketch out what is observed experimentally, back over here on the tablet. If I take the volume of the gas, as a function of the temperature of the gra, gas in degrees centigrade, here is in fact the function that we have already gotten from our data. This is for sample n1 at pressure P1. What would happen if we increased the number of moles of the gas? It turns out, in fact, that we get a new straight line relationship for n2 larger than n1. At the same pressure. Notice that the slope of the line changes. Notice also that the y-intercept of the line changes. The volume at zero degrees centigrade is not the same for these two samples. You might think for a moment, why did the graph go up? It's because from Avogadro's hypothesis, the more moles there are, the larger the volume is going to be. What if we were to change the pressure of the gas instead? Let's draw a graph of that one as well. Volume versus temperature. Here's our original data. I'll reproduce it again here. n1 and P1. Now if we're going to increase the pressure of the gas and in this particular case what we've discovered, we get a new straight line relationship, not parallel to the first one and again with a different y-intercept and different than either of the previous y intercepts that we observed. What that tells us then, is that although we get a straight line relationship, both the slope and the y-intercept vary with the sample size and with the pressure. What's an interesting surprise, though, is if we go to the trouble to try to extrapolate what this data might look like as we go back in the direction towards zero volume both of these two curves, it turns out, extrapolate back to the same point on the temperature axis. That particular point is minus 273.15. Likewise, let's put all of these on the same graph together, but give ourselves room to extrapolate below the temperature equal to zero degrees centigrade. Here is the data that we're looking at on the graph on the screen, n1 and P1 Here is the graph if we increase the number of moles, here is the graph if we increase the pressure, while holding the number of moles constant. If I extrapolate each of these curves back to the point where the volume is zero, each of them in fact extrapolate back to exactly the same point, regardless of the pressure, regardless of the sample size. All of these straight line relationships go back and intersect the t axis at minus 273.15 degrees centigrade. The x intercept does not vary with either the sample size or the pressure size. That suggests that there's something rather important about this x-intercept or temperature intercept, minus 273.15. It seems to be a constant. Furthermore, relative to a temperature at minus 273.15, all of these are proportionalities, the volume is proportional to the temperature, relative to minus 273.15. That suggests that we should define a new temperature scale, we should just shift the zero over. Rather than taking the zero to be zero degrees centigrade, we'll take the zero to be this new temperature, minus 273.15 degrees centigrade. To do that is actually relatively simple. We'll simply define a new temperature by adding 273.15 to every temperature in degrees centigrade, as shown in this equation. Notice that minus 273.15 becomes 0 on the new temperature scale, and 0 becomes 273.15 on the new temperature scale. This new temperature scale is called the Absolute Temperature Scale and the units are called Kelvin for Lord Kelvin. In this new set of data now, notice that when we've shifted the temperature over, we're going to wind up with graphs which are straight lined proportionalities passing through zero. This is actually a result which is referred to as Charles' Law. That the volume is proportional to the absolute temperature as we have defined it here. Here is our proportionality with a slope which is alpha, the proportionality constant we've now called alpha. If we extrapolate this particular straight line back to zero it passes through the 0,0 so this is in fact a proportionality. You might ask the question, why do we have to extrapolate? Why don't we just take some more data down here, and see if we can in fact get data that would fit all the way down? The answer is straightforward. As we lower the temperature of the gas, eventually we reach the point that it will condense into a liquid, or possibly a solid. And as a consequence we can't actually collect data on the volume of the gases once the temperature is too low. So in fact we do have to extrapolate but that extrapolation does carry us back through zero, zero. So here we have Charles' Law. It says that the volume is proportional to the absolute temperature but of course, the data that we have drawn here are for a single sample, at a single pressure. So alpha then is going to depend upon what is the sample size and what is the pressure. Here's Charles' Law expressed as an equation now. The volume is proportional to the temperature and the proportionality constant depends upon what is the sample size and what is the pressure. As we've seen back over here, though, that proportionality constant will vary if we vary either n or vary the pressure. What happens if we vary n? How will alpha change? Well, we know. For one thing, we've observed experimentally. That the slope becomes greater so alpha must go up. How much does it go up? Let's answer that question by using Avogadro's hypothesis. Let's fix the pressure and fix the temperature. And now let's say double the number of moles here. I know from Avogadro's hypothesis that the volume's proportional to the number of moles when the pressure and the temperature are constant. That means, if I doubled n, I have doubled V. But if I doubled V then I have to in this equation, have doubled alpha. So when I double n, I double alpha and therefore I can conclude alpha is proportional to n by Avogadro's hypothesis. Well that unlocks the n part of the problem what about the pressure part of the problem. If we vary the pressure instead, how is alpha going to change? To find out let's use Boyle's law. This time let's hold the number of moles of the gas and the temperature constant. Let's do like we did before. Let's say we doubled the pressure. What do we think is going to happen to the proportionality constant? Well, from our data we know that the slope of the line actually decreases when we increase the pressure. But that is consistent from what we've learned from Boyle's Law. Because if I doubled say the pressure here I know that the volume must be cut in half because the pressure and the volume are inversely proportional. And according to this equation, if the volume is cut in half, then alpha is also cut in half. Therefore, if I double the pressure, I've cut alpha in half, and what does that tell me? It tells me that this proportionality constant alpha is inversely proportional to the pressure. Let's put all this together now. Here is our expression of Charles's Law with the volumes proportional of the temperature with the proportionality constant depends on n and P. We know that alpha is inversely proportional to P and proportional to n. We can express that in this particular equation, alpha is proportional to n so it's in the numerator inversely proportional to P so its in the denominator and the new proportionality constant now, we're just going to call that r because it turns out conventionally that's what chemists call it. It doesn't have to have any other particular name. If I just take this alpha and insert it into the equation above, which is Charles's law, what do I wind up with? I have v is equal to r times n divided by P multiplied by T. That in combination just becomes PV is equal to the nRT. Something which chemists refer to as the ideal gas law. It summarizes the relationships of the pressure and volume together, the pressure and the temperature together, the volume and the temperature together and the volume of the number of moles together. In sum then what we've observed is that the ideal gas law is includes all of the experiemental observations that we've made over these two lectures. Boyle's law, Charles' law and Avogardo's law. Where Boyle's law is the pressure volume relationship. Charles' law is the volume temperature relationship. And Avogadro's Law is the volume relationship with the number of moles. Why do we call this thing the Ideal Gas Law? The answer has to do with the fact that we have observed that this relationship is independent of the type of gas molecules that we have captured. It wouldn't matter whether we had contained helium or neon or oxygen or nitrogen or fluorine gas. Or perhaps complex hydrocarbons like methane, or ethane, or propane, or butane into our cylinder, we get exactly the same relationship for pressure, volume, number of moles, and temperature. It's in that regard that we call this ideal. It begs the question somewhat, why would this be true? Why would the relationship of these physical variables be exactly the same for all gases regardless of what the structures of the molecules are? We're going to deal with that question at length in the following concept development study. In the next lecture though first, we're going to consider some applications of the Ideal Gas Law predicting pressures particular, particular samples of gases. Or even using pressure volume and temperature data to determine the number of moles of the gas. We'll pick that up in the next lecture.
