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(III) What is the mathematical relation between water's boiling temperature and atmospheric pressure? (a) Using the data from Table $2,$ in the temperature range from $50^{\circ} \mathrm{C}$ to $150^{\circ} \mathrm{C},$ plot $\ln P$ versus $(1 / T),$ where $P$ is water's saturated vapor pressure (Pa) and $T$ is temperature on the Kelvin scale. Show that a straight-line plot results and determine the slope and $y$ -intercept of the line. $(b)$ Show that your result implies$$P=B e^{-A / T}$$where $A$ and $B$ are constants. Use the slope and $y$-intercept from your plot to show that $A \approx 5000 \mathrm{K}$ and $B \approx 7 \times 10^{10} \mathrm{Pa}.$

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(a) $-5000 \mathrm{K}$ , 24.91(b) $B e^{-A / T}$ $\left(7 \times 10^{10} \mathrm{Pa}\right) e^{-5000 / T}$

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this problem, We're going to find a method medical relationship. Been tweet point of water and the external pressure. We're gonna start by making a plot off the net. The pressure in Pascal's versus one over the temperature. And Kelvin, we're gonna show that the plot is a straight line. We're gonna find the slope in the whiner sect of that light. After that, we're going to write the byte. That equation in the form P equals a times exponents of minus a over tea, and we're gonna find the values of A and B so we can start by analysing our table over here. And the first thing that we notice is the the units are in Celsius when they should be in Kelvin. So let's just switch it up to Kelvin by adding 273 into each one of these data points. So, after adding 273 we're going to get these values. You can complete it by yourself, and we can already we can already start, um, find, uh, doing one over tea for the temperature and taking a natural log for the pressure. And if you're playing in your calculator, you're going to get these values. With these values, we can start planning your graph and let me pick up my graph over here. And let's start with the first data point. Just gonna bay surprises or three point for one should be around the end over here near the middle. Um, so somewhere around here, So for the second point we're going is going to be syrup Windsor. Three of around around the line, almost a 10. So somewhere around here, a little bit to the site. We can keep doing this for all other values. And let's just pick up the less one for the sake of time. And it's a little bit over her, Jean. And your point is your old 2326 So around here could do another point just around the middle we can find Ah, zero point is your old two. Um, let's use this point over. Here is your 0.0.275 she said in five. It should be around here and 11 point 15 so little bit about the 11 somewhere around here. So the graph looks like a straight line. Let's draw the straight line and let's calculate the slope and different points to make sure that it is a straight line indeed. The former for the slope. As you may know it the changing Why? Why? To minus y one divided by the change in X Exoo minus next one. This is this force to data points. So let's start by subtracting these two and we're going to get zero point for a 11 two and then invited by the difference in the X. So this value minus this value over here and it's going to be negative nine point 297 times can to the minus five. If we divide these two, we're gonna find that slope here is gonna be approximately minus 5000. We can do this again for another data point. Let's say just less data point and this one over here before the third to last. So if you calculate that it's going to be 1.55 divided by 3.16 nine times 10 to the minus four and again, we're going to get minus 5000. If you want a positive video and completed by yourself, go ahead. The slope shouldn't change too much between these values and can pick any two sets of numbers to do this. Um, so let's move on to finding the white intercept for the Y intercept. We can use the equation of a line, which is why equals experts be This is a beat. Sorry. Well, let me do that again. Xmas Bi. We already found a So why equals minus and let's be And we can substitute 5000 x. Sorry about that. 5000 x busby. And then we can substitute any data points in here. And that's just you just used, um, bring them one. I'm gonna use 10.34 over here with why went three for a equals minus 5000 times. Serial 0.0 29154 It was day. If you solve this equation, you're gonna see that the slope be he's going to be equal approximately 25 again. You can use any of these data points too, if you want to do it. Not a double check. Make sure that you got the right number, but this is the slope over here. So phone slope. Oh, I'm sorry. We found the White Intersect, and we also found this lope. So we're going to do next. So we did this. We did this and we found the intercept. So next we're going to proved that we can write that. Oh, yeah, Let's write that. Unless the final equation just to be consistent, So lug and a p going to be equal to minus 1000 Ex waas won over tea plus 25. So I want to show that vision in this format, and we can do it by two different ways. We can either use this equation and manipulated to get it to this format, or we can manipulate this format and trying to get something that looks like a wreck eight equation. And for this case, I think this one's a simpler one. So let's just start with p equals be times e too in minus a over tea. And since we want to move into a natural log, we're going to have to make sure this be becomes exponential somehow. So what I'm gonna do here is a variable substitution in a very substitution variable substitution. I'm just gonna assign another variable, just so it's easier for me to work with. And so I'm trying to move the speech to something else, and it has to be an exponential. So be is gonna be equal to an exponential to say. Okay, so this equation now becomes P equals each of the K times e to the minus a over tea. If we use the power rule over here, we find that p equals hey to the K minus a over t. Okay, now that we have an explanation over here, we can put the luxury of them. The natural log on both sides on the right hand side, just simply national love p. And then on the right hand side, the natural log is gonna cancel out with exponential. And we're going to be left with K minus a over tea. So at this point, the question really looks a lot like this one. But we can already when you pull it a little bit more and make it even more like So let's do this right now. Well, in a piece perfect on this side, and then we want to have one over tea and something multiply it so minus a times one of her tea, and then were you just have to some with what's left over here, which is Kay. And we can already see that we have the same equation on the same format and which showed that you can be rated on that format. So what we do next, this is we're going to calculate A and B, But here we have a K. So just by looking, let's start with the easier one, which is a and just by looking at it, we can already tell that is gonna be equal to 5000. And here we have kay, and we know that Kay's equal to 25 just by quick inspection. So okay, is equal to 25 Now, we're gonna go back to the variable substitution that we did earlier, and we're gonna plug in this cave that we found over here, and we're gonna find what B is so be is gonna be equal to hey 2 25 which in turn it's gonna mean that be is going to be equal to seven times Turn to the 10 looking at the units on this first, form it over here. We know that here we need to have us calls as the units. And here we have Kelvin, since there's no Kelvin on this side, we have to make sure to cancel all this Calvin, which means that we have to assign Hey Kelvin units. And then since we have already canceled this out, the right hand side has to have the same unit as the left hand side. So that means that be it's going to have Let's go, all units knowledge, you can plug it into your answer. Over here we have that A is going to be equal to 5000. Kelvin and B is going to be equal to seven times 10 to the 10 Pascal's and that is it for this problem. We just finished. We just finished all of it and we're done.

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