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A 0.1052 g sample of $\mathrm{H}_{2} \mathrm{O}(1)$ in an 8.050 L sample of dry air at $30.1^{\circ} \mathrm{C}$ evaporates completely. To what temperature must the air be cooled to give a relative humidity of $80.0 \% ?$ Vapor pressures of water: $20^{\circ} \mathrm{C}, 17.54 \mathrm{mmHg} ; 19^{\circ} \mathrm{C}, 16.49 \mathrm{mmHg}$ $18^{\circ} \mathrm{C}, 15.48 \mathrm{mmHg} ; 17^{\circ} \mathrm{C}, 14.54 \mathrm{mmHg} ; 16^{\circ} \mathrm{C}$, $13.63 \mathrm{mmHg} ; 15^{\circ} \mathrm{C}, 12.79 \mathrm{mmHg} .[$Hint. Go to Focus On feature for Chapter 6 on the MasteringChemistry site, www.masteringchemistry.com, for a discussion of relative humidity.

So in this problem were given a table of data of temperatures of water. The water boils out at selected pressures pete. And we're giving this model at T equals 87.97 plus 34.96 Ln. Of P plus 7.91. I'm the square root of p. Okay. Where t the temperature that we calculate from this model is in degrees Fahrenheit and the pressure is in pounds. Always stood this way. Pounds per square inch. Okay. So the first thing we're asked to do yes, to graph this with the same window as the table and graph the data as well. So let's go over to r graphing calculator over here. Oops. Wrong thing. Sorry. Table My Table Data 1st. And I have I've 10 14 point 96 Just one atmosphere. So that didn't work, did it? 14.696. Okay. In 20 30 40 60 80 100. Okay, men say this is 1 62.24. This one is 193.21. Then of course to 12 And 2 27 0.96 And to 50.33 And to 67 0.25 and 2 92 0.71 and 3 12 point oh three and 327 0.81 Thanks. So there's our data now. Data up here on the screen. There's the data showing up right there. Okay. You know them? We need to graph the function. So here we go. Will say why? One equals 87.97 of 34 points and six Ln X one plus 7.91. And then we'll say X one we need that to the on half power. Right? So we get the square root of it. Okay, Well, maybe this is supposed to be why and this is supposed to be X and this is supposed to be X. Okay. We'll see. Mhm, mm hmm. To the Okay. Here we go. We got it, didn't we? All right. So, the model it's the day pretty well, didn't it? That's pretty darn close through that data. Almost like every single point is going through there. Okay. So next were asked to use it to estimate the pressure at which the boiling point of water. So when T is 300°F. P. Is what? Well, it would be right there, wouldn't it? 300 Okay, 67.3, isn't it? And thats PSC. Okay. Then. We're asked When P is 74 P. S. I calculate the temperature. Okay. So T is 87.97 Plus 34.96 Ln. Of 74 plus 7.91 Times The Square Root of 74. And when we calculate this out, Let me see, 87.97 Was 34.96. Ln. of 74 Plus 7.91 Square of 74. I get 306 point for eight degrees F. And when I look at the graph At a pressure of £74, so here's 50 60 70 34 3 or 6.485. Right? So there we go. We're dead on. Okay. So there's that. We found that and we did the graph just possessed.

In this question, we will learn about the basics of pleasure and uh universe like that must feel pressure. So then the given question, there are some steps which are the steps of an experiment, but not in a sequential order. So we have to find the correct sequence of these steps. Right? So the experiment to prove the conclude that atmosphere air exist, exerts pressure. We can take a metallic container. Mhm. Mhm. Right. So the experiment would be, first of all, we can take a metal container which is half filled with water. So this is a statement B Yeah. So mightily container half field with water. The next step would be they we can hit this container for some time and we can place a cap on the container family. So hit this and keep a gap over this container. The next step would be to stop hating the container and call it to very low temperatures. Yeah. Mhm. Right. And the next step would be see step that the container gets compressed because of the pleasure, which is accepted by the year from the outside of the container late. Mhm. So this is the experiment to blow that atmosphere. Uh It's just a pleasure. Right. Right. The experiment is first, we'll take a metallic container which is half filled with water. We'll hit this continent and we will keep a kappa. Wait. Now we will stop hitting this. And we'll call this to a very low temperature. Now, when we are calling this, the container will get compressed because the pressure outside will be higher than the Pleasure which is inside the container. They so this means that the L which is outside the container is accepting a pressure over the complete already container. Right? So this is the experiment. And the correct sequence would be B. D. You see this is matching with the correct option too. Right.

Right here, we want to look at the tables um that were given. So we have P and T. So this is P for pressure and pathways branch and T for the temperature. So we see that he is going to be five han, I will call this 1520 and then this is going to go up to 30, 40 60 80 100. And the corresponding temperatures are going to be about 162. Um we're just going around so 160-193 to 12, 2 28 2 50 Uh to 67 92 1393 weeks today. So you see that with this, it follows this general trend are fit, might be even better if we chose the exact values, but we see it will have this general trend and we can use a cubic model though to get an even better fit or at least a potentially better fit. So this is the cubic model and it's even better. Um So we see this is our general trend, but we see that this model would not be accurate for exceeding pressures of 100 because they would say that the temperatures is way too high.

So that gives this data here and they want us to answer a couple of questions about it. So the first thing they tell us to do is to find some regression model for this data. So I'm gonna be using Google sheets to do this. If you use something else, then your answer might be slightly different. But the equations should still be in the same ballpark of what we're going to get. So to do this in Google Sheets, once you get the data kind of input like I have over here, go ahead and highlight. This will come up here and hit Mawr, go to insert chart. And so there is going to give us this chart here now just to kind of get ready for the next part over here and set up. If it doesn't have this, like, just click on it, it set up. And then we'll come over here and changes to a scatter plot. And now to actually get the regression will come over here to customize hits Siri's scroll down a little bit. We're going to do ad trendline and so we want a cubic function. So we'll click on this goto polynomial and we want a degree three. So now that's going to be our degree. Three polynomial, and then if we want to actually get the equation, will come over here, do label and then use equation. And so then this up here is going to be our, um, regression line that they believe we have. And so they remember this E to the negative zero for Israeli just saying 2.99 times 10 to the negative fourth power. So I just kind of keep that in mind if you're using this. So now So we've done a and we've already also knocked out part because we have the data as well as the scatter plot. So they say they want us to use this data to estimate the pressure required to make the bulletin point of water exceed 300. So let's just go ahead and look so 300 is around like, right here. So it looks like it's like, right there. And if we were to come down, So maybe something around like 70 um, is what we would need. So, yeah, for part C, we would say about 70 and then we want to explain why exceeding £100 per square inch would not be a good model. Will. In order for us to figure that out, let's go ahead and expand this so we can double click down here. So let's make our maximum or minimum zero, and then we can make our maximum A. I don't know, let's say like, 200. So we have this and why we would say that it might be bad for exceeding past this. Well, anything past 100 it starts to just really, like increased significantly. But we can kind of see how the data here it looks like it's like flat lining, so, like it's increasing kind of a bit, and then it looks like it should just kind of keep on going out this way. So I wouldn't think it would have this steep increase in the actual temperature. I would think it would kind of like flatten out a little bit more. So now, at least that's what I would think