Question 3gas mlcontainer changes its volume according to the law, PV_3600, where P is the number of units of pressure and V is the uumber of units of the volume: G...

At a temperature of $20^{\circ} \mathrm{C}$, the volume $V$ (in liters) of $1.33 \mathrm{g}$ of $\mathrm{O}_{2}$ is related to its pressure $p$ (in atmospheres) by the formula $V=1 / p$ a. What is the average rate of change of $V$ with respect to $p$ as $p$ increases from $p=2$ to $p=3 ?$
b. What is the rate of change of $V$ with respect to $p$ when $p=2 ?$

Okay, we are given the equation P V To the one and 4/10 power is a constant K. I'm going to rearrange this to get the variables on either side. There we go. And now I'm going to take the derivative with respect to time -1.4 -1 is negative 2.4 DV over DT Okay. So in part a the rate of change of Russia with volume interesting. Mhm. So I think what I did, we're going to need for part B but rate of change of pressure with volume is going to be DP over DV. So going back to this then D P over D V. I just have to take the derivative of the right side with respect to V, which would be negative. 1.4 V to the negative 2.4 Power Whoops. Uh times K. So let's put in the numbers -1.4 V is 20 Oh wait a minute. We're not putting in Viet because this is the rate of change, suspect two V. So okay we need to figure out K. So let's go back to the original equation. P times V. To the 1.4 power equals K. So Okay. Is 20. Yeah. To the 1.4 power. Because that's v. 20 is v times K. I mean times P which is one atmosphere. Okay. 22. The 1.4 power. Yeah Is 66.29. Okay so I can take that 66.29 and put it up here And then via the negative 2.4 power is just over v. To the 2.4 power. Okay 66 29 Times -1.4 Is going to be -92.8. So switch to black here. D. P. Over DV is just taking this equation. Um The numerator becomes -92.82 And V The 2.4 Power Alright, I did not expect that question because that's different from what we've been doing uh in this chapter in this section. So let's look at part B. The volume. So they're telling us DV over DT is to and V is 30. So they're asking us to figure out DP over DT. So D. P over D. T. Is negative 1.4 30. To the negative 2.4 power times DV over DT which is to Times K. Which we figured out over here is 66.29. Hey, where'd that go? All right now, we'll put that into a calculator. Yeah, -1.4 times 30. Mhm. To the negative 2.4 Power Times two times 66.29, Which gives me negative 0.053 atmospheres per minute. Um And it is decreasing because it's negative. Yes. Yeah.

This question asks us to determine at what rate is the volume decreasing At this given instance, what we know is that Boyle's law is that sea is equivalent to P. V. We no see is a constant. Therefore, we can differentiate this equation. Remember, the derivative of a constant is simply zero. Now plug in what we know 1 50 times DV over DT plus 600 times 20 is equivalent to zero. This indicates that Devi over DT is equivalent to negative 12,000 divided by 1 50 which is negative 80 centimeters cubed permanent.

If we take the derivative with respect to time of PV equals C, we get P Thames TV, T T plus v Thames, dp DT equals zero. And so the pressure is 150. Killer Pascal's DVD T is unknown. The volume is 600 centimeters cute, and the change in pressure is 22 of Pascal's per minute. So that's going to give us. David ET is equal to negative 600 times 20 over 150 or negative 80 centimetres cute per minute.

Right. Welcome back. We're in. Let's say section, Chapter three, Section nine. Page 2. 49 number 33. Okay, so we have Boyle's law, which states that P v is a constant Pete times V. It is a constant. Okay, So Penis pressure of the gas V is volume of a gas. Uh, if it's compressed at a constant temperature. Okay. Suppose that at some particular instant V is 600 cubic centimeters she sees on DPI 150 kill Paschal's on bond on the pressure is decreasing at a rate of 10 killer killer passes per minute. Right now, supposed to certain instance, we have volume of 400. I look at the one question Sorry. Uh, and the pressure is increasing. Had 20 killer Paschal's per second. Okay, so d p d v d p d. T. Is 20 killer Paschal's per minute. Okay, fine. DVT. Okay, well, I guess we have to do the product rule. So the first time to do of the second, plus a second determine the product to functions that are changing right? Has it through the first equals in this case through the constant zero. So to figure out the PVT. We have to know P which is 1 50. We need to know DVT. No, I'm sorry. We're trying to find anything, so find the prime Getting DPD to be fine. We need to know V which is 600 on. We need to know what dp DT which is 20. So to step linear equation algebra One days are back again. That's attract Wow. Who would that be? Mhm. 10 times. 606,000 times two is 12,000. So subtract 12,000. Both sides it and development 1 50. Oh, it's a negative. 12,000. You have about 1 15. So we just by 10? No, uh, let's see. It is by 10 reduced about three of this. It's a negative 80. Um um Well, let's see these and keeping centimeters and teasing minutes. There you go. Alright then. Right. Helpful was helpful. You with your homework. See you next thing right