The graph in Fig. E $19.4$

Question

The graph in Fig. E

19.4

shows a

p V

-diagram of the air in a human lung when a person is inhaling and then exhaling a deep breath. Such graphs, obtained in clinical practice, are normally somewhat curved, but we have modeled one as a set of straight lines of the same general shape. (Important: The pressure shown is the gauge pressure, not the absolute pressure.) How many joules of net work does this person's lung do during one complete breath?

Graph of lung pV diagram with straight lines showing inhaling and exhaling phases, pressure in mm Hg, volume in liters.

Accepted Answer

Hey, everyone in this problem, we're told that a model of the lungs can be made using a balloon inside a container fitted with a movable piston. The balloon is connected to the atmosphere using a pipe air is sucked into the balloon. When the piston is lowered and expelled. When the piston is raised, we have this idealized curve with straight lines for the sucking and expulsion shown in the diagram given. And the curve is labeled using gauge pressure instead of absolute pressure. So we have to be aware of that and we're asked to determine the network done by the balloon in one second. So we have our diagram. We have the pressure and millimeters of mercury on the Y axis, the volume in liters on the X axis. We have this diamond shaped curve connecting points, MNO and P. And we're given four answer choices all in jewels. Option A 0.24 option B 14.9 option C 1.87 and option D two. So what we need to do is think about the relationship between the network that we're looking for and what we've been given in our diagram and recall that the work is just going to be the area enclosed by our curve. Yeah. So if we're looking for the work, this is now an area problem. OK. So let's take a look at our diagram and we're gonna calculate the area kind of approximately as well as we can from looking at it. And we're gonna start with these squares, that kind of look like full squares. OK? So we have a full square here. We have very close to nine full squares through here. OK? We have another full square through here that gives us a lot. OK. So right now we have 11 full squares and let's keep piecing this together. So let's switch colors. We're gonna go in red here for a minute. And if we look down at the bottom, we have these three pieces at the bottom here. We're gonna draw those in a red box and those kind of make up one square. Hi. So we're gonna say that this is another full square. So we outlined in red and we can do the same on the other side in ball. OK? So we have this big square almost complete and then this teeny triangle. So we can color those in, in blue. That's gonna give us another full square looking at them some more. We have kind of almost a third of a triangle on the right hand side, we have kind of two thirds of it, of the square. Sorry, a third of the square and two thirds on the left hand side. So that's gonna give us one full square and we can keep going. Ok? And we're gonna keep going. We're gonna match these up. Hi again, a square almost complete the little triangle piece that's missing. That gives us a full square, these triangles in the bottom left if we put those together and then maybe this little piece on the right hand side that gives us another full square. OK. So another full square and then we kind of have these ones at the top. So the two at the very top, we can put those together kind of into one full square. And then finally, we have these two sort of half pieces and we can combine those into a full square as well. OK. So those will be the last two pieces and that's going to be um a full square as well. OK. So we're just breaking this area up looking at the squares. And when we do this, if we add up all of the squares, we found, what do we have? We have 1112, 1314, 1516, 17, approximately 18 squares. OK. So we have 18 squares in our diagram. So the area is going to be 18 units. And the question is, what are these units? Right? So again, we're looking for the work, we know it's the area enclosed and we now know that that's 18 squares. OK. So we have 18 now in the X direction, OK. Each square is represented by 0.1 of a liter. So our square is gonna be 18 multiplied by 0.1 L. And then in the Y direction, each square is represented by one and the unit is millimeters of mercury. Remember we were warned about that. We're using gauge pressure, not absolute pressure. So then we're gonna multiply by one millimeter of mercury. So what we have are 18 squares, we actually have 18 multiplied by 0.1 L multiplied by one millimeter of mercy. Now we want to get a work and we want this work to be in jewels like the answer choices which means that our measurement of volume needs to be in meters cubed. Our measurement of pressure needs to be in pascal those standard units. So we're gonna go ahead and convert these before we even do this calculation. And that's gonna look like this. So we have 18 multiplied by 0.1 L and to go from liters to meters cubed, we're gonna multiply and we're gonna multiply by 0.001 meters cubed her leap. Bye. When we do that, the units of leaders will divide it. We're gonna be left with meters. Cute. Now we multiply by one millimeter of mercury multiplied by approximately 133.3224 pascals per millimeter of mercury. OK? So just taking those conversion factors. Those are conversion factors you can find in a table in your textbook. OK. And again, the unit millimeter mercury is gonna divide it. We're gonna be left with meters cubed multiplied bypass scales. And when we work this all out on our calculator, we get that this is actually about 0.24 Pasco meter skipped. Now we have Pascale meters cube. We want this to be in jewels. They it turns out Pascale meters cubed is equivalent to a jewel. But if you don't remember, you can kind of break this down. So what can we do? We'll recall that a pascal is equivalent to a Newton per meter squared. We have Newton per meter squared multiplied by meter cubed. OK. So this gives us 0.24 Newton meters and we know that a Newton meter is equivalent to a jewel. And so we get that the final work that we were looking for is about 0.24 jewels comparing this to the answer choices we were given. We can see that this corresponds with option A thanks everyone for watching. I hope this video helped see you in the next one.

Key Concepts

pV Diagram

Work Done by a Gas

Gauge Pressure vs. Absolute Pressure

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