. BIO Work Done by the Lungs. The graph in Fig. E19.4 shows a pV-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.) (b) The process illustrated here is somewhat different from those we have been studying, because the pressure change is due to changes in the amount of gas in the lung, not to temperature changes. (Think of your own breathing. Your lungs do not expand because they've gotten hot.) If the temperature of the air in the lung remains a reasonable 20°C, what is the maximum number of moles in this person's lung during a breath?

Question

Accepted Answer

Hey, everyone in this problem. A model of the lungs is made of a balloon, a glass container and a movable diaphragm. The straw is used to connect the balloon to the atmosphere. The balloons are inflated when the diaphragm is pulled down and deflated. When the diaphragm is released, we're showing an ideal PV diagram for the process and we can see that we have four key points indicated. We have point M and O and P that we have the volume in liters and the X axis, the pressure and millimeters of mercury on the y axis. We're told that the actual PV diagram would have curves and not the straight lines that connect these four points. OK. Notice that the pressure is gauge pressure and not absolute pressure in this diagram. And we're told that this process differs from other processes and that the pressure variations are caused by changes in the quantity of gas rather than changes in temperature. Now, if the model operated at a temperature of 25 degrees Celsius or asked to determine the greatest number of moles of gas attained in the world, we have five possible answer choices here. Option A 0.279 mole. Option B 0.351 mole. Option C 0.351 milli mole. Option D 0.0371 mole and option E 0.0351. So we're looking to find the number of moles, we have information about pressure volume temperature. So this is exactly where we start to think about our ideal gas block or the ideal gas equation. And I recall that we know that the pressure P multiplied by the volume B is equal to the number of moles and multiplied by the gas constant are multiplied by the temperature T. And this is gonna relate all of those variables of interest. Now, in this problem and we have teeth and are constant. We know that the temperature is kept constant at 25 degrees Celsius, the ideal gas constant, that's just a constant. And so the maximum number of moles and oops maximum number of moles hm is going to occur when the pressure P multiplied by the volume V is maxim. Right now, looking at our diagram, this maximum is going to occur at one of our four points. Hey, these are our kind of boundaries. This is where that maximum is gonna occur. So we're gonna go ahead and we're gonna evaluate P multiplied by V at all four of these. Now, for this particular problem, we can already get a sense of where that maximum is gonna be, and we're multiplying two values together, 0.0 has the highest volume and the highest pressure of any other point. And so when we multiply those two together, we're gonna expect that we get that highest result. OK? So we're expecting this is gonna be point. Oh But let's test all of the points to double check. And so that we know how to do this if we weren't given um a diagram where it was a little more straightforward. All right. So let's go ahead and check that. A we have our gauge pressure, we're gonna call it PG which is one millimeter. Oh My. Now let's recall to convert from gauge pressure to absolute pressure. We're gonna go and subtract, sorry. So our pressure is gonna be the age pressure plus the atmosphere of pressure. So for each of these values, we're gonna add our atmospheric pressure which is 768 millimeters of period. So our gauge pressure one millimeter of mercury. And so our pressure P for this point M is gonna be 761 millimeters of mercury adding our volume V 0.2 liters we can see from our diagram. So PV is gonna be equal to multiplied by 0. which gives us 152. millimeters oh multiplied by the leader. All right. So that's for, we do the same for 0.9. OK. So at point N what do we have our gauge pressure that we're given is five millimeters of mercury. We're gonna add our atmos atmospheric pressure to that 760. OK. And so we get that our pressure is 765 millimeters of mercury. We're given a volume V of 0.4 liters. And so this quantity PV that we're interested in is gonna be multiplied by 0.4 liters, which gives us a result of 306 millimeters of mercury multiplied by liters. All right. So is greater than M already. We can see that. OK. So we can eliminate m and it's definitely not gonna be where we have the greatest number of moles we're gonna carry on doing the rest of them. OK. At 0.0 our gauge pressure, according to our diagram is going to be seven meters of mercury giving us an absolute pressure. When we add the atmospheric pressure of 760 millimeters of mercury of 767. I mean, there's a mercury and we have a volume V of 0.9 liters multiplying them together. We get that PV is equal to millimeters of mercury multiplied by 0.9 liters which gives us 690.3 millimeters of mercury multiplied by liters. So we can see this result is greater than both M and N. So now we can eliminate N as well. Oh Is the highest option at this point? And we're just gonna check that last point piece gauge pressure. According to the diagram, two millimeters of mercury converting to absolute pressure is gonna give us millimeters of mercury. Their diagram shows a volume of 0.7 liters multiplying the two together 762 multiplied by 0.7. We get a result of 533. millimeters of mercury multiplied by liters. All right. So O is the highest result as expect. OK. So PV is the highest. So let's go up to er to our gas equation. Again, our ideal gas law. OK. Again, we said that R is constant and T is constant. So the maximum number of moles N is gonna occur where P multiplied by V is maximum. And we said that that was gonna occur at one of these points. We've evaluated all of them. We found that at O we have the maximum of PV, which means we're gonna have our maximum number of. So now we're gonna use 0.0 and our ideal gas equation to calculate the number of moles N. All right. So to refresh our memories, we have PV is equal to N RT. OK. At 0.0 we have a pressure of 767 millimeters of mercury. We want to convert this into our standard unit of pascal. So what we're gonna do is multiply 767 multiply by our conversion factor of 133.3 cal. OK? And this is gonna give us 102,000, 241.1. So that's the pressure we're gonna use. Substituting this in, we have 102, 0.1 pascals multiplied by our volume. I have to be a little bit careful. OK. Our volume is 0.9 liters, but we need this in our standard unit of meters cubed. What we're gonna do is divide by liters per meter cube. And this is gonna leave us with meters cubed. The units of meters are gonna divide it. All right. And this is gonna be equal to N the number of moles multiplied by the ideal gas constant 8.314 jewels per mole. Alvin. We are running out of space here. I apologize for that. And in here we're also gonna multiply by the temperature and our temperature is 25 degrees Celsius. We want this in Calvin. So we add 273 to convert to Calvin. All right. So let me write this out just once more so that it's clear what we have without getting cut off. We have 102,241 0. multiplied by 0. divided by 1000 m cut. This is equal to N multiplied by 8. jewels for mole Calvin multiplied by 298. All right, now, if we simplify on both sides, we have 92. meters. Cute is equal to 2477. jewels per mole multiplied by N in that unit of Calvin divided up. And this gives us an end value if we divide by 2477 of 0.03714. All right. So that's the greatest number of moles we can have in this situation. If we go up to our answer choices and compare what we found, we see that this corresponds with answer choice, the but not OK. So answer choice, D 0.0371. Thanks everyone for watching. I hope this video helped see you in the next one.

Verified Solution

Key Concepts

Ideal Gas Law

Gauge Pressure vs. Absolute Pressure

Breathing Mechanics