The total lung capacity of a typical adult is 5.0 L. Approximately 20% of the air is oxygen. At sea level and at a body temperature of 37°C, how many oxygen molecules do the lungs contain at the end of a strong inhalation?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let's read and highlight all the key pieces of information in the problem that we need to use in order to solve it. OK. So a cylinder and an auto cycle engine in a ship has a volume of 1828 liters. The cylinder sucks in air from the atmosphere at sea level at a temperature of 15 degrees Celsius, assuming air contains 78% nitrogen, calculate the number of nitrogen molecules in the cylinder when the cylinder has sucked maximum capacity. OK. So our end goal is to calculate the number of nitrogen molecules. Awesome. So we're given some multiple choice answers. Let's read it all, read them off and see what our final answer might be. A is 4.66 multiplied by 10 to the 25th. Power B is 1.3 multiplied by 10 to the 25th. Power C is 60.3 multiplied by 10 to the 25th power and D is 3.63 multiplied by 10 to the 25th power. OK. First off, let's assume that air is an ideal gas. So then we can recall and use the ideal gas law, it's called equation one which is pressure multiplied by the volume is equal to lower case N, the number of moles multiplied by the universal gas constant multiplied by temperature. Awesome. So since we know our end goal is to find the number of molecules and nitrogen, we must first solve for the number of moles to determine the number of molecules will be. So let's rearrange equation one, the ideal gas law to solve for the number of moles. So when we do that, you should get that the number of moles is equal to the pressure multiplied by the volume divided by the universal gas constant multiplied by temperature. But this is defined. So we'll call it equation two. So equation two finds a number of moles of air, what we need to find the number of moles of nitrogen. So in order to do that, we'll call it equation three, we need to account for the, the air contains 78% nitrogen. So to do that, to find the number of moles of nitrogen, specifically, we're gonna call it lowercase N subscript capital N subscript two. So N two representing nitrogen. So the mo number of moles of nitrogen is equal to, we have to write the 78% as a fraction. So 78 178 divided by 100 multiplied by pressure multiplied by volume divided by the universal gas constant multiplied by temperature. Awesome. So in order to find a numerical value for the number of moles of nitrogen, we need to quickly make some quick conversions which you could do this at the very end. But I prefer to do it earlier on, but let's do that really quick. So we need to convert degree Celsius to Kelvin because in order to properly use our equation. And so all the units cancel out. It has to be in Kelvin. So to do that. And so we have to recall and remember it's a quick conversion. So we take our temperature in degree Celsius, which in this case is 15 and all you have to do is just add 273. So when you do that, when you add that up, you should get 288 Kelvin where the temperature and also we're given the volume as liters 1828 liters, we need to convert that to meters cubed. So to do that, it's really simple, let's write it down really quick is we just need to quickly use dimensional analysis to convert the liters to meters cubed. So we need to recall that in one cubic meter, there is 1000 liters. So then the liters will cancel out. And when you plug that into a calculator, you should get 1. meters cubes. So cubic meters awesome and then it didn't give us directly in the, in the actual problem itself. It didn't give us a numerical value for the pressure at sea level. Here, we're gonna write as pressure capital P subscript s. So pressure at sea level, the numerical value for that when you quickly look it up as 101,325 pascals. Awesome. So now that we have all of our numerical values, let's plug them into equation three to find our number of moles of nitrogen. So let's do that. So when we plug in our numerical values, so 78 100 it's multiplied by OK. 101, pascals multiplied by the volume. 1.828 m cubed divided by universal gas constant, 8.3145. And it's oh, this is the numerical value for the universal gas constant and it's in units of jewels per mole multiplied by Calvin. And then our temperature was 288 Kelvin. Awesome. So when you plug that into a calculator, your final answer should be 60.33 moles. Awesome. So now I'm trying to solve for the number of molecules N two, we're gonna denote it as capital N subscript, capital N two. So number of molecules of nitrogen and the equation to solve that, we'll call it equation four. We take our number of moles nitrogen and we multiply it by capital N subscript A Ava Gao's number. So let's note that Ava Ger's number and a, the numerical value for that is 6. multiplied by 10 to the 23rd power and its units are mole to the power of negative one. So let's remember that Avogadro's number will always be some elemental substance, whether it be atoms, molecules, et cetera Permal. But in this case, we're solving for molecules. So we're gonna have it as molecules Permal. So let's plug in our numerical values to solve for the number of molecules of N two. So our number of moles was 60.33 mole multiplied by Avogadro's number 6.22 multiplied by 10 to the 23rd. As we said, it was molecules Permal because we're trying to find the number of molecules moles cancel out. And then when you plug that into a calculator, the number of molecules of nitrogen will be 3.63 multiplied by 10 to the 25th power molecules. All right, we found our final answer. So let's go back to the top here. So that means our final answer will be D 3.63 multiplied by 10 to the 25th power. So that's our number of nitrogen molecules in the cylinder. Thank you so much for watching. I can't wait to see you in the next video. Bye.