An inflated bicycle inner tube is 2.2 cm in diameter and 200 cm in circumference. A small leak causes the gauge pressure to decrease from 110 psi to 80 psi on a day when the temperature is 20°C. What mass of air is lost? Assume the air is pure nitrogen.

Video transcript

Hey, everyone. Let's go through this practice problem. A buoy has a gauge pressure of 32 P si when the temperature is degrees Celsius, the buoy is a volume of 10 liters and contains pure helium gas due to a puncture. The gauge pressure drops at 25 P si on a day when the temperature is degrees Celsius, what mass of helium gas leaked out of the buoy assuming the volume remains constant. And we're given four choices to choose from. Option a 0.9 g, option B 0.7 g, option C 1.7 g and option D 1.9 g. Now, first, it's good to know that when we want to relate an amount of a substance in this case, an amount of helium gas to other variables like pressure and temperature and volume, that's a good sign that we want to use the ideal gas law which states that the pressure multiplied by the volume of gas is equal to its number of moles multiplied by the ideal gas constant multiplied by its temperature. In the case of this problem, we specifically want to at some point be solving for the number of moles of the gas, since that will allow us to then find the mass of that gas. So algebraically solving the ideal gas, the I, the ideal gas law for the number of moles, we find that N is equal to the pressure multiplied by the volume divided by the ideal gas constant multiplied by the temperature. Since the problem is asking us for the mass of the gas that was leaked out of the buoy, we're gonna want to solve ultimately for some change in this end value between the two different states of the, of the problem, the two different states of the buoy with state one being before it started leaking when it had a temperature of 25 degrees Celsius and state two being after it leaked when the temperature was 20 degrees Celsius. So let's go through our variables before we start plugging things. In. First of all, the problem tells us to assume that the volume remains constant, which means that V sub one is equal to V sub two. And we're told conveniently what that volume is. It's 10 liters. However, in order to use this volume in formulas, we'll want to convert it into cubic meters. And we can do that by using the conversion rate where one liter is equal to 1000 cubic meters. And we put that into a calculator. And we find that there is a volume in the buoy of 0. cubic meters. Next, let's go through the pressures. First, we're told that the initial gauge pressure is equal to 32 P si. So that'll be our P sub one since that's the pressure before it started leaking. But there is a problem with this pressure variable because the problem specifically tells us that this is a gauge pressure and the pressures in the ideal gas law only work if they are absolute. So we're going to have to take this gauge pressure and add the atmospheric pressure or one atmosphere. Now, obviously, these units don't match up. So we'll want to convert both of these into Pest Kells. So that's 32 P SI and we'll convert this into pascals by using the conversion factor where 6894. pascals are equivalent to one P SI. And then to that, we add one atmosphere of pressure, which is of course, 1.013 multiplied by 10 to the power of five pascals. And if you put this into a calculator, we find that P sub one is equal to about 321, pascals. Now let's do the exact same thing with P sub two, the pressure after it leaked where the pressure, we're told that the gauge pressure is 25 P si. So that's 25 psi multiplied by the same conversion rate. 6894. pascals divided by one psi. And then once again, we're gonna have to add the atmospheric pressure, 1.013 multiplied by 10 of the power of five pascals put that into a calculator. And we find that this is equal to about 273, pascals. Lastly, the temperatures also need to be accounted for. So we're told that t sub one, the initial temperature is 25 degrees Celsius. So we're gonna want to convert this into Kelvins in order to use this in the ideal gas law. So that's 25 plus Kelvins or 298 Kelvins using the exact same process for T sub two, the second temperature on 20 Kelvins 20 degrees Celsius, we find a second temperature of 200 Kelvins. Now let's apply the Ideal gas Law. Remember we're looking for a change in the number of moles. So we're gonna do N sub one, the number of moles initially minus N sub two, the number of moles left to find the change in the moles. So using the Ideal gas law up ahead, that's going to be P sub one multiplied by the volume. And I'm not going to add a subscript on the volume because it's constant in both cases, divided by the ideal guess the constant multiplied by the initial temperature then minus P sub two V divided by RT sub two and to simplify this a little bit. I'm just going to factor out the V and the R. So this becomes V divided by R multiplied by P sub one divided by T sub one minus P sub two divided by T sub two. Now, we want to put this into a calculator along with our values. So that's going to be a volume of 0.01 cubic meters divided by an ideal gas constant which is 8.314 Jews per mole. Kelvin. And this is being multiplied by P sub one divided by T sub one. So that's 321,932 pascals divided by 298 Kelvins minus P sub two, which we found earlier was 273, pascals divided by 293 Kelvins. And if we put this into a calculator, then we find a change in the number of moles of about 0.17 moles. Now that we've found the number of moles that leaked out, we want to convert this into a mass. We know from the problem statement that we're working with the helium gas. And we also know that for helium, there are 4 g per mole. This is something you can look up in a periodic table of elements, for example, but we'll use this as a conversion factor on the number of moles. We found So 0.17595 moles multiplied by the conversion factor of 4 g per mole. And if we put this into a calculator, then we find a mass of about 0.7 g. So that means that this is our answer to the problem. And if we go back to our multiple choice options and look at the choices we're given, we can see that option B says 0.7 g. So that is our answer to the problem and that's it for this video. I hope this video helped you out. If you think you need more practice, please check out some of our other videos which will give you more experience with these types of problems. But that's all for now. I hope you all have a lovely day. Bye bye.