At an altitude of 11,000 m (a typical cruising altitude for a jet airliner), the air temperature is -56.5°C and the air density is 0.364 kg/m^3 . What is the pressure of the atmosphere at that altitude? (Note: The temperature at this altitude is not the same as at the surface of the earth, so the calculation of Example 18.4 in Section 18.1 doesn't apply.)

Video transcript

Welcome back everybody. We are taking a look at the service ceiling of an aircraft, A. K. A, the highest altitude that an aircraft can operate at. And we are told that 8000 m up that service ceiling And we're told a couple different things, we're told that the temperature is negative 36.9 degrees Celsius. We're told that the density at that altitude is 0.53 kg per meter cubed. And since we're dealing with air, I'm just gonna write this down. We know that the molar mass of air is 28. times 10 to the negative third kilograms per. And we are tasked with finding what the atmospheric pressure is. Well, we're giving a lot of terms here but I'm just gonna write down some formulas that we know and we'll eventually arrive at something to where we can find our pressure here. So we know by row here, apologies. They may look a little similar, but a row or our density is just mass divided by volume. I can actually replace the numerator with the fact that mass is just equal to the number of moles, times the molar mass divided by volume here. Now, I'm also going to look at this term and replace this term with something else according to the ideal gas law, we know that PV is equal to N R T. Now, if I divide R T on both sides, these terms are going to cancel out and we are left with n is equal to this. So I'm going to sub this in for this end. Right here we get then uh let's see here the pressure, which is what we're looking for times the volume divided by the ideal gas constant, times the absolute temperature times the molar mass, divided by volume is equal to our density. Now the volume term is going to cancel out on top and bottom and we are left with that, our density is equal to pressure times the molar mass divided by our ideal gas constant times temperature. I'm gonna rewrite this down here, we have that our density is equal to pressure as well as all over our T. Now, what I'm gonna do is I'm gonna multiply both sides by our T over the molar mass, R. T over the molar mass. All of these terms are going to cancel out and we have now solved for our pressure and we have stated that our pressure is equal to the ideal gas constant times the let's see, absolute temperature, times our density. I'm actually gonna move this over just a little bit. Here, we are All divided by our molar mass. So now that we have that formula, let's just go ahead and plug in all our numbers that we have. So pressure is equal to our density of 0.53 times our ideal gas constant of 8.3145 Times are absolute temperature. Now absolute temperature means we need to have this in Kelvin. So I'm gonna take our temperature and add 273.15 to convert this to Kelvin. This of course, is all divided by our molar mass of air, which is 28.8 times 10 to the negative third. And when you plug all of this into your calculator, you get an atmospheric pressure of 3.61 times 10 to the fourth. Haskell's corresponding to our answer choice of C. Thank you all so much for watching. Hope this video helped. We will see you all in the next one.