Martian Climate. The atmosphere of Mars is mostly CO2 (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0°C in summer to -100°C in winter. Over the course of a Martian year, what are the ranges of (b) the density (in mol/m^3) of the atmosphere?

Video transcript

Welcome back everybody. We are looking at a container filled with S. 02 and inside this container there's a movable piston. We're told that the pressure is constantly kept at 924 pascal's and as the piston moves, the temperature ranges from 100 degrees Celsius to negative 10 degrees Celsius. And we are tasked with finding what the range of the density is as the temperature fluctuates. So a K. We are going to find a lower bound for density and we are going to find an upper bound for density. Well, we're dealing with pressure and temperature and we're dealing with this this closed container. First thing that pops out to me that we should use is the ideal gas law. So E. V equals N R T. Is our law. And we are looking for density in most per meters cubed. As you can see by all of our answer choices. So I need to isolate N over V. So I'm going to divide both sides by V. R. E. And what you'll see here is that the V. S will cancel out on the left and the R. T. S. Will cancel out on the right. Leaving us with our dot desired equation, which is N over V is equal to the pressure over the ideal gas constant, times our temperature. And what we're going to do is we're going to plug in all of our values. But we're going to do it twice for each of the bounds of the temperature. Let's go ahead and start with the lower bound of density here. So our lower density bound. So this is going to be equal to. And hold on. Let me just backtrack real quick if I haven't made it clear and over V this is equivalent to density. There we go. So for our lower bound we have our pressure of 920 for pascal's divided by our ideal gas constant of eight 0.314 or times are temperature of 100 degrees Celsius. But we need to work in kelvin. So I'm gonna add 273 to this. When you plug this into your calculator, you get 0. moles per meter cubed. Great. So now let's go ahead and do the same thing for the upper bound of density, we are going to have a pressure of 924 on top. But by our ideal gas constant of 8.314 times our temperature of negative 10 plus 2 73. To convert to kelvin. This gives us an upper bound of 0.423 moles per meter cubed. Now, I'm going to write this in terms of limits here and we get that the range for our density is as follows our density is greater than or equal to 0.298 moles per meter cubed And our density is less than or equal .4-3 moles are meter cubed. And this of course corresponds to our final answer of answer choice. D thank you all so much for watching hope. This video helped. We will see you all in the next one.