The density of a gas of unknown molar mass was measured as a function of pressure at 0 C, as in the table that follows. (a) Determine a precise molar mass for the gas. [Hint: Graph d>P versus P.]

Video transcript

Hey everyone in this example, we need to analyze the following pressure density data that is obtained for an unknown gas at 60°C. We're told to use this data to calculate the molar mass of our gas. And we need to draw a graph of density versus pressure or density divided by pressure versus pressure. So what we should first recall is that for an ideal gas, the value for density divided by pressure is going to be constant at all pressure values. We also want to remember that due to the non zero molecular volumes and attractive forces that are between our gas molecules. Our value for density divided by pressure is actually therefore not going to be constant. And so what we would recall is that at low pressure gasses are going to behave ideally Only at zero pressure. So based on the data given to us, we want to find the value for density divided by pressure at zero pressure. So this is what we're going to deduce from our graph. And as the prompt states to generate our graph, we want to plot density divided by pressure versus pressure as the prompt states. So we're going to go ahead and do so by entering the data given to us here in our example and we will generate the following graph. So I'll make some room below here. So this is the graph that we will generate in our calculators. After we input the data above for density divided by pressure versus pressure into our calculators. We're going to have this graph here. And as you can see we have our density divided by pressure values on the y axis and our pressure values on our X axis. And all of our data values are plotted on our graph in the line here. We are also, we also have noted on our graph our equation of our line which also comes from our graph on our calculators. And as we can see we have a y intercept here of .8409. And because we know that our density divided by pressure value is our Y axis. We can understand that our Y intercept is going to be our value for density divided by pressure at zero pressure. So to write this more clearly, we can say that this is equal to 0.8409. So we are going to use this value to help us find our molar mass for our unknown gas. And now that we know this value. Our next step is to recall our gas density formula where we recall that row or density is equal to our pressure of our gas, multiplied by its polarity, which is then divided by the gas constant R times the temperature in kelvin. And when we look above to our prompt, we are given a temperature here of 60°C. So we're going to take our temperature And we're gonna add to .15 Kelvin or just to 73.15 to get our temperature in Kelvin. And this is going to give us a value equal to 333.15 Kelvin for our temperature. So now that we have our temperature value, we want to reorganize our gas density formula to solve for molar mass. So we're gonna isolate molar mass. And in doing so we're going to take our gas constant R. Times temperature and multiply it by our value for density divided by pressure. So we're gonna do so below. And sorry, what we would get for our molar mass of our unknown gas is that it's equal to r. Gas constant R. Which we should recall is 0.8206. We have units of leaders times a T. M's divided by moles times kelvin. And just to make more room, I'm just going to move our solution for our molar mass here below our graph. So next we're going to multiply by our temperature in Kelvin recall that we converted that to 333.15 Kelvin Were then multiplying this value by our value for density divided by pressure, which we already actually know from above. So we can just plug that in as 0.8409. And we have units of g divided by leaders for density. Which leaders in the denominator is going to be multiplied by ATMs for our pressure value in the denominator. So now that we have our full equation written out, we want to sell for the mass of our unknown gas. And we're just going to input all of this into our calculators. And in doing so we're also going to focus on canceling out our units so we can get rid of a. T. M's. We can also cancel out leaders as well as kelvin. And we're left with grams per mole for our molar mass, which is what we want. So what we should get for our molar mass of our unknown gas. And sorry about that. So for our molar mass of our unknown guests, we would get a value equal to 22.9 g per mole. And this is going to complete this example here as our final answer for our molar mass of our unknown gas given in the prompt. So I hope that everything we reviewed was clear. If you have any questions, just leave them down below and I will see everyone in the next practice video