A solution of LiCl in a mixture of water and methanol 1CH3OH2 has a vapor pressure of 39.4 mm Hg at 17 °C and 68.2 mm Hg at 27 °C. The vapor pressure of pure water is 14.5 mm Hg at 17 °C and 26.8 mm Hg at 27 °C, and the vapor pressure of pure methanol is 82.5 mm Hg at 17 °C and 140.3 mm Hg at 27 °C. What is the composition of the solution in mass percent?

Video transcript

hi everyone for this problem, it reads the vapor pressure of a solution of potassium bromide and water and ethanol is 151 millimeters of mercury at 50 degrees Celsius and 374 millimeters of mercury at 70 degrees Celsius. What is the mass percent of potassium bromide, water and ethanol in the solution? If water has a pressure of 92.5 millimeters of mercury at 50 degrees Celsius and 234 millimeters of mercury at 70 degrees Celsius, while ethanol has a vapor pressure of 220 millimeters of mercury at 50 degrees Celsius and 541 millimeters of mercury at 70 degrees Celsius. So, we have a lot of information presented here, and the question that we want to answer is the mass percent of potassium bromide water and ethanol in the solution. So we need three mass percentages here. And what we're going to do is we're going to let x equal the mole fraction of water, and we're going to let y equal the mole fraction of ethanol. So with that information, we can set up two equations. Okay, and our first equation will label this equation one equation, one is going to equal our water vapor pressure times the ethanol vapor pressure equal to the potassium bromide vapor pressure. Okay, so it will look like this. So for water we're going to have 92. times X, which we said x is going to equal the mole fraction of water? And this is going to be plus 220, which is our ethanol why Is equal to 151. Okay, and we're going to have a second equation which will say is equation two. We're going to let equation too be equal to 234 times X. Will the mole fraction plus 541 times y is equal to 374. Okay, so with that let's take a look at equation two for equation to if we solve for X, we're going to rewrite this to solve for X. And what we're going to get is X is equal to 374 minus 541 times why? Over 234. So we rearrange the equation to to solve for X. So what we can do here is substitute X from equation two, equation one. So let's go ahead and write that. So we're going to substitute X. From equation 22, equation one. Okay, so when we do that, what this will look like is 92.5 times -541 times y Over plus why Is equal 251? So that's what this looks like. Once we do the substitution. So now we just want to simplify this because we want to solve for why? Okay, so let's go ahead and simplify this. So the left side is going to become 92.5 times 374. Over 92.5 times 541 times why? Over 234 plus 220 Y. Is equal to 1 51. Okay, so let's go ahead and simplify the left side where It will become 147. 213. times y plus 220 Times Y is equal to 151. Okay, so we're going to simplify this by this becoming 6.14316 times why Is equal to 3.15812. So now we can solve for Y. Okay, we're gonna divide both sides And when we do that, what we get is why is equal to 0.51409. Okay, so now that we know what why is we can solve for X. Okay, and X. Because we're gonna use this equation here, so let's go ahead and solve for X. So X is equal to 374 minus 541 times Y. Which we said why we just saw four is 0. divided by 234. So what this means is X is equal to 0.40994. So we know what why is and we know what X is. And so we can use this to solve for the mole fraction of potassium bromide because the mole fraction of potassium bromide is equal to one minus X minus Y. Okay. And because we know what both X and y is, what this becomes is one minus X, Which is 4.0 994. So one minus four point excuse me, It's -0.40994 minus why? Which is 0. is equal to 0.07597. Okay, so that is the mole fraction of potassium bromine and the mole fraction equals the number of moles of each component because the total moses one. Alright, so now we have all the information that we need and we can calculate the mass of each. So that's going to be what we're going to do next. We're going to calculate the mass of potassium bromide, water and ethanol. Given the amount of moles, that's going to be our starting point, we're going to go from moles to grams. So to find the mass of potassium bromide, we just calculated what the moles is. Okay, it is this value right here. Alright, so that is our starting point. So we have 0.7597 moles of potassium bromide. We need to go from moles of potassium bromide to grams. And we can do that using the molar mass of potassium bromide. Okay. And one mole of potassium bromide. The mass is 119.002g. Okay, so our moles is going to cancel and we're left with g. So our moles are are mass of potassium bromide. Once we do this calculation is equal to 9.04058g. Okay, so now let's go ahead and do the same thing. But for the mass of water for a massive water, we calculated the moles of water and that was signified by X. And so we have 0. moles of water. And we need to go for moles of water, two g of water and one mole of water. The molar mass is 18 point 016g. Okay, so moles cancel. And we're left with grams. And so the massive water is 7. g. So now let's go ahead and do the same thing for the last thing, which is our ethanol mass of ethanol. So we calculated the molds of ethanol, which was why that was 0.51409 moles. So we need to go from moles of ethanol, two g of ethanol and we'll do that using its smaller mass and one mole of ethanol. There is 46.7 g of ethanol moles cancel. And we calculate this to be 23.68413 g of ethanol. So we have the information that we need to calculate the total mass. We're gonna add these all up to get the total mass. Okay, so the total mass Is going to equal 9. 9.04058g plus 7. 548 g plus 23. g. This gives us a total mass of .110187 g. So now that we know the total mass, we can calculate the mass percent because our mass percent is going to be the mass of each over the total mass. Alright, so let's go ahead and first calculate our mass percent of potassium bromide. So it's going to be the part over the whole. Okay, so the part, the mass that we calculated for potassium bromide is 9. 058 g. And the whole which is the total mass is 40. g. And this is going to be multiplied by 100. Okay, so that means the mass percent for potassium bromide is 22.5%. And now we'll go ahead and do the same thing for water and ethanol. So our mass percent for water is going to be the part over the whole. So the mask for water was 7. 548 g. That was the part and the whole, which is the total mass is 40.110187 g. Multiplied by 100. The mask% 18.4%. And lastly lastly we have the mass percent for ethanol. Okay, so the mask for Ethanol was calculated to be 23.68413g. And this is gonna be over the total mass of 40. g. This is multiplied by 100 and we get 59.1%. Okay, so let me just highlight the final answers. So the mass percent for potassium bromide is 22.5%. The mass percent for water is 18.4% and the mass percent for ethanol is 59.1%. That is it for this problem. I hope this was helpful.