A driver with a nearly empty fuel tank may say she is 'running on fumes.' If a 15.0-gallon automobile gas tank had only gasoline vapor remaining in it, what is the farthest the vehicle could travel if it gets 20.0 miles per gallon on liquid gasoline? Assume the average molar mass of molecules in gasoline is 105 g/mol, the density of liquid gasoline is 0.75 g/mL, the pressure is 743 mm Hg, and the temperature is 25 °C.

Video transcript

Welcome back everyone. We're told that a driver who is almost out of fuel may claim to be running on fumes. What is the maximum distance to travel? A taxi could travel if it's 30 gallon, gas tank contained only diesel vapor and gets 20 MPH, sorry, 20 MPG on diesel. Assume that the diesel has a density of 200.8 30 g per mil leader and an average molar mass of 167 g per mole, A pressure of 246 mm of mercury and a temperature of 40°C. So if we know that our tank of the car only contains diesel, we want to figure out the mass of diesel present in that tank to figure out how far that diesel can take the car or the driver. So we're going to first begin by taking the mass or rather the volume of the tank, which from the prompt is given to us as 30 gallons and we want to convert from gallons to leaders. So we're going to recall that we have a conversion for one gallon equivalent to 3.7854 liters. So canceling out gallons. This gives us our volume of our tank being 113.562 liters for the tank for the gas tank of the car. And so next we want to recognize that to figure out the massive diesel in our tank, we need to get molds of diesel first. And so we're going to have to recall our formula which relates pressure times volume to most of our gas times the gas constant R times temperature in kelvin. We're isolating for molds of our gas specifically for our molds of diesel gas. In this case we would say that that is related to pressure times volume divided by the gas constant R times temperature, meaning that we need to figure out the temperature in kelvin where from the prompt were given a temperature of 40 degrees Celsius and so adding 2 73 we would get a kelvin temperature of 313 kelvin. And so now going into our formula for the moles of diesel gas, we would plug in our pressure given in the prompt as 246 mm of mercury Multiplied by our volume of our tank. Given or not given, but which we calculated from gallons two leaders as 113.562 liters. And we're going to divide this by r. Gas constant R which we recall is 0.8206 leaders times A t M's divided by moles, times kelvin. And just to be clear, this reminds me, since we see a t M's in the diameter with r gas constant R. We need to convert that pressure from millimeters of mercury to a t M. So we're going to do that conversion and our numerator. So we're going to have this multiplied by our conversion factor. Where we would recall that we have 760 millimeters of mercury equivalent to 1 80 m. So this is a conversion factor in our notes that we should remember. And this allows us to cancel out millimeters of mercury and leave us with ATMs in our numerator. So again, going back to our denominator, we have our temperature in Kelvin, which we converted to 313 Kelvin And this should say K for Kelvin. So now focusing on canceling out our units, we can get rid of leaders, we can get rid of ATMs, we can get rid of Kelvin and we're left with moles as our final unit, which is what we want. So carefully plugging this all into our calculator. We're going to get a value of 1. Moles of our diesel that should be in our tank. Now, now that we have moles of diesel, we want grams of diesel. So we're going to go from 1.4311 moles of our diesel by multiplying by the molar mass of diesel given in the prompt. So we would have molds of diesel, specifically one mole of diesel in the denominator equivalent to g of diesel as given in the prompt for its molar mass. So now canceling out mold of diesel were left with grams of diesel and now we would calculate a mass equal to 238. or 2 38.9985 g of our diesel. And now that we have this mass of the fuel. Diesel, we can see how far it can take us by using stock geometry to convert from g to miles. So we're going to begin with the 238. g of our diesel. And we're going to first get rid of the unit grams by recalling that to go from grams to first milliliters. We have an equivalent of 0.83 or rather 0.830 g equal to one millimeter. And this comes from that density given in the prompt as a conversion factor. So this allows us to get rid of grams and now we can get rid of milliliters by multiplying by our conversion factor. Which we recall from our lecture where milliliters, two leaders are prefix milli tells us that we have equivalent to one Leader 10 to the third power milliliters. So canceling out male leaders were left with leaders now and we want to get rid of leaders by recalling our second conversion factor from lecture where we have 3. leaders equivalent to one gallon. So canceling out leaders were now going to get rid of our units of gallons by multiplying by our last conversion factor to get two miles, where we would recall that one gallon is equal to miles. And so now canceling out gallons were left with miles as our final unit. And this is going to yield the distance that our gas tank of diesel will take us or take the driver at least equal to 1.5 to 14 miles. And so rounding this to about three sig figs, We would have 1.52 miles exactly as our final answer. And so this would be again the distance that Our taxi driver can travel with the 30 gallon gas tank only containing diesel, getting 20 MPG on diesel. So I hope that everything I referenced was clear. If you have any questions, please leave them down below and I'll see everyone in the next practice video.