Which has a higher average speed, a Ferrari at 145 mph or a gaseous UF6 molecule at 145 °C?

Video transcript

Welcome back everyone we need to identify which would have a greater mean speed, an F1 car moving at 230 mph, or methane, gas molecules held at 120°C. So we should recall our formula for root mean square speed or velocity represented by this symbol here recall that it's set equal to the square root of three, multiplied by r, gas constant. R multiplied by r temperature in kelvin, divided by our molar mass of our substance represented in kilograms Permal. We should also recognize that our root mean square speed is recorded in units of meters per second. So taking that speed of the car given in the prompt, we have 230 MPH and we want to convert to meters per second by first converting from miles to kilometers. So we're going to recall our conversion factor. That one mile is equivalent to 1.6093 kilometers. And now we can cancel out miles and focus on getting rid of the kilometer units so that we can get two m. So we're going to have one kilometer in the denominator and recall that it's equivalent to to the third power meters. Now we can cancel out kilometers and focus on getting our units of seconds. So we're going to multiply by our next conversion factor where we would recall that one hour in the numerator so that ours can cancel out is equivalent to 3600 seconds. So now we can get rid of ours. We're left with units of meters per second. And this is going to yield a velocity for the car of 103 m per second. So this is going to be our mean speed of the car. But now we need the mean speed Of our methane gas molecules. So we're going to first begin by calculating the molar mass of methane from the periodic table, where we would first find carbon in group four a and see that it corresponds to a molar mass of 12.01 g per mole. And now we're going to add this to our molar mass of hydrogen, found in group one a on the periodic table corresponding to a mass of 1.08 g per mole, which is multiplied by our four hydrogen atoms. And so taking the total here, we're going to have a molar mass for methane equal to 16.042 g per mole. And now we want to convert two kg per mole because we're going to use our mean square speed formula to find the mean speed of methane molecules. So we're going to multiply by grams and then kilograms in the numerator where we would recall that one g is equal to 10 to the negative third power kilograms. Getting rid of our units of grams were left with kilograms is our final unit. And this is going to give us a molar mass for methane as 0.016, 4, 2 or sorry, 0. kilograms Permal. So now that we have this molar mass, let's go into our root mean square speed formula. So we would say that our root mean square speed of our methane gas molecules are equal to the square root of three. Multiplied by r gas constant R Which we recall is 8.314 joules divided by moles times kelvin multiplied by our temperature where in the prompt were given a temp of 120 degrees Celsius. We need this in Kelvin. So we're going to add to 73.15 Kelvin And now we're just going to divide by our molar mass which we just calculated to be 0.016042 kg per mole for our methane molecules. Now we're going to simplify this and this parentheses should be complete here. So we're going to simplify this and say that our root mean square speed of our methane molecules are equal to 24.942. We have units of joules divided by most times Kelvin when we multiply the product here. And now we're going to multiply this by our temperature which we convert to 393.15 Kelvin all divided by 0.016042 kg per mole. Now we also should recognize that we have a conversion factor where one jewel is equivalent to one kg times meter squared divided by seconds squared. So we can actually interpret our numerator as a kilogram times meter squared divided by seconds squared divided by moles times kelvin. And this allows us to cancel out kilograms as well as cancel out our units of moles as well as kelvin. And we still have that square root symbol, so we need to also carry that over and let's simplify in our next line so that we can say that our root mean square speed of methane Is now equal to 781.8 and what will be left with is one unit of m and one unit of seconds when our square root term is applied. And this leads us with units of meters per second as our final units. For our mean square speed of methane and we can round this 236 fix as 782 m per second. And so for our final answer, we have our root mean square speed of methane As 782 m/s versus our mean square speed of our F one car, which we calculated to be 103 m/s. So we'll say the main square speed of our F one car was 103 m/s And we would say that this is going to be a greater mean square speed than the F1 car. So our final answer is going to be that the methane gas molecules have the higher mean square speed. So this would be our final answer, highlighted in yellow for the methane gas gas molecules, having the higher mean square speed at 120°C. If you have any questions, please, leave them down below, and I will see everyone in the next practice video.