Question

Question #b61a4

Answer 1

`"C"_4"H"_8` and `"C"_4"H"_10`.

Explanation:

Start by using the ideal gas law equation to find the total number of moles of gas present in the mixture.

`color(blue)(ul(color(black)(PV = nRT)))`

Here

• `P` is the pressure of the gas
• `V` is the volume it occupies
• `n` is the number of moles of gas present in the sample
• `R` is the universal gas constant, equal to `0.0821("atm L")/("mol K")`
• `T` is the absolute temperature of the gas

Rearrange to solve for `n`

`PV = nRT implies n = (PV)/(RT)`

Plug in your values to find

`n = (2.46 color(red)(cancel(color(black)("atm"))) * 5.0color(red)(cancel(color(black)("L"))))/(0.0821(color(red)(cancel(color(black)("atm"))) * color(red)(cancel(color(black)("L"))))/("mol" * color(red)(cancel(color(black)("K")))) * (27 + 273.15)color(red)(cancel(color(black)("K"))))`

`n ~~ "0.5 moles"`

Now, you know that this mixture is `84.5%` carbon by mass, which means that it contains

`28.4 color(red)(cancel(color(black)("g mixture"))) * overbrace("84.5 g C"/(100color(red)(cancel(color(black)("g mixture")))))^(color(blue)("= 84.5% carbon")) = "23.998 g C"`

Use the molar mass of carbon to determine how many moles of carbon are present in the sample

`23.998 color(red)(cancel(color(black)("g"))) * "1 mole C"/(12.011color(red)(cancel(color(black)("g")))) = 1.998 ~~ "2 moles C"`

Notice that the two gases contain equal numbers of moles of carbon, since

• `"C"_x"H"_8 implies "1 mole C"_x"H"_8 = xcolor(white)(.)"moles C"`
• `"C"_x"H"_10 implies "1 mole C"_x"H"_10 = xcolor(white)(.)"moles C"`

Now, if you take `n` to be the number of moles of `"C"_x"H"_8` and present in the sample, you can say that the sample contains

`(0.5 - n)color(white)(.)"moles C"_x"H"_10`

You can also say that the sample contains

`overbrace((n * x)color(white)(.)"moles C")^(color(blue)("from C"_x"H"_8)) + overbrace([(0.5 - n) * x ]color(white)(.)"moles C")^(color(blue)("from C"_x"H"_10)) = "2 moles C"`

This is equivalent to

`color(red)(cancel(color(black)(n * x))) + 0.5 * x - color(red)(cancel(color(black)(n * x))) = 2`

which gets you

`0.5x = 2 implies x= 2/0.5 = 4`

Therefore, you can say that the two gases are

`"C"_4"H"_8" "` and `" " "C"_4"H"_10`