Question #956b2

The idea here is that the volume of a gas, #V#, is proportional to the number of moles of gas, #n#, present in the sample #-># think Avogadro's Law here.

#V prop n#

The balloon is said to contain #2# parts helium gas to #1# part air, which implies that the volume of helium gas is twice as large as the volume of air.

Consequently, you can say that the balloon contains twice as many moles of helium gas than moles of air. In other words, if you take #n# to be the number of moles of air, you can say that the balloon contains #2n# moles of helium gas.

Mathematically, you can show that this is the case because

#V_"He" prop n_"He" " " -># for helium

#V_"air" prop n_"air" " " -># for air

gets you

#V_"He"/V_"air" = n_"He"/n_"air"#

And since you know that

#V_"He" = 2 * V_"air"#

you will end up with

#n_"He" = (2 * color(red)(cancel(color(black)(V_"air"))))/color(red)(cancel(color(black)(V_"air"))) * n_"air"#

#n_"He" = 2 * n_"air"#

#color(white)(a/a)#

Now, the partial pressure of a gas that's part of a gaseous mixture depends on the mole fraction of the gas in the mixture #-># think Dalton's Law of Partial Pressures here.

The mole fraction of helium, #chi_"He"#, is calculated by dividing the number of moles of helium by the total number of moles of gas present in the balloon.

Since we've said that for #n# moles of air, the balloon must contain #2n# moles of helium, you can say that the total number of moles of gas will be

#n + 2n = 3n#

This means that you have

#chi_"He" = (color(blue)(cancel(color(black)(n))) color(red)(cancel(color(black)("moles"))))/(3color(blue)(cancel(color(black)(n))) color(red)(cancel(color(black)("moles")))) = 1/3#

The partial pressure of helium in the balloon will be

#P_"He" = chi_ "he" * P_"total"#

Here #P_"total"#, the total pressure in the balloon, is equal to #"100.1 kPa"#.

Plug in your values to find

#color(darkgreen)(ul(color(black)(chi_"He"))) = 1/3 * "100.1 kPa" = color(radkgreen)(ul(color(black)("33.37 kPa")))#

The answer is rounded to four sig figs, the number of sig figs you have for the total pressure in the balloon.