The density of air #20# #km# above Earths surface is #92# #g##/##m^3.# The pressure of the atmosphere is #42# #mm# #Hg#, and the temperature is #-63# #°C#. What is the average molar mass of the atmosphere at this altitude?
If the atmosphere at this altitude consists of only #O_2# and #N_2# , what is the mole fraction of each gas?
If the atmosphere at this altitude consists of only
1 Answer
Here's what I got.
Explanation:
The idea here is that you need to start from the standard form of the ideal gas law equation
#color(blue)(PV = nRT) " " " "color(purple)((1))#
and try to manipulate it by incorporating density and molar mass.
As you know, molar mass is defined as mass per unit of mole. This means that you an write the number of moles of substance by using its molar mass and its mass
#color(blue)(M_M = m/n implies n = m/M_M)#
Plug this into equation color(purple)((1)) to get
#PV = m/M_M *RT #
Rearrange this equation to solve for
#M_M = m/V * (RT)/P " " " "color(purple)((2))#
Now focus on density, which is defined as mass per unit of volume.
#color(blue)(rho = m/V)#
Plug this into equation
#M_M = rho * (RT)/P#
Plug in your values and solve for
#92"g"/color(red)(cancel(color(black)("m"^3))) * (1color(red)(cancel(color(black)("m"^3))))/(10^3"L") = "0.092 g/L"#
This will get you
#M_M = 0.092"g"/color(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")))) * (273.15 + (-63))color(red)(cancel(color(black)("K"))))/(42/760color(red)(cancel(color(black)("atm"))))#
#M_M = "28.723 g/mol"#
Now, the average molar mass of air is calculated by taking the weighted average of the molar masses of its constituent gasses.
More specifically, each gas will contribute to the average molar mass proportionally to the mole fraction it has in the mixture.
In your case, air is said to contain oxygen gas,
#chi_(O_2) + chi_(N_2) = 1#
The mixture contains two gases, so their mole fractions must add up to give
#chi_(O_2) xx M_(O_2) + chi_(N_2) xx M_(N_2) = "28.72 g/mol"#
The molar masses of oxygen gas and nitrogen gas are
#M_(O_2) = "31.999 g/mol" " "# and#" "M_(N_2) = "28.0134 g/mol"#
Use these two equations to get
#chi_(O_2) = 1 - chi_(N_2)#
#(1 - chi_(N_2)) * 31.999 color(red)(cancel(color(black)("g/mol"))) + chi_(N_2) * 28.0134 color(red)(cancel(color(black)("g/mol"))) = 28.723 color(red)(cancel(color(black)("g/mol")))#
#31.999 - 3.9856 * chi_(N_2) = 28.723#
#chi_(N_2) = 3.276/3.9856 = 0.822#
This means that you have
#chi_(O_2) = 1- 0.822 = 0.178#
Now, you need to round these answers to two sig figs, the number of sig figs you have for your given values
#M_"M air" = color(green)("29 g/mol")#
#chi_(O_2) = color(green)(0.18)#
#chi_(N_2) = color(green)(0.82)#