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Partial Pessure and Vapor Pressure

Dalton's Law of partial pressure and the mole fraction:


Here's the simple idea: If I have a mixture of gas in a bottle, the total pressure exerted by the mixture is equal to the sum of the pressures exerted by all of the individual gases. This makes sense because we treat all the gases as though they are all "ideal." So, if I have one mole of gas, it does not matter if it is one mole of nitrogen, or 0.19 mole of oxygen, 0.8 mole of nitrogen and 0.01 mole of several other gases, the pressure is the same. The term "n" in PV=nRT does not specify which gas is in there. So, to get the total pressure, you use the total number of moles.
So, if we have a mixture of gases, each one contributes to the total pressure. And, each contributes the amount proportional to the number of moles they contribute. The notes from the board are at the bottom. The equation is this:

`P_T=P_(g1)+P_(g2)+P_(g3)…+P_(gn)`

Substituting '(nRT)/(V)' for P in the equation above, you can write:

`P_T=(n_(g1)RT)/(V)+(n_(g2)RT)/(V)+(n_(g3)RT)/(V)…+(n_(gn)RT)/(V) `

and then pull out the sum of the number of moles of gas

`P_T=(n_1+n_2+n_3…+n_n)(RT)/(V)`

That is, you sum the "n" values and multiply all of them by RT/V.
You could also individually calculate the pressure contributed by each of the gases, which is called the "Partial Pressure", with the equation

Mole Fraction


This is just like percent composition, but we skip the "multiply by 100%" part of the calculation. Mole fraction is called (greek letter "chi") and is calculated as:
`chi=n_(g1)/(n_(gT))=P_(g1)/(P_T)`

Notice that there are no units for this. It is "dimensionless." If you look at the bottom of the notes from the board, you will see the derivation that shows that also. So, if you know any three, you can calculate the last. How do you use this

Vapor Pressure


A topic related to partial pressure is "vapor pressure." If you have a bottle of water, or any liquid, some of the molecules of liquid will evaporate into the space above the liquid. So, the "air" above the liquid in our water bottle will also include some water vapor. Vapor pressure is just the partial pressure of the water gas (or other vapor in the case of other liquids) above the liquid. Vapor pressure is proportional to temperature. The higher the temp, the higher the vapor pressure. There are tables and equations that allow us to know the vapor pressure of various liquids at various temps. So, for example, water vapor pressure at 25
oC (298K) is 0.031 atmospheres.

Temperature is a measure of average kinetic energies of the molecules. In any average, there is a range of values. Some molecules will be above the average, some below. Some of the ones above may be moving fast enough that they "escape" the attraction of water molecules for each other and become a gas. The higher the temperature, the more molecules will reach "escape velocity." In a sealed bottle, some of the molecules that are in gas state will collide and lose some energy, resulting in them sticking together and condensing back into liquid form. This will reach an equilibrium in which there is no net change in the number of moles in the gas form.
Whatever that number of moles is will give you the vapor pressure with the equation

So, suppose you seal up your water bottle and let it sit. You may notice after a while little bubbles forming on the inside of the bottle where the water is. Those are bubbles of water vapor. When you open the bottle, you may notice that there was just a little bit more pressure in it (you would hear a slight "ffft" noise when you open it. That little bit more would be 0.031 atm if the temperature were 298K. That is, the pressure would increase inside the bottle by the vapor pressure of water at that temp.

Boiling:


Boiling is when vapor pressure reaches the surrounding pressure. At that temperature, essentially all of the molecules reach "escape velocity" and go into gas form. When water is at 100
oC (393K), its vapor pressure is 1.00 atm. That means it will boil on a typical day at sea level. At higher altitudes at which atmospheric pressure is lower, water boils at lower temperatures. In Boulder Colorado, the typical atmospheric pressure is about 0.83 atmospheres. Water vapor pressure at 95o C is 0.83 atm. So, in Boulder, water boils at about 95oC.
You can also increase the temperature at which water boils by increasing the pressure. This is how a pressure cooker works. If I can increase the pressure in a pressure cooker to about 2.0 atmospheres, the water will be boiling at 125
oC, which will cook the food faster.

Here's a Problem
The vapor pressure of water is about 0.00276 atm at 263K (-10
oC). On the other hand, it is 0.054 atm at 308 (a pretty hot day). Assuming a that atmospheric pressure is 1.00atm in each case, what is the mole fraction (chi) of water in air at each temperature given?
(got any ideas as to why your skin gets so dry and chapped when it is cold?)



Here’s a hard one:
A typical human breath takes in about 0.50L of air (maximal volume of the lungs is about twice that). Assume a temperature of 290K and atmospheric pressure of 1.00 atm. How many moles of air is that? Assuming the mole fraction of oxygen in air is 0.20, how many moles of oxygen do you take in with each breath?

Now, transport to the top of mount Everest, where the atmospheric pressure is 0.31atm. Assume the mole fraction of oxygen in air is the same. How many moles of air would you breath in with each breath on Mt Everest (in a typical breath)?
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