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Partial pressure - liquids

Page history last edited by Joe Redish 9 years, 5 months ago

Working Content Thermodynamics and Statistical Physics > Kinetic Theory: The ideal gas law 

  

7.1.2

 

Prerequisites:

 

In our discussion of pressure in dilute (nearly ideal) gases, we learned that each molecule in the gas that bounced off a wall felt a force from the wall, and therefore, by N3, exerted a force on the wall. The pressure (force on the wall per unit area) was proportional to the concentration -- the number of molecules per unit volume hitting the wall. At constant T, the pressure could therefore be used as a stand-in for concentration (= number density).

 

When there is a mixture of gases, each molecule of each gas contributes the same amount to the pressure, so the total pressure is the sum of the partial pressures created by each gas separately. 

 

Gases can also be dissolved in liquids. In order to make contact with the way concentrations are described in gases, we would like to use the same language. But a problem arises that can lead to confusion. The key physics that makes the ideal gas law work is that it is dilute. Molecules are far apart, collide rarely, and mostly travel in straight lines (ignoring gravity). This leads to the ideal gas law: p=nkBT where n is the concentration.   

 

But in liquids, molecules are close to each other. Basically they touching and interact with each other all the time. This means the the ideal gas law does NOT hold for liquids -- not even for gases dissolved in liquids. 

 

We might say, well, let's just use the same equation anyway. This would say the partial pressure is the concentration the gas would have if there were no liquid. I've struck this through since this is NOT what is done. Rather, a somewhat more sophisticated choice is made. It is defined as follows.

 

The partial pressure of a gas dissolved in a liquid is taken to be that partial pressure of gas that would be in equilibrium when that gas is in contact with the liquid.

 

Although this sounds a bit confusing, it makes sense if you consider that one way to measure the concentration of dissolved gas in a liquid is to let it come to equilibrium with a small open space above the liquid and then measure the concentration (partial pressure) in the gas. 

 

Let's consider an example. Consider oxygen (O2) dissolved in water. In the picture at the right, we show a container of water with a surface open to the air above the water. The dissolved oxygen has a concentration of nwater molecules per cm3 and the air has a concentration of nair molecules per cm3.  Only the oxygen molecules are shown (but the water is shown as blue). 

 

The partial pressure of the oxygen in the air (nair) is, by our discussion of gases, proportional to the number density of oxygen molecules by

 

poxygen = nairkBT.

 

Molecules of oxygen are continually crossing the surface from both sides.The equilibrium value occurs (the numbers stabilize) when equal numbers leave and enter the water. But because the molecules of oxygen interact strongly with the water molecules (but not strongly with the molecules of air, the equilibrium value does NOT occur when the two concentrations are the same. 

 

Let's define the ratio of the two concentrations at equilibrium at H. (Note that H is dimensionless since it is the ratio of two of the same kinds of quantities.) It will depend on the properties of the liquid and what gas we are considering. The equilibrium concentrations determine H by

 

H = nair/nwater 

 

We define the partial pressure of the oxygen in water  to be

 

pwaternairkBT

 

-- the amount of oxygen in the air that would be in equilibrium with the oxygen in the water. To relate this to the actual concentration of the oxygen in the water, we have to substitute for nair

 

nair = Hnwater 

 

to get

pwater = (Hnwater )kBT = (HkBTnwater

 

This relates the partial pressure of the oxygen above the water to the concentration (number density) of oxygen in the water. 

 

Chemists (and biologists) tend to prefer to use molar concentration rather than number of molecules. To convert the number of molecules to the number of moles we have to divide by Avogadro's number, NA. The number of moles per cubic centimeter is called the molar concentration and is typically written c. We therefore have

 

cwater =  nwater/NA       or     nwater = NA cwater 

 

so

pwater = (HkBTnwater (HkBNATcwater 

 

Note that changing from number density (n) to molar density (c) just changes kB into kBNA = R, the familiar gas constant from chemistry. The combination HRT is referred to as Henry's constant, kH.

 

The final result typically quoted in chemistry is

 

pwater = kH cwater 

 

that is, what we define to be the partial pressure of oxygen in water is proportional to the molar concentration of oxygen in water. This is called Henry's law and the constant of proportionality is Henry's constant. Of course this is easily generalized to any liquid and any dissolved gas. Note also that although it's called "Henry's constant", it actually depends on what dissolved gas we are talking about, the temperature, and the properties of the liquid. 

 

The discussion of Henry's law and the Henry constant is somewhat confused by the fact that different communities measure pressure in different units and different communities measure concentrations in different units. As a result, there are lots of different values for a single "Henry constant." Although the (unitless) constant "H" we defined above is not commonly used, the relation nair = Hnwater is probably a good way to think about what Henry's law is telling you.

 

Joe Redish 10/30/14

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