Partial pressure - liquids

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 Newton's 3rd law, 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: $p=nk_BT$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. It's much harder to measure the actual concentration of gas inside a liquid directly.

But besides being reasonable from a measurement point of view it makes a lot of sense biologically. A critical point in many places in biology is exchanging gases between a gas (air) and a liquid (water). Animals need to take in oxygen from the air into their fluids and put out carbon dioxides. Complex structures such as lungs, alveoli, and gills are evolved to facilitate this.

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 $n_{water}$ molecules per cm³ and the air has a concentration of $n_{air}$ 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 ($n_{air}$) is, by our discussion of gases, proportional to the number density of oxygen molecules by

$$p_{air} = n_{air}k_BT$$

[Careful! Since we are only talking about the oxygen pressure on this page we won't bother to write $p_{oxygen\;in\;air}$ or $n_{oxygen\;in\;air}$. That just seems to cumbersome. But don't confuse $p_{air}$ with the total air pressure. Throughout, we are always talking about the pressure and density of only the oxygen.]

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

$$H = \frac{n_{air}}{n_{water}}$$ 

It's not trivial to figure this ratio out. It basically has to be measured. 

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

$$p_{water} = n_{air}k_BT$$

Note that this is the amount of oxygen in the air that would be in equilibrium with the oxygen in the water. This is NOT equal to the concentration of oxygen in air that produces this pressure. To relate this to the actual concentration of the oxygen in the water, we have to substitute for $n_{air}$

$$n_{air} = Hn_{water}$$

to get

$$p_{water} = (Hn_{water})k_BT = n_{water}(Hk_BT)$$

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, N_A. The number of moles per cubic centimeter is called the molar concentration and is typically written c. We therefore have

$$c_{water} = n_{water} / N_A  \quad \mathrm{or} \quad n_{water} = N_A c_{water}$$

so

$$p_{water}  = n_{water}(Hk_BT)  = c_{water}(HN_Ak_BT)$$

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

The final result typically quoted in chemistry is

$$p_{water}  = k_H c_{water}$$

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. 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. This is not easily calculated. It has to be looked up in a table obtained from measured values.

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 $n_{air} = Hn_{water}$ is probably a good way to think about what Henry's law is telling you.

While the "concentration" meaning of partial pressure is the primary biological consideration for dissolved gases, there are contexts in which the "leads to a force" meaning of partial pressure also can have biological implications. 

When divers descend deep below the water the pressure of the gases they are inhaling has to be increased to match the increased pressure from the water. As a result the concentration of dissolved gases in the blood (particularly nitrogen) can become much higher than those concentrations that are in equilibrium with air at normal pressure. If the pressure is not dropped slowly so that the nitrogen in the blood can be expressed to the air through the lungs, bubbles of nitrogen can form in the blood. 

Now, partial pressure is not just concentration! Inside a gas bubble, the pressure of the gas exerts forces on the walls of the bubble and, as the bubble expands, on the walls of the blood vessels, doing physical damage ("the bends")!

Joe Redish 10/30/14