# Henry's Law with varying temperature

I am trying to determine what the derived partial pressure would be for CO2 in the global atmosphere from historical inconstant temperatures and CO2 concentrations of the global atmosphere:

Is it possible to do this? If so, how?

• An expression for partial pressure, or some sort of useful single value? Did you mean partial pressure of CO2 or something else? – rch Jul 11 '14 at 16:57
• @rch Thank you for looking rch! I don't think I was concentrating when I wrote that. I'll edit... – user7075 Jul 11 '14 at 17:00

Henry's law works for small concentrations of ideal mixtures at equilibrium. Henry's Law constant varies with temperature according to the Van't Hoff equation:

$$k(T)=k(T^{\circ})exp\left[-C\left(\frac{1}{T}-\frac{1}{T\circ}\right)\right]$$

Here $C$ is a constant related to the enthalpy of solvation for each gas, and $^{\circ}$ represents the standard state, T = 298 K and 1 atm.

Using standard state data, you could get an estimate of the partial pressure at a given temperature. However, two key assumptions that are implicit to Henry's law and the Van't Hoff equation do not hold for global atmospheric gas composition:

1. The atmosphere and the oceans are not at equilibrium (not even close when you consider atmospheric layers and convection)
2. $\ce{CO2}$ in seawater is not an ideal mixture (very non-ideal since $\ce{CO2}$ reacts with water).

This is one of the many reasons that climate modeling is difficult - the oceans act as huge carbon sinks with a very complex and not-very-stable relationship between dissolved $\ce{CO2}$ content, temperature, composition of rock on the sea floor, convection currents in the atmosphere and the ocean, amount of shellfish (no kidding), and probably lots of other factors (I'm not a climate expert).

The net result is that while Henry's law can get you a quick idea of how much $\ce{CO2}$ could be dissolved at a given temperature under pseudo-equilibrium conditions (or what the corresponding partial pressure would be for a given concentration in water), it isn't enough to get you anywhere close to a good prediction of what the global atmospheric composition will be, particularly for $\ce{CO2}$ (due to its reactivity with water and incorporation into biomass).

Now, if you really just want to convert $\ce{CO2}$ concentration in ppm to a partial pressure, you can do that.

$\ce{CO2}$ in air is close enough to ideal that we can use the ideal gas law. For an ideal gas, the partial pressure is the same as whatever the pressure would be if it were in a container by itself:

$$P_1 = \frac{n_1RT}{V}$$

You can find V for a given mass of air, find the mass of $\ce{CO2}$ from the concentration, then convert to moles and plug it into the above.

You currently have numbers in ppm (of ${CO_2}$); ppm values for gas tend to be equivalent to ${\mu mol/mol}$ units. Can't you just use that, the the definition of partial pressure, and the atmospheric pressure and be set (though it varies by altitude and a little over time too)? Some forum has a similar homework-level question. Not sure if I'm missing something with that.

I take it you wanted to use Henry's Law to relate concentration to partial pressure. typically that's used in the realm of vapor-liquid systems; not sure how it fares in a mixture of many gases that I'd qualify the atmosphere as. See things to note: (Chemwiki)

Looking at a (pretty crude IMHO) pressure vs altitude graph of the atmosphere (Wikipedia: Atmospheric pressure): you can see the pressure even within the troposphere ranges from ~12kPa all the way to ~100kPa (sea level: ~101 kPa). The troposphere has 80% of the atmosphere's mass and you want the whole atmosphere anyways, so just take 101kPa as atmospheric pressure? I also assume atmosphere's mass is constant over the span of 400 million years - no clue on whether I can neglect losses and gains from weather, the biosphere, and space over this time span. Then you will have a bunch of carbon dioxide partial pressures over time.

On short-term constancy of the atmosphere's mass, see SE.Physics. The mean even changes by season. It's comparatively small comparing 1.5E+15 kg to the total of 5.15E+18 kg.