Gases as Solutes

For many gases, equilibrium solubility in water decreases as temperature rises at fixed gas partial pressure. That is a substance-specific equilibrium trend, distinct from dissolution rate. Many solid solubilities increase with temperature, but this is not universal; use the measured curve or supplied equilibrium model.

For a dilute dissolved gas at equilibrium, Henry’s law relates its concentration to that gas’s partial pressure, with a temperature-dependent coefficient. Raising its partial pressure at fixed temperature often increases its dissolved concentration. The temperature trend is represented by the coefficient and the dissolution thermodynamics; it is not explained simply by subtracting the liquid’s vapor pressure from outside total pressure.

It might be hard to understand how a gas can get into water, but our bodies actually rely upon this process every few seconds. Can you guess why? Here’s a hint: fish rely upon a very similar process to survive.

The simple act of breathing would not be useful to us if not for the solubility of gases, in this case Oxygen. When you breathe, your lungs fill with Oxygen-rich, Carbon-Dioxide-poor air. The alveoli in your lungs are carrying blood that has a lot of carbon dioxide, and when they interact with the air in your lungs, the carbon dioxide dissipates while the Oxygen dissolves into the blood. If gases weren’t soluble, then this process would not work and we would not be able to acquire oxygen by breathing.

If you’ve ever had a fish tank, chances are you needed to have an aerator in there. The typical aerator relies on putting forth a froth of bubbles at the bottom of the tank, allowing them to rise up through the water. During this rise, the gas in the bubbles is dissolving into the water. This process oxygenates the water, allowing the fish to breathe.

Another cool example of gases dissolving in water is in rainwater. When rain condenses in the atmosphere, it begins to grow in size until it becomes heavy enough to fall. When it does fall, it has quite a ways to travel. While is falls, the carbon dioxide in the surrounding air dissolves into the drop. Once carbon dioxide dissolves in water, it becomes acidic. The process is as follows:

Dissolved CO₂ participates in equilibria: CO₂(aq)+H₂O(l)⇌H₂CO₃(aq). Only a fraction is present as the hydrated carbonic-acid species. Write an equilibrium arrow rather than implying every dissolved CO₂ molecule converts completely.

Carbonic acid is a weak acid: H₂CO₃(aq)+H₂O(l)⇌H₃O⁺(aq)+HCO₃⁻(aq). Bicarbonate has charge −1. The equilibrium position depends on the stated conditions; the arrow does not imply complete ionization.

Bicarbonate can undergo a second acid equilibrium: HCO₃⁻(aq)+H₂O(l)⇌H₃O⁺(aq)+CO₃²⁻(aq). Bicarbonate is also capable of accepting a proton. Detailed coupled polyprotic calculations are enrichment; the AP skill here is identifying conjugate species and explaining equilibrium.

By releasing the Hydrogen ion, the carbon dioxide causes the drop to become acidic. This process will be concentrated upon in a future chapter.

Due to the carbon dioxide’s influence, the rainwater is naturally a little acidic. This can become even more acidic if there are other acidic gases present, and this leads to acid rain. More on that in a future chapter.

Okay, okay. You know that gases dissolve in water, and that they can influence us in many different ways, but what actually happens when they dissolve? What kind of processes allow the gas to become more soluble or less soluble? What kind of gases are soluble and what kind aren’t?

Well that’s a long list of questions with a lot of answers to provide. We can start by introducing a quantifiable way of measuring a gas’ solubility. It is called Henry’s Law. It looks like this:

$p_A=k_H\cdot [A]$

The pA refers to the partial pressure of the gas (A) outside of the solution. The [A] is simply the molar concentration of the gas in solution. The kH is Henry’s Constant, and it provides a link between the two. Henry’s law can be written a multitude of ways, and therefore it is important to look at the units of the Henry’s Constant you are using. Here’s a few examples of Henry’s Constant units and the appropriate equation:

$p_A = k_H\cdot[A]_{aq}\phantom{phnatomphanton}k_H = \frac{L\cdot atm}{mol}$

$[A]_{aq} = k_H \cdot p_A \phantom{phantomphanton}k_H = \frac{mol}{L\cdot atm}$

$[A]_{aq} = k_H\cdot [A]_{gas}\phantom{phantomphan}k_H = \text{unitless}$

We will be using the first form of Henry’s Law, so all the Henry’s constants you will be using are going to be in L-atm/mol.

Now you might be asking yourself: “So Henry’s Law relies only on the partial pressure of a gas? I thought you said that it is affected by temperature as well?” That is a great question.

In fact, temperature does play a strong role in Henry’s Law. It presents itself in Henry’s Constant. Henry’s Constant is so reliant on temperature it is actually a bit of a misnomer to call it a constant. Henry’s Constant is actually a function of temperature and gas. Typically you will be given the Henry’s Constant at 25ºC, the scientific number for room temperature. That’s about 76ºF, if you were curious.

In order to find the Henry’s constant at any given temperature, you will need to know two things: The Enthalpy of Solvation for the gas ($\Delta H_{solv}$) and Henry’s constant at 25ºC. You will also need to convert your desired temperature to Kelvin, as it relies on the absolute temperature. The relationship is as follows:

Henry-law temperature enrichment, using the explicit convention p=k_p/cc: ln[k_p/c(T)/k_p/c(T₀)]=(ΔH_diss/R)(1/T−1/T₀), when the dissolution enthalpy is approximated constant. For an exothermic dissolution ΔH_diss<0. Warming then increases this pressure-per-concentration coefficient and decreases c at fixed p. A coefficient using the reciprocal convention has the opposite temperature sign.

The 298 comes from the fact that the standard Henry’s Constant is given at 25ºC, which is equal to 298K. That is also why the subscript for the Henry’s Constant on the right side of the equation says 298. Remember that you will need to input T in Kelvin for this equation to work.

If the pressure-per-concentration form is written k(T)=k(T₀)exp[−C(1/T−1/T₀)], define C=−ΔH_diss/R, in kelvin. Positive C corresponds to an exothermic dissolution in this convention. State that definition instead of saying C=ΔH/R while placing an extra negative sign into the same equation.

Reviewed enrichment relation: k_p/c(T)=k_p/c(T₀)exp[−C(1/T−1/T₀)], with C=−ΔH_diss/R and absolute temperatures. It is a supplied model for a defined gas, solvent and temperature range, not a universal prediction at arbitrary pressure or for strongly reacting gas solutions.

As you can see, the Henry’s constant can vary considerably between gases. If you go back to our equation, we see that a higher Henry’s Constant will correlate to a lower solubility. The most soluble gases in this table, Ammonia, Sulfur Dioxide, and Hydrogen Chloride, all easily affect the pH of water. This reflects the high solubility of acids and bases in water, even for gases. Now it’s time to start using these numbers to solve problems.

Example Problem 2.3.1
What is the solubility of Oxygen from air to water at sea level, if the temperature is 15ºC?