CASE STUDY: The pH of Natural and Acid Rain

Source:

adapted from Consider a Spherical Cow: A Course in Environmental Problem-Solving, by John Harte, and The Dynamic Environment, by Leonard J. Stolzberg. Narrative and case study design by Robert R. Gotwals, Jr.

WARNING! This case study requires a solid chemistry background AND the willingness to tackle a large, complex system that requires solid planning PRIOR to building the model .

Goal:

To learn about tropospheric chemistry, in particular man's effect on acid rain.

Background Reading:

One of the most important environmental problems is that of acid rain. An acid is any substance that has a pH value of less than 7.0. The question we want to answer is: what would be the pH of rain be in the absence of anthropogenic (man-made) sources of sulfuric and nitric acids? To be able to understand this model, you need to understand a little about acid and acid rain.

Acid rain is defined as the mixing of atmospheric pollutants with natural precipitation (rain, snow, sleet, etc.) to form an acidic aqueous (water) compound. When this acidic water is deposited on the Earth in the form of rain or snow, the acid causes substantial damage to farm crops, plants and trees, buildings and statues, and living creatures who live in ponds, lakes, and rivers.

Acid rain is produced by the emission of various pollutants into the atmosphere by various industries. Coal-burning plants that produce electricity, for example, emit large quantities of sulfur-containing chemicals into the air. These chemicals undergo a series of chemical reactions in the lower atmosphere. The products of these reactions combine with natural rainwater and snow, and are then deposited as acid rain.

A quick overview of acidity might be helpful. There are a number of definitions for an acid, as you know if you have taken first-year chemistry! We'll use a fairly simple definition. An acid is created when a hydrogen ion (a hydrogen atom with a positive charge, written H⁺) mixes with ordinary water (H₂O) to form a hydronium ion (H₃O⁺). The reaction looks like this:

Reactants		Product
H⁺ + H₂O	------->	H₃O⁺

If we know how much of the hydrogen atom we have (that is, if we know its concentration, written as [H⁺], we can calculate its pH. pH is a measure of the acidity of a liquid solution, and is shown on a scale from 1 to 14:

ACIDIC	NEUTRAL	BASIC
____________	___\|______	________________
1 2 3 4 5	6 7 8 9	10 11 12 13 14

According to the graphic above, an chemical with a pH less than 7 is considered to be an acid, a neutral substance (like pure water) has a pH of 7, and any liquid above a pH of 7 is called basic. We are concerned, of course, with rainwater that has a pH of less than 7!

To build this model, we need to investigate the chemistry of acid rain in a little more detail. The primary pollutant generated by coal-burning plants is sulfuric acid, or H₂SO₄. When sulfuric acid is deposited into the atmosphere, it dissociates, or breaks apart, into two new products:

Reactant		Products
H₂SO₄	----->	H^{+ +}HSO_4^-

Notice the charges on the right -- a plus charge on the hydrogen and the minus charge on the sulfur compound add up to zero, which is the charge on the sulfuric acid. More importantly, notice that we now have a hydrogen ion (H⁺) on the loose! This ion can now react with water to form acid rain, but we're not quite done yet!

We need to investigate two "k" numbers: K and k! The letter "K" is called the equilibrium constant. This is a unitless number that measures the tendency of chemical to break apart. Chemists prefer to say: K is a measure of the tendency of a reaction to go to the right, that is, to follow the reaction arrow to form products. The table below shows this tendency:

If K > 1, there is a HIGH probability that the reaction will go to the right

If K = 1, the reaction is at equilibrium, and it will stay just where it is

If K < 1, there is a LOW probability that the reaction will go to the right

An example: what is the value of K for these non-chemical situations?

The probability that you can take a week off of school whenever you feel like it? My guess is that K is very low, a number like 0.0000000000001!
The probability that it will rain sometime in the next month? K is probably very high, something like 1,0000.

In the reaction above, K is 1,0000. Since this is a number substantially bigger than 1, we can say that it is absolutely true that H₂SO₄ will dissociate into the two products. If sulfuric acid gets into the atmosphere, we can guarantee that it will break apart.

In most books it should be noted that you will generally see the term "K_a", where the little "a" stands for acid. K_a is called the "dissociation constant of an acid", and you look them up in tables of dissociation constants found in many chemistry books.

The second "k" is the rate constant for a reaction. This "k" tells you not whether or not the reaction will occur (that's the job of K_a!), but how fast it will go IF it is allowed to do so! In each of the acid rain reactions, there will be TWO values of k -- one in the forward direction and one in the reverse direction. All of the reactions of interest here are reversible -- the products can recombine to form the original chemical. Basically, a chemical can break apart (dissociate) but then, if conditions are right, connect back together again! We show this with an arrow going in both directions:

	k_-1
	<-----
H₂SO₄	----->	H^{+ +}HSO_4-
	k₁

Remember that if K_a is large, there is not much of a chance that things will recombine, but it is still a possibility!

Each of the reactions of importance will produce a hydrogen ion, which contributes to the overall acidity of the water. If we know the concentration of the hydrogen ion, written as [H⁺], we can calculate the pH of the water using this equation:

pH = -log₁₀[H⁺]

Building the Model:

In this model, we wish to look at the contribution of two different species to the pH of rainwater. Basically, this model involves the simultaneous equilibria of two acids in rainwater -- carbonic acid dissolved from CO₂ and sulfurous acid from dissolved SO₂. For clarity purpose, we'll call them Reaction Group A and Reaction Group B:

Reaction Group A: H₂CO₃

H₂CO₃ <===> HCO_3^- + H⁺

HCO_3^- <====> CO_3^- + H⁺

Reaction Group B: H₂SO₃

H₂SO₃ <===> HSO_3^- + H⁺

HSO_3^- <====> SO_3^- + H⁺

For each reaction there is an equilibrium constant K_a. For each reaction, there is also a forward rate constant "k" and a reverse rate constant. We need to start the model with some initial concentration of each diprotic acid (the one with two hydrogens in it!)

This is actually ONE model done twice. If you can build the model JUST for Reaction Group A, it is a simple matter of copying that model for Reaction Group B!

Initial Conditions:

Use these numbers as your starting points:

[H₂CO₃] = 1.38 x 10^-5 moles/liter

[H₂SO³] = 2.05 x 10^-6 moles/liter

Initial concentrations of all other chemicals are set at 0 moles/liter

Equilibrium constants for Reaction Group A:

K_a1=10^-6.35

K_a2=10^-1.77

Equilibrium constants for Reaction Group B:

K_a1=10^-10.33

K_a2=10^-7.21

Rate constants for Reaction Group A:

k₁=[H₂CO₃]

k_-1=[HCO₃]*H/K_a1

k₂=[HCO₃]

k_-2=[CO₃]*H/K_a2

Rate constants for Reaction Group B:

k₁=[H₂SO₃]

k_-1=[HSO₃]*H/K_a1

k₂=[HSO₃]

k_-2=[SO₃]*H/K_a2

You should run this model until it gets to a steady-state condition. Experiment with the dt and integration methods to get the best results.

Make plots of both reaction groups. You might also wish to use a numeric display window to look at the pH of the reactions over time.

Experimenting with the model:

Prior to the Industrial Revolution, the atmospheric concentration of carbon dioxide was about 80% of its current value. Use your model and adjust the initial carbonic acid concentration to 80% of the value used in the initial conditions. Determine the approximate pH of pristine precipitation corresponding to the pre-industrial atmosphere.
The emission of sulfurous acids is a major factor in the decrease in pH of rainwater -- the well-known phenomenon of acid rain. Experiment with increasing the amount of H₂SO₄ to see the effect on pH.

Developed by
Graphic of Shodor Logo