## Tuesday, May 2, 2017

### The T equation

In science we have many numbers, constants, equations, formulas, laws and principles that have the name of someone who invested a long time studying the phenomena related to the former. It is hard to know how long they stayed looking and learning about what they were studying. In most cases it doesn't matter. When dealing with pressure we have the unit Torricelli (torr) in honor of Evangelista Torricelli who invented the barometer. I don't know but it is not hard to think that the invention took many long hours to take place and to improve until he was able to have a working instrument. While he was doing this he was also thinking about pressure. How can it be defined? How can it be related to the forces involved? How can it be related to the area? et cetera. [By the way the pressure of the atmosphere at sea level is about 760 torr.] So the names associated to these constants, units, laws, et al. are in a way a representation of the effort of those individuals and the societies where they lived.

There is also the fact that naming things makes it easier to remember. It has been studied that when someone is presented with two individuals, one named Baker, and the other being a baker. It is easier to remember the fact that one is a baker rather than the name of the other. If you want to know more about this read the excellent book by John J. Medina "Brain Rules".
One very useful equation in buffer chemistry is the Henderson-Hasselbalch equation:

pH = pKa + Log(Base/Acid)

that relates the pH of a solution made with a weak acid or base and its conjugate acid or base. As it is known in chemistry by definition the mathematical operator p stands for the -Log.
So the pH can be calculated from the concentration of the Hydronium ion H3Oby calculating the -Log,

pH = -Log[H3O+].

The Ka or equilibrium constant for the acid base reaction is calculated from the concentrations of the products and reactants in equilibrium using the following relationship:

Ka = [H3O+][Base]/[Acid] with this relationship and using the properties of Log functions such as Log (AxB) = Log A + Log B. One can derive Henderson-Hasselbalch equation.

Now, traditionally when one is trying to calculate what is the change in pH when a small amount of acid or base is added to a buffered solution one calculates the pH before and after the change occurred, it easy to do by using Henderson-Hasselbalch equation twice, and calculating the change by difference.
I have developed a shortcut by doing the following: First I make the point that I know both concentrations, the initial and the final concentration of both acid and base. I will call them Ai, Bi, Af, and Bf. (The final concentrations of course can be easily calculated as we know the initial concentrations and the amount of acid or base added to the solution. Let's not waste time here with an example of how to do it.)
The change on pH of course can be written as the difference between pHf - pHi

ΔpH = pHf - pHi

If we use the Henderson-Hasselbalch equation twice in the previous equation and use the properties of Logarithms we can get to the following condensed equation to calculate the change in pH:

ΔpH = Log (Bf * Ai)/(Bi * Af)     This is the T equation!

This very simple equation states that the change in pH is the Log of the product of the final base times the initial acid divided by the initial base times the final acid. Even though we should be aware of the values of the initial and final concentrations the fact of the matter is that as long as we have the acid and base cross multiplied, i.e. if the base is the initial the acid must be the final, the only difference if we do them vice versa is that the sign of the difference will change from negative to positive or vice versa. Which in reality doesn't matter because we normally want to know the absolute value of the change in pH. We know that if we add a base the pH will increase a bit, and if we add an acid the pH will decrease a bit. But what we are interested is in the absolute value, the magnitude of the change.