How to calculate pH?

Have you ever wondered why some things taste sour and others do not? Or what is more acidic, vinegar or lemon juice? Technically, what causes the sour or acidic taste of things is a high concentration of free hydrogen or hydronium ions in whatever you are drinking or eating. The greater the number of free hydronium ions a solution has, the more acidic it is. pH is exactly that, the measurement of how strong an acid or its opposite, a base is.

Acids are substances that donate protons or hydrogen ions when they are in a solution and have a sour taste, while bases donate OH ions, also known as hydroxide ions, and tend to have a bitter taste and a slippery or soapy feel to touch.

More specifically pH quantifies the concentration of hydrogen ions present in a solution per litre of the solution, in other words, the molarity of hydronium ions. Generally, you would use some pH indicator solution or colored paper to determine the pH of a solution, the color change of the pH indicator is an easy way to compare different solutions and determine which ones are more acidic or basic. The pH scale goes from 0 to 14, and in this scale 7 is a neutral pH. Below 7, substances are considered acids and above 7 substances are considered basic. In this article we will see how pH is calculated and where this measurement comes from.

How to calculate pH?

The mathematical definition of pH is simply the negative logarithm of the molarity of hydronium ions in a solution. In general you will need to know the concentration of you acid or base in a solution to be able to calculate the pH.

pH= -log [H+] or more precisely pH= -log [H3O+ ]

But these two equations are considered equivalent.

Acids and Bases

Knowing the pH is very important for many industries, such as food production, cleaning and self-hygiene products, clothing, printing, and even art preservation. For example, to ensure the shelf life of canned goods in the supermarket, that soaps or shampoos do not irritate the skin of those using them, or that cleaning products do not corrode the surfaces that you might be trying to clean.

Although in this last case in the cleaning product industry is where you will find the strongest acids and bases because that corrosive quality is what we use to remove things like grease and grime. The reason for this is because both strong acids and bases are very reactive, precisely for that tendency they have to donate or accept protons or hydrogen ions, this makes acids positively charged while bases by accepting hydrogen ions are negatively charged. In other words, the stronger the acid or base the more corrosive it is, that is why maintaining or regulating the pH of many products is fundamental. It is also worth to note that strong acids and bases dissociate  completely in solution, releasing all its hydronium or hydroxide ions. 

But how do we know at what point substances stop being corrosive? What would be that limit when substances are neutral? Well, the answer is in water. Water is very interesting because it is composed exactly and solely by the two ions that we have been talking about! And which cause many substances to behave like acids or bases, therefore water is technically both an acid and base at the same time, which actually makes it neutral. 

Dissociation constant of water

Water has a very familiar molecular structure, is H2O, however, water molecules are constantly dissociating themselves into hydronium ions and hydroxide ions, these ions when in contact with each other form a water molecule again, following this chemical reaction:

2H2O(l) <=> H3O+ + OH

In the equation above you can see that we need two molecules of water and that one of them loses a hydrogen to form the positively charged hydronium ion H3O+, while the other one loses it and becomes the negatively charged OH or hydroxide ion. This reaction is constantly happening in water albeit in very small amounts or concentration. This dissociation happens very fast as well, two water molecules will exchange a proton forming the two ions and revert to water in fractions of a second. This allows for the properties of water to remain constant and its pH neutral despite behaving both as an acid and a base. 

Based on the balanced equation, you can see that the number of hydronium ions will be exactly the same as the hydroxide ions. This means that the concentration of hydronium ions will be exactly the same as the concentration of hydroxy ions in pure water at all times (at 25 °C, because pH changes with temperature).

The equilibrium  constant equation for the concentration of both ions is known as the water dissociation constant:

Kw= [H3O+][OH-]= 1X10-14 mol/L

Since the concentration of each ion would be the same, we can calculate each one as:

Kw= [H3O+][OH-]= [x][x]= 1X10-14 

Naming the unknown concentrations as x. Then we can simplify like this:

Kw= [H3O+][OH-]= [x]2= 1X10-14 M

And resolve x noting that the ions will be in the same concentration:

Ka=[H3O+]=[OH-]=1 X10-14=1×10-7 M

We can conclude then that concentration of hydronium ions in pure water is always 1×10-7 M, if we calculate the negative logarithm of that concentration, we will have calculated the pH:

pH=-log[H3O+]=-log 1×10-7 =7

The pH of water is 7, and since this equilibrium is maintained, water is then electrically neutral and non-corrosive, and this is why acids and bases fall on either side of this number as the concentration of hydronium ions increases or decreases. At this point you will be probably wondering why acids fall below the 7 treshhold if they have higher hydronium concentrations than bases. It would be expected that the pH would be higher. The answer is in the negative logarithm. The reason for using the negative logarithm is because the concentration of the ions varies greatly, several orders of magnitude, but always in very small amounts, in the order of 1×10-14mol/L. and it is always in values less than 1.  This is why using the logarithm transforms those values into more manageable and especially more comparable numbers, the negative is used solely to turn a positive value as the logarithm of numbers below 1 will always be a negative number.

pH is generally estimated by using either some color indicator such as phenolphthalein or malachite green, or to measure it more precisely, a pH meter is generally used. The pH meter works by measuring the voltage (or electrical potential) of our sample and compares it to a reference solution, to later deduce the difference between them too. 

Example

Imagine that you are a chemical+ engineer in a factory and that you are preparing a solution of hydrochloric acid to perform some other reactions. Hydrochloric acid is always found in aqueous solution and it’s the perfect example of a substance that donates protons as it dissociates completely when dissolved in water, it has the molecular formula HCl. So, let’s say that you have 0.3L of 0.08 M hydrochloric acid, and that you want to dilute it in 0.5L of water and find the pH of that. Fortunately, every seller of chemical products must inform you of the concentration of these solutions.

So, let’s calculate the molarity of your resulting solution using our known concentration (M), and the volumes we want to dilute (V1 and V2):

M1V1=M2V

Reorganising the equation, we can calculate the molarity of the resulting solution:

M1V1 / V2= M2

Let’s put in our values:

0.08M*0.3L/0.5L=0.048M

Now let’s calculate the pH of that solution, since we know the concentration of acid, we know the concentration of hydronium ions.

pH=-log[0.048M]

pH=1.31

As you can see this solution is very acidic so you can expect for it to be very corrosive.