pH Formula – Definition, Equation, Applications

pH Formula: There are numerous acid-base indicators, such as natural and artificial indicators. Among them, one indicator (universal indicator) gives different colours at different pH values. Very often, in chemical laboratories, we use pH paper to check whether a substance is acidic or basic. It changes its colour based on the pH scale that ranges from the value \(0\) to \(14\). Hence, it is used to measure the pH of a solution. After dipping the pH paper in the solution, the colour produced is compared to a colour chart. As it follows a standard ‘pH-scale,’ it is also called a ‘Universal Indicator.’ As a result, pH can be defined as a scale for determining the acidity or basicity of a substance in an aqueous solution.

Before learning about pH, we should have a brief idea of acids and bases. According to Arrhenius concept, Acids are the substances that ionise in water to produce \({{\rm{H}}^ + }\) or \({{\rm{H}}_3}{{\rm{O}}^ + }\) ions while bases are the substances that ionise in water to produce \({\rm{O}}{{\rm{H}}^ – }\) ions. Based on the dissociation of these ions, the pH of a substance is measured. In this article, let’s learn everything about the pH formula in detail.

What is pH?

The term “pH” is derived from the German word “potenz”, which means “power,” combined with hydrogen that is symbolised as \({\rm{H}}\), so the full form of pH is “power of hydrogen.” Danish chemist Soren Sorensen first introduced the concept of pH in \(1909\).

A strong acid releases \({{\rm{H}}^ + }\) ions rapidly in an aqueous solution, and a strong base accepts \({{\rm{H}}^ + }\) ions rapidly in an aqueous solution. The concentration of \({{\rm{H}}^ + }\) ions discussed here are very few. Thus, Sorensen proposed an equation that would measure even a very small \({{\rm{H}}^ + }\) concentration in a solution.

Definition of pH

pH is defined as a logarithmic scale that measures the pH of a substance based on even a minimum quantity of \({{\rm{H}}^ + }\) ions by using a mathematical expression, according to which pH is equal to the negative of the base-ten logarithm of the concentration of \({{\rm{H}}^ + }\) or \({{\rm{H}}_3}{{\rm{O}}^ + }\), where the concentration of \({{\rm{H}}^ + }\) is in terms of moles/litre. A pH scale ranges from \(0\) to \(14\). Where pH equal to \(7\) represents neutral. While a pH value of less than \(7\) indicates acidic, and a pH value greater than \(7\) indicates a basic substance.

pH Formula in Chemistry

As proposed by Sorensen, pH is dependent on the concentration of hydrogen ions or hydronium ions that are formed when an acid is added to water, it releases \({{\rm{H}}^ + }\) ions that combine with \({{\rm{H}}_2}{\rm{O}}\) and form \({{\rm{H}}_3}{{\rm{O}}^ + }\) ions. As pH is the negative of the base-ten logarithm of the concentration of \({{\rm{H}}^ + }\) or \({{\rm{H}}_3}{{\rm{O}}^ + }\), more the concentration of these ions in a solution, the lower will be its pH value. Thus, strong acids have a low pH value because they readily ionise to release hydrogen or hydronium ions. In chemistry pH formula is represented as:

\({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\)

Or

\({\rm{pH}} = – \log \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\)

Calculation of Concentration of Ions from pH value

If the pH value of a solution is already given, and we want to calculate the concentration of ions, then the pH equation can be transformed as:

\(\left[ {{{\rm{H}}^ + }} \right] = {10^{ – {\rm{pH}}}}\)

The pOH scale – Apart from the pH scale, there also exists a pOH scale that is less popular than the pH scale. As the pH scale, pOH is the negative of the logarithm of the hydroxide ion concentration in an aqueous solution:

\({\rm{pOH}} = – \log \left[ {{\rm{O}}{{\rm{H}}^ – }} \right]\)

or

\(\left[ {{\rm{O}}{{\rm{H}}^ – }} \right] = {10^{ – {\rm{pOH}}}}\)

pH of Water

A pure sample of water is almost neutral, i.e., neither acidic nor basic. It undergoes negligible auto-ionisation or self-ionisation.

Thus, the concentration of \({{\rm{H}}^ + }\) ions in water are:

\(\left[ {{{\rm{H}}^ + }} \right] = 1.0 \times {10^{ – 7}}\,{\rm{moles}}\,{\rm{per}}\,{\rm{litre}}\)

the pH of water can be calculated as:

\({\rm{pH}} = – \log \left[ {1.0 \times {{10}^{ – 7}}} \right]\)

\({\rm{pH}} = – ( – 7)\)

\({\rm{pH}} = 7\)

Ionic Product Constant of Water

Water shows amphoteric behaviour. In self ionisation process, the water dissociates as:

According to the ‘Law of Mass Action‘, the Equilibrium constant \(\left( {{{\rm{K}}_{{\rm{w}})}}} \right.\) is:

\({{\rm{K}}_{\rm{w}}} = \frac{{\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\left[ {{\rm{O}}{{\rm{H}}^ – }} \right]}}{{\left[ {{{\rm{H}}_2}{\rm{O}}} \right]}}\)

Since water is the solvent and is in excess amount. So, its concentration can be estimated as \(1\).

\({{\rm{K}}_{\rm{w}}} = \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\left[ {{\rm{O}}{{\rm{H}}^ – }} \right]\)

Thus, at room temperature,

\({{\rm{K}}_{\rm{W}}} = 1.0 \times {10^{ – 14}}\)

In an aqueous solution, it can be inferred that, if:

\(\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right] = \left[ {{\rm{O}}{{\rm{H}}^ – }} \right],{\bf{pH}} = {\bf{7}}\) (Neutral Solution)

\(\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right] > \left[ {{\rm{O}}{{\rm{H}}^ – }} \right],{\bf{pH}} < {\bf{7}}\) (Acidic Solution)

\(\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right] < \left[ {{\rm{O}}{{\rm{H}}^ – }} \right],{\bf{pH}} > {\bf{7}}\) (Basic Solution)

Relation between pH, pOH, and \({\rm{p}}{{\rm{K}}_{\rm{w}}}\)

\( – \log {{\bf{K}}{\bf{w}}} = {\bf{p}}{{\bf{K}}{\bf{w}}}\)

\( – \log {{\rm{K}}_{\rm{w}}} = – \log \left( {\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\left[ {{\rm{O}}{{\rm{H}}^ – }} \right]} \right)\)

\( – \log {{\rm{K}}_{\rm{w}}} = \left( { – \log \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]} \right) + \left( { – \log \left[ {{\rm{O}}{{\rm{H}}^ – }} \right]} \right)\)

Thus, \({\rm{p}}{{\rm{K}}_{\rm{w}}} = {\rm{pH}} + {\rm{pOH}}\)

As we know that \({{\rm{K}}_{\rm{w}}} = 1.0 \times {10^{ – 14}}\). Thus, \({\rm{p}}{{\rm{K}}_{\rm{w}}} = 14\).

Hence, \({\rm{pH}} + {\rm{pOH}} = 14\)

The pH of Acid and Base in Buffer Solution

Buffer solutions contain a weak acid, and it’s a conjugate base, and vice-versa.

The pH of Acid Buffer Solution

Let’s take an example of the dissociation of acid in its aqueous solution. Acid \({\rm{HA}}\) is dissociated to form \({{\rm{H}}^ + }\) ion and its conjugate base, i.e., \({{\rm{A}}^ – }\).

The equilibrium constant for such ionisation reaction is denoted as \({{\rm{K}}_{\rm{a}}}\).
\({{\rm{K}}_{\rm{a}}} = \frac{{\left[ {{{\rm{H}}^ + }} \right]\left[ {{{\rm{A}}^ – }} \right]}}{{[{\rm{HA}}]}}\)

On rearranging the above equation and taking the negative logarithm of LHS and RHS, we get:
\( – \log \left( {\frac{{[{\rm{HA}}]}}{{[{{\rm{A}}^{ – 1}}]}}} \right) – \log {{\rm{K}}_{\rm{a}}} = – \log \left[ {{{\rm{H}}^ + }} \right]\,\,\,\,\,\,…{\rm{(i)}}\)

We know that \({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\) and \({\rm{p}}{{\rm{K}}{\rm{a}}} = – \log {{\rm{K}}{\rm{a}}}\)

Therefore,
\({\rm{pH}} = {\rm{p}}{{\rm{K}}_{\rm{a}}} + \log \left( {\frac{{\left[ {{{\rm{A}}^ – }} \right]}}{{[{\rm{HA}}]}}} \right)\)

It can also be written as:
\({\rm{pH}} = {\rm{p}}{{\rm{K}}_{\rm{a}}} + \log \left( {\frac{{[{\rm{Salt}}]}}{{[{\rm{Acid}}]}}} \right)\)

pOH of Basic Buffer Solution

Let’s take an example of the dissociation of a base in its aqueous solution. A base, denoted as \({\rm{B}}\) in its aqueous solution, reacts with water to form \({\rm{O}}{{\rm{H}}^ – }\) ion and its conjugate acid, i.e., \({\rm{B}}{{\rm{H}}^ + }\).

The equilibrium constant for such ionisation reaction is denoted as \({{\rm{K}}_{\rm{b}}}\).
\({{\rm{K}}_{\rm{b}}} = \frac{{\left[ {{\rm{B}}{{\rm{H}}^ + }} \right]\left[ {{\rm{O}}{{\rm{H}}^ – }} \right]}}{{[{\rm{B}}]}}\)

On rearranging the above equation and taking the negative logarithm of LHS and RHS, we get:
\( – \log \left( {\frac{{[{\rm{B}}]}}{{\left[ {{\rm{B}}{{\rm{H}}^ + }} \right]}}} \right) – \log {{\rm{K}}_{\rm{b}}} = – \log \left[ {{\rm{O}}{{\rm{H}}^ – }} \right]\,\,\,\,\,{\rm{(i)}}\)

We know that \({\rm{pOH}} = – \log \left[ {{\rm{O}}{{\rm{H}}^ – }} \right]\) and \({\rm{p}}{{\rm{K}}{\rm{b}}} = – \log {{\rm{K}}{\rm{b}}}\)

Therefore,
\({\rm{pOH}} = {\rm{p}}{{\rm{K}}_{\rm{b}}} + \log \left( {\frac{{\left[ {{\rm{B}}{{\rm{H}}^ + }} \right]}}{{[{\rm{B}}]}}} \right)\)

It can also be written as:
\({\rm{pOH}} = {\rm{p}}{{\rm{K}}_{\rm{b}}} + \log \left( {\frac{{[{\rm{Salt}}]}}{{[{\rm{Base}}]}}} \right)\)

pH formula for Salt of Weak acid and Strong base

In the reaction between a weak acid and a strong base, the acid will be fully neutralised, and the conjugate base of the acid is left. Hence, the pH of a weak acid and a strong base has a pH greater than \(7\).

\({\rm{pH}} = \frac{1}{2}\left[ {{\rm{p}}{{\rm{K}}{\rm{w}}} + {\rm{p}}{{\rm{K}}{\rm{a}}} + \log {\rm{C}}} \right]\)

Where \({\rm{p}}{{\rm{K}}{\rm{w}}} = 7,\;{{\rm{K}}{\rm{a}}}\) is the dissociation constant, and \({\rm{‘C’}}\) is the concentration of salt formed.

Thus,
\({\rm{pH}} = 7 + \frac{1}{2}\left[ {{\rm{p}}{{\rm{K}}_{\rm{a}}} + \log {\rm{C}}} \right]\)

pH formula for Salt of Weak base and Strong acid

In the reaction between a strong acid and a weak base, the base will be fully neutralised, and the conjugate acid of the weak base is left. Hence, the pH of a weak base and a strong acid has a pH less than \(7\).

\({\rm{pH}} = \frac{1}{2}\left[ {{\rm{p}}{{\rm{K}}{\rm{w}}} – {\rm{p}}{{\rm{K}}{\rm{b}}} – {\mathop{\rm logC}\nolimits} } \right]\)

Where \({\rm{p}}{{\rm{K}}{\rm{w}}} = 7,\;{{\rm{K}}{\rm{b}}}\) is the dissociation constant, and \({\rm{‘C’}}\) is the concentration of salt formed.

Thus,
\({\rm{pH}} = 7 – \frac{1}{2}\left[ {{\rm{p}}{{\rm{K}}_{\rm{b}}} + {\mathop{\rm logC}\nolimits} } \right]\)

Applications of pH as Universal Indicator

1. The pH of a solution is also determined by using pH paper that gives different colours at different pH values. A piece of this Universal Indicator paper is dipped in the solution. The pH value is then found by comparing the colour obtained on the pH strip with a colour chart.

• 2. pH value is necessary for pharmaceutical industries where a medicine’s pH value needs to equal our body fluids pH otherwise, it does not react when consumed by a patient.
• 3. In chemical reactions, the pH environment needs to be in equilibrium for proper results; for this purpose, buffer solutions are used to maintain pH equilibrium.
• 4. The pH meter is a device used to measure the exact pH value of a solution. It consists of a pH electrode connected to a meter. When the electrode is dipped into the solution, the meter shows the exact pH value either on a scale or digitally.

Summary

In a nutshell, pH is a measurement of a substance’s acidity or basicity based on the hydrogen ions or hydroxide ions it releases in an aqueous solution. It was derived by a Danish chemist Soren Sorensen, who expressed it mathematically as \({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\) or \({\rm{pH}} = – \log \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\). From this formula, we can conclude that the concentration of hydrogen ion or hydronium ion is inversely proportional to the pH value, i.e., more \({{\rm{H}}^ + }\) concentration, lesser will be the value on the pH scale.

The range of a pH scale lies between \(0\) to \(14\). Where \(7\) represents neutral. While pH value of less than \(7\) indicates acidic and a pH value greater than \(7\) indicates a basic substance. This is because, acidic solution releases hydrogen ions in its aqueous solution, and the basic solution releases hydroxide ions in its aqueous solution. pH and pOH are related to each other as \({\rm{pH}} + {\rm{pOH}} = 14\). pH is very useful in our daily life as most of the products used by us have a particular pH. Even our blood has a pH of \(7.4\). If the pH balance of our body is disturbed, it may be fatal. The pH scale is the most accurate and reliable among all acid-base indicators that help determine whether a substance is acidic, basic, or neutral.

FAQs on pH Formula

Q.1. What is the pH Formula?
Ans: pH is equal to the negative of the base-ten logarithm of the concentration of \({{\rm{H}}^ + }\) or \({{\rm{H}}_3}{{\rm{O}}^ + }\) where the concentration of \({{\rm{H}}^ + }\) is in terms of moles/litre. The pH formula is as- \({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\) or \({\rm{pH}} = – \log \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\).

Q.2. What is meant by the pH calculation formula?
Ans: The amount of hydrogen ion activity in a solution is measured by pH, and this helps to judge its alkalinity or acidity. The pH value is due to the activity of \({{\rm{H}}^ + }\) ion in the solution as pH is defined as a logarithmic scale that measures the pH of a substance based on even a minimum quantity of \({{\rm{H}}^ + }\) ions by using a mathematical expression \({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\).

Q.3. How do you calculate pH and pOH?
Ans: The pH of acid and pH of a base is dependent on the \({{\rm{H}}^ + }\) ions and \({\rm{O}}{{\rm{H}}^ – }\) ions released in an aqueous solution, respectively. pH is calculated by using a formula \({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\) or \({\rm{pH}} = – \log \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\), and pOH is calculated by the formula: \({\rm{pOH}} = – \log \left[ {{\rm{O}}{{\rm{H}}^ – }} \right]\). pH and pOH are related to each other as \({\rm{pH}} + {\rm{pOH}} = 14\).

Q.4. What is an Acid’s pH?
Ans: From the formula \({\rm{pH}} = – \log \left[ {{{\rm{H}}^ + }} \right]\) it is understood that concentration of hydrogen ion or hydronium ion is inversely proportional to the pH value, i.e., more \({{\rm{H}}^ + }\) concentration, lesser will be the value on the pH scale. Thus, acid has a pH value of less than 7.

Q.5. What is the pH of 0.001 M HCl?
Ans: HCl is a strong acid. The pH of HCl is \(3\).

Q.6. Find out the pH of the solution in which the concentration of hydronium ions is 8.0 ×10^-8 M.
Ans: By the formula of \({\rm{p}}{{\rm{H}}_i}{\rm{pH}} = – \log \left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\), thus, \(\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\) is given as \(8.0 \times {10^{ – 8}}{\rm{M}}\). On substituting this value in the pH formula, \({\rm{pH = – log}}\left[ {{\rm{8}}{\rm{.0 \times 1}}{{\rm{0}}^{{\rm{ – 8}}}}} \right]\), on solving this equation, we get \({\rm{pH}} = 7.09\).

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