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Where would you place the following substances on the scale if you had to guess?

- Balsamic vinegar.
- Beer.
- Seawater.
- Hand soap.

You can see that the pH scale runs from 0 to 14. It runs even further, but values below 0 or above 14 are rare. You may have heard that pH is a measure of a substance's acidity. That's true, but we will take the definition a little further at this level of knowledge.

## What is pH?

The** pH** is a measure of the hydrogen ion concentration of a solution.

In **Brønsted-Lowry ****Acids and Bases**, we defined an **acid** as a proton donor. Protons are just hydrogen ions, ${\mathrm{H}}^{+}$. The stronger an acid is, the better it is at donating protons, the lower its pH. Using the scale above, we can see that balsamic vinegar is a much stronger acid than soap, for example – it donates more protons in solution.

Danish chemist Søren Peder Lauritz Sørensen at the Carlsberg Laboratory invented the pH scale in 1909. He was a brewer and wanted to carefully control the acidity of his beer to promote healthy yeast growth but prevent the growth of undesirable bacteria. However, he worked with tiny acid concentrations, and the calculations got messy. He, therefore, decided to log his answers and then take the negative of that value to get a positive answer.

The H in pH stands for hydrogen, but interestingly, no one is quite sure where the p comes from. Although Sørensen himself was Danish, he worked in a French laboratory dominated by German scientific work. The words for both 'power' and 'potential' start with a p in all three languages, so it could be any of them. However, Sørensen could have simply referred to the test solution as p – hence, pH.

We can represent pH with the following equation:

$\mathrm{pH}=-{\mathrm{log}}_{10}\left[{\mathrm{H}}^{+}\right(\mathrm{aq}\left)\right]$

So if we know the concentration of hydrogen ions in a solution, we can calculate the pH of the solution.

Practice finding the${\mathrm{log}}_{10}$key on your calculator. It makes typing equations much faster.

**The higher the concentration of hydrogen ions in a solution, the lower the pH**, and vice versa. A pH of less than 7 is acidic, whilst more than 7 is alkaline. You may have been told in the past that a pH of 7 is **neutral**, but in reality, neutral has a different definition.

A **neutral solution** has equal concentrations of hydrogen and hydroxide ions.

## What are strong acids and bases?

You may recall from the previous article that **acids** are proton donors. In solution, they **dissociate** to form negative ions and positive hydrogen ions. This dissociation is also called **ionisation**. It is a reversible reaction, as shown below:

$\mathrm{HX}\leftrightharpoons {\mathrm{H}}^{+}+{\mathrm{X}}^{-}$

However, some acids are very good at giving up their hydrogen ions -– so good that the reaction is essentially one-way. We call these acids **strong acids**.

A **strong acid** dissociates completely in solution.

The following equation arises:

$\mathrm{HX}\to {\mathrm{H}}^{+}+{\mathrm{X}}^{-}$

Similarly, we can obtain **strong bases.**

A **strong base** is a base that dissociates completely in solution.

If we add a strong base to water, we get the following equation:

$\mathrm{B}+{\mathrm{H}}_{2}\mathrm{O}\to {\mathrm{BH}}^{+}+{\mathrm{OH}}^{-}$

### The pH value of strong acids and bases

Strong acids have a low pH because they have a high concentration of hydrogen ions in the solution. Examples are hydrochloric acid ($\mathrm{HCl}$), nitric acid (${\mathrm{HNO}}_{3}$) and sulfuric acid (${\mathrm{H}}_{2}{\mathrm{SO}}_{4}$).

Strong bases have a low concentration of hydrogen ions in the solution. As a result, they have a high pH. Examples of this are all group 1 and group 2 hydroxides, such as sodium hydroxide ($\mathrm{NaOH}$).

## How do we find the pH of strong acids?

Remember that a strong acid dissociates completely in an aqueous solution. Let us take hydrochloric acid as an example.

Find the pH of $0.1$ moles of hydrochloric acid dissolved in $0.5{\mathrm{dm}}^{3}$ of water.

If we put $0.1$ moles of hydrochloric acid in $0.5{\mathrm{dm}}^{3}$ of water, the acid will completely dissociate into $0.1$ mole of hydrogen ions, ${\mathrm{H}}^{+}$, and 0.1 moles of chloride ions, ${\mathrm{Cl}}^{-}$.

To find the concentration, we divide the number of moles by the volume of the solution. So, to find the hydrogen ions concentration in this particular solution, we do $0.1\mathrm{\u1365}0.5=0.2\mathrm{mol}{\mathrm{dm}}^{-3}$.

Now what? How can we find the pH value?

Well, let's go back to our original equation for pH:

$\mathrm{pH}=-{\mathrm{log}}_{10}\left[{\mathrm{H}}^{+}\right(\mathrm{aq}\left)\right]$

We now know the concentration of hydrogen ions in the solution. Therefore, we can substitute this into our equation as shown:

$\mathrm{pH}=-{\mathrm{log}}_{10}(0.2)=0.70$

Note that pH is always given to two decimal places. It also has no units.

### Finding [H+] from pH

What if you know the pH of a strong acid and want to find its hydrogen ion concentration? We can rearrange the equation for pH to make $\left[{\mathrm{H}}^{+}\right]$ the subject. First, switch the minus sign around:

$-\mathrm{pH}={\mathrm{log}}_{10}\left(\right[{\mathrm{H}}^{+}\left]\right)$

Next, take antilogs of both sides:

${10}^{-\mathrm{pH}}=\left[{\mathrm{H}}^{+}\right]$

There you go! To find the hydrogen ion concentration, simply substitute your value for pH into the equation. Here's an example to help you understand more clearly:

The pH of a $0.5{\mathrm{dm}}^{3}$ solution of hydrochloric acid is 0.75. Find the number of moles of HCl in the solution.

Now, we know the pH. Let's substitute it into our equation:

${10}^{-0.75}=\left[{\mathrm{H}}^{+}\right]\phantom{\rule{0ex}{0ex}}\left[{\mathrm{H}}^{+}\right]=0.1778$

This gives us the concentration of hydrogen ions in the solution. But, what do we know about hydrochloric acid? Hydrochloric acid is a strong acid and dissociates entirely in solution. Therefore, the concentration of hydrochloric acid is also $0.1778$. To find the number of moles of hydrochloric acid dissolved in the solution, we can multiply the concentration by the volume:

$0.1778\mathrm{x}0.5=0.089\mathrm{mols}$ to two decimal places.

Remember to check the units of your numbers before you do any calculations. $\left[{\mathrm{H}}^{+}\right]$ is always given in $\mathrm{mol}{\mathrm{dm}}^{-3}$, so make sure that you convert your volume to ${\mathrm{dm}}^{3}$ as well.

## How do we find the pH of strong bases?

Calculating the pH of strong bases is a little more tricky than working out the pH of a strong acid – there is an extra step. To do this, you need a value known as ${\mathrm{K}}_{\mathrm{w}}$. ${\mathrm{K}}_{\mathrm{w}}$ is also known as the **I****onic Product of Water**. It has the following equation:

${\mathrm{K}}_{\mathrm{w}}=\left[{\mathrm{H}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]$

${\mathrm{K}}_{\mathrm{w}}$ varies depending on temperature. At a fixed temperature, ${\mathrm{K}}_{\mathrm{w}}$ always remains the same. For example, you'll usually work with acids and bases at room temperature – about 25℃. At this temperature, ${\mathrm{K}}_{\mathrm{w}}$ takes the value $1.00\mathrm{x}{10}^{-14}{\mathrm{mol}}^{2}{\mathrm{dm}}^{-6}$ .

If we know the concentration of hydroxide ions in the solution, we can use ${\mathrm{K}}_{\mathrm{w}}$ to find the concentration of hydrogen ions. We can then calculate the pH of the solution, as we did above.

Example: Calculate the pH of a $0.1\mathrm{mol}{\mathrm{dm}}^{-3}$ solution of sodium hydroxide, $\mathrm{NaOH}$.

Have a go at answering the question by yourself first. But if you are stuck, let us go through it together now.

Since sodium hydroxide is a **strong base**, it completely dissociates in solution into hydroxide ions and sodium ions, ${\mathrm{OH}}^{-}$ and ${\mathrm{Na}}^{+}$ respectively, as we explored above. The concentration of hydroxide ions in the solution is thus also $0.1\mathrm{mol}{\mathrm{dm}}^{-3}$. We can use this value, alongside ${\mathrm{K}}_{\mathrm{w}}$, to find the concentration of hydrogen ions in the solution.

${\mathrm{K}}_{\mathrm{w}}=\left[{\mathrm{H}}^{+}\right]\left[{\mathrm{OH}}^{-}\right]$

Divide both sides by $\left[{\mathrm{OH}}^{-}\right]$:

$\frac{{\mathrm{K}}_{\mathrm{w}}}{\left[{\mathrm{OH}}^{-}\right]}=\left[{\mathrm{H}}^{+}\right]$

Substitute our values in:

$\frac{1.00\mathrm{x}{10}^{-14}}{0.1}=\left[{\mathrm{H}}^{+}\right]=1.00\mathrm{x}{10}^{-13}$

We can then put this into our equation for pH:

$\mathrm{pH}=-\mathrm{log}(1.00\mathrm{x}{10}^{-13})\phantom{\rule{0ex}{0ex}}\mathrm{pH}=13.00$

## How do we find the pH of mixtures?

Calculating the pH of an acid or base is all well and good, but in chemistry and everyday life, you'll more often encounter **mixtures** of acids and bases. These react with each other in** neutralisation reactions**. For example, if you suffer from heartburn, you might take magnesium hydroxide tablets to neutralise excess stomach acid. In this example, some of the hydroxide ions from the base, magnesium hydroxide, react with hydrogen ions from the acid, hydrochloric acid, to form water. They'll continue to react until one of the reagents is used up. This reagent is the** limiting reactor,** whilst the other reagent is in** excess**.

To determine the pH of a mixture, you need to determine which reagent is in excess. You can then calculate the concentration of the hydrogen ions or hydroxide ions remaining and then calculate the pH as before. We'll now go through an example to help you understand the process.

Example: A mixture contains $50{\mathrm{cm}}^{3}$ of $0.100\mathrm{mol}{\mathrm{dm}}^{-3}{\mathrm{H}}_{2}{\mathrm{SO}}_{4}$ and $25{\mathrm{cm}}^{3}$ of $0.150\mathrm{mol}{\mathrm{dm}}^{-3}\mathrm{NaOH}$. Calculate its pH.

First of all, we need to convert volume into ${\mathrm{dm}}^{3}$:

$50{\mathrm{cm}}^{3}{\mathrm{H}}_{2}{\mathrm{SO}}_{4}=0.05{\mathrm{dm}}^{3}\phantom{\rule{0ex}{0ex}}0.05\mathrm{x}0.100=5\mathrm{x}{10}^{-3}\mathrm{moles}{\mathrm{H}}_{2}{\mathrm{SO}}_{4}$

Then we can calculate the number of moles of hydrogen ions and hydroxide ions in the solution. However, you'll notice that H2SO4 contains two hydrogen atoms per mole, meaning each mole dissolved in the solution produces two moles of aqueous hydrogen ions. So we have $(5\mathrm{x}{10}^{-3})\mathrm{x}2=1\mathrm{x}{10}^{-2}$ moles of H+.

Let's do the same thing with $\mathrm{NaOH}$.

$25{\mathrm{cm}}^{3}\mathrm{NaOH}=0.025{\mathrm{dm}}^{3}\phantom{\rule{0ex}{0ex}}0.025\mathrm{x}0.150=3.75\mathrm{x}{10}^{-3}\mathrm{moles}\mathrm{NaOH}.$

Each mole of $\mathrm{NaOH}$ dissociates to produce just one mole of hydroxide ions, so we have $3.75\mathrm{x}{10}^{-3}$ moles of hydroxide ions.

But this is a neutralisation reaction, and so the hydrogen ions react with the hydroxide ions, as we mentioned above. They react in a 1:1 ratio. We have fewer hydroxide ions than hydrogen ions, which means that some hydrogen ions won't react and will remain in the solution. To find out this value, subtract the number of hydroxide ions from the number of hydrogen ions:

$(1\mathrm{x}{10}^{-2})-(3.75\mathrm{x}{10}^{-3})=6.25\mathrm{x}{10}^{-3}$

This is the number of hydrogen ions remaining in the solution. We can now calculate the pH as in the examples above, first by working out $\left[{\mathrm{H}}^{+}\right]$ using the total volume, which is $0.05+0.025=0.075{\mathrm{dm}}^{3}$, and then by taking logs:

$\left[{\mathrm{H}}^{+}\right]=(6.25\mathrm{x}{10}^{-3})\mathrm{\u1365}0.075{\mathrm{dm}}^{3}=0.083\mathrm{mol}{\mathrm{dm}}^{-3}\phantom{\rule{0ex}{0ex}}\mathrm{pH}=-\mathrm{log}(0.083)=1.08$

The following flowchart summarises how you calculate pH for strong acids, bases, and mixtures of the two.

How do we find pH without doing calculations? We can use a **universal indicator **for a rough measurement, which is a mixture of dyes that change colour at different pH levels. For a more accurate measurement, we can use a **pH meter**. It measures the difference in electrical potential between a reference probe and a pH probe, varying depending on the concentration of hydrogen ions in the solution. This method is helpful for continuous readings, such as during a chemical reaction.

## pH - Key takeaways

- The pH scale measures the concentration of hydrogen ions in a solution. The following equation gives the pH:$\mathrm{pH}=-\mathrm{log}\left(\right[{\mathrm{H}}^{+}\left]\right)$
- A pH below 7 is acidic, whilst a pH above 7 is alkaline.
- A neutral solution has equal concentrations of hydrogen and hydroxide ions.
- Strong acids and bases dissociate completely in solution.
- To find the pH of a strong acid, a base, or a mixture of the two, you first calculate the concentration of hydrogen ions in the solution. Then use the formula above to find the pH.

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##### Frequently Asked Questions about pH

What is pH?

pH is a measure of hydrogen ion concentration in solution. It tells you how acidic or alkaline a substance is.

How do you calculate pH?

To calculate pH:

- Calculate the moles of hydrogen ions in the solution.
- Calculate the concentration of hydrogen ions in the solution.
- Take a negative log of the hydrogen ion concentration.

For a further explanation and worked example, check out the rest of this article.

How do you measure pH?

You measure pH using a universal indicator or a pH probe.

What is pH balance?

The term pH balance describes the way your body maintains a constant pH. When parts of your body become too acidic or too alkaline, their systems stop functioning properly. Your body has therefore developed several mechanisms to prevent this.

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