Abstract: pH is a unit of measurement often used in fundamental chemistry concepts. “How to Calculate pH” explains it’s categories the scientific mathematics and role pH has in our lives.

Terms to Be Familiar With:

• pH
• pOH
• Hydrogen ion
• Hydroxide ion
• Acid
• Base

What is pH?

The term “pH” is an abbreviation for the “potential of hydrogen.” pH is a unit of measurement which represents the concentration of hydrogen ions in a solution. This unit was introduced by biochemist Søren Peter Lauritz Sørensen in 1909. It was an easy way to represent the concentration of hydrogen ions in a solution during titrations. When an acid or base is added to water, that compound dissociates into ions. For acids one of those ions is a hydrogen ion (H⁺) and for bases one of the ions is a hydroxide ion (OH^–). This description of acids and bases is known as the Arrhenius Theory. The concentration of hydrogen ions are often described by the pH scale as a numeric value.

The pH Scale: Acidic, Neutral, and Basic

The pH scale describes the acidity of the solution: acidic, neutral, or basic A solution with a pH less than 7 is an acid, exactly 7 is a neutral solution, and above 7 is a base. Bases have less hydrogen ions but more hydroxide ions, represented by the pOH or “potential of hydroxide ions.” Table 1. The pH Scale

Acidic	Neutral	Basic
Less than 7	7	Greater than 7

Many other scientists studied the proprieties of acids and bases from the ideas of Sorensen and Arrhenius and came up with their own definitions. A notable theory is known as the Bronsted-Lowry Theory. The Bronsted-Lowry Theory is a concept involving acid and bases which suggest that acids act as proton donors. Since neutral hydrogen atoms are usually made of one proton and one electron, a positive hydrogen ion is often referred to as a proton. These protons carry a positive charge and are given away to the bases. Bases, with that logic, are proton acceptors. Bases, carrying lone pairs of electrons, attract positive hydrogen ions (protons). In the lab, pH can be determined by a pH indicator such as pH paper. pH paper usually contains a weak acid or a weak base which will respond by changing color at a specific pH. This method is used frequently as a cheap, quick way to determine pH rather than using pH meters which need frequent calibration and maintenance. Keep in mind that very low or very high pH value solutions can be very caustic and should be handled with care. Image Source: Flickr

Practice Problems

Determine if the following solution is acidic, neutral, or basic.

1. pH = 1.00
2. pH = 10.00
3. pH = 6.99
4. pH = 7.02
5. pH = 8.00
6. pH = 13.00
7. pH = 2.00

Answers:

1. Acidic
2. Basic
3. Neutral
4. Neutral
5. Basic
6. Basic
7. Acidic

Concentrations of H⁺ and OH^–

Concentration is the amount of solute in respect to the amount of total solution. A high amount of solute equals a high concentration, where a lower amount of solute would equal a low overall concentration. When an acid or a base is placed into a solvent, that compound will dissociate into ions. The concentration of H⁺ (hydrogen ions) in the solution will determine the acidity or basicity of the solution. A high concentration of H⁺ will signify an acidic solution and a low concentration of H⁺ will signify a basic solution. In hydrochloric acid for example (a common acid that is an aqueous solution of HCl), HCl molecules have dissociated into two kinds of ions, H⁺ and Cl^–. This dissociation produces a high H⁺ concentration, which is a property of an acidic solution. The same can be seen in basic solutions where there is a low H⁺ concentration, due to the high OH^– (hydroxide ion). For example, when NaOH (sodium hydroxide, a common base), is placed in water, it dissociates into two kinds of ions, Na⁺ and OH^–. The high OH^– concentration, which corresponds to a low H⁺ concentration, is a property of basic solution. Table 2. Relationship Between pH and pOH

	Concentration of H+	Concentration of -OH	Example(s)
Acid	High	Low	HCl, HCOOH, HNO3
Base	Low	High	NaOH, MgO, CaCO3
Neutral	Equal to OH–	Equal to H+	Water

The determination of pH and pOH will be calculated by using the concentration of hydrogen ions and hydroxide ions respectively. pH and pOH also have a relationship so that if you do not have enough information to determine one, you can use the concentration of the other. This will be done through Sorensen’s equation for calculating pH.

How to Calculate pH

Note: Please use a scientific calculator. pH is determined by the concentration of H⁺, which is frequently summarized as [H⁺]. This can be calculated by the following equation: pH=-log { \left[ { H }^{ + } \right] } or pH=log { \left( \dfrac { 1 }{ \left[ { H }^{ + } \right] } \right) } Conversely, the hydrogen concentration can be found by a given pH. The [H⁺] can be calculated by the following equation. \left[ { H }^{ + } \right] ={ 10 }^{ -pH } The determination of the concentration of hydrogen ions and pH will later be used to show the relationship between pH and pOH.

Key Equations:

pH=-log { \left[ { H }^{ + } \right] } or pH=log { \left( \dfrac { 1 }{ \left[ { H }^{ + } \right] } \right) } \left[ { H }^{ + } \right] ={ 10 }^{ -pH }

Example 1: Calculate the pH of a 0.200 M HCl solution.

HCl solutions are strong acids, so we can already expect a pH less than 7. Using the 0.200 M HCl as the [H+] (concentration of hydrogen ions) the solution is as follows: pH=-log { \left[ { H }^{ + } \right] } = log(0.200) =0.70 A 0.70 pH indicates a very acidic solution.

Example 2: Calculate the pH of a 0.100 M nitric acid solution.

Nitric acid has a chemical formula of HNO₃. HNO₃ is another strong acid, so the pH of this solution will also be less than 7. Using the 0.100 M nitric acid as the [H⁺] (concentration of hydrogen ions) the solution is as follows: pH=-log { \left[ { H }^{ + } \right] } = log (0.100) = 1.00 A 1.00 pH indicates a very acidic solution. In examples 1 and 2 we are able to use the concentration of the acid as the concentration of the H⁺ ion because every acid molecule dissociates, thereby releasing an H⁺ ion. These types of acids are referred to as “strong acids”. For “weak acids”, most of the acid molecules do not dissociate, so we would have to use more complex methods to calculate pH for solutions of weak acids. We’ll stick to strong acids and bases in this article, and explore weak acids and base in another article.

Example 3. What is the hydrogen ion concentration of a solution that has a pH of 4.30?

This example provides the opposite information. Here, we are given the concentration of H⁺ in a solution and are asked to determine the pH. 4.30 = -log { \left[ { H }^{ + } \right] } -4.30 = log { \left[ { H }^{ + } \right] } \left[ { H }^{ + } \right] ={ 10 }^{ -4.30 }M=5.01x{ 10 }^{ -5 }M

How to Calculate pOH

pOH is determined by the concentration of OH^–, [OH^–]. This can be calculated by the following equation: pOH=-log { \left[ { OH }^{ — } \right] } or pOH=log { \left( \dfrac { 1 }{ \left[ { OH }^{ — } \right] } \right) } Conversely, the hydroxide concentration can be found by a given pOH. The [OH^–] can be calculated by the following equation. \left[ { OH }^{ — } \right] ={ 10 }^{ -pOH } pOH is a different way of describing acidity and basicity, so be careful not to mix it up with pH. The descriptions for solutions based on the pOH scale are given in Table 2. Table 3. The pOH Scale

Basic	Neutral	Acidic
Less than 7	7	Greater than 7

The determination of the concentration of hydroxide ions and pOH will be later used to show the relationship between pH and pOH.

Key Equations:

pOH=-log { \left[ { OH }^{ — } \right] } or pOH=log { \left( \dfrac { 1 }{ \left[ { OH }^{ — } \right] } \right) } \left[ { OH }^{ — } \right] ={ 10 }^{ -pOH }

Example 1: Calculate the pOH of a 1.20 M NaOH solution.

This is calculated similarly to the determination of pH. Instead of determining the pH, we will be determining the pOH with use of -log[OH^–]. NaOH will dissociate completely in solution, so we can use the concentration of NaOH as the concentration of OH^–. pOH=-log { \left[ { OH }^{ — } \right] } = log(1.20) = -0.08 A -0.08 pOH indicates a very basic solution.

Example 2: Calculate the pOH of a solution with a hydroxide concentration of 5.23 x 10^-5 M.

pOH=-log { \left[ { OH }^{ — } \right] } = log(5.23 x 10^{ -5}) = 4.20

Example 3. What is the hydroxide concentration of a solution that has a pOH of 11.30?

This example provides the opposite information. Here, we are given the concentration of OH^– in a solution and are asked to determine the pOH. This is done similarly to the determination of hydrogen concentration from a pH. 11.30 = -log { \left[ { OH }^{ — } \right] } – 11.30 = log { \left[ { OH }^{ — } \right] } { \left[ { OH }^{ — } \right] }={ 10 }^{ -11.30 }M=5.01x{ 10 }^{ -12 }

Calculate the Relationship Between pH and pOH

In the section “Concentrations of H⁺ and OH^–“ we discussed that the high concentration of hydroxide ions left little room for hydrogen ions and vice versa. The relationship of pH and pOH is that both values will equal 14.00. This can be represented in the following equation: pH + pOH = 14.00 You can check your work by adding the pH and pOH to ensure that the total equals 14.00. This also is an excellent representation of the concept of pH neutrality, where equal concentrations of [H⁺] and [OH^–] result in having both pH and pOH as 7. pH+pOH=14.00 pH=14-pOH pH=14-pH

Example 1: What is the pH of a solution that has a pOH of 12.40?

Keep in mind that the relationship of pH and pOH equals 14. pH+pOH = 14.00 pOH = 12.40 pH= unknown pH +12.40 = 14.00 pH =1.60 Check Your Work: 1.60 + 12.40 = 14.00

Example 2: What is the pOH of a solution that has a [H⁺] of 0.100 M HCl?

First, determine the pH and use that value with the relationship of pH and pOH. pH+pOH = 14.00 pH = -log[0.1000] = 1.00 1.00 + pOH = 14.00 pOH = 13.00 Check Your Work: 1.00 + 13.00 = 14.00

Example 3: What is the pOH of a solution that has a pH of 3.40?

Keep in mind that the relationship of pH and pOH equals 14.00. pH+pOH = 14.00 pH = 3.40 pOH = unknown pOH+3.40 = 14.00 pOH=10.60 = pH 10.60 Check Your Work: 10.60 + 3.40 = 14.00

The Importance of pH

pH is all around us. It is important that vital solutions such as water, stomach acid, and blood maintain a consistent pH. Water, with a neutral pH of around 7, determines the solubility of many compounds. Without the appropriate pH of water, many chemical reactions would not occur. This can also be seen through naturally occurring phenomena such as acid rain. Highly acidic precipitation can cause erosion and other hazardous environmental outcomes. Image Source: EPA pH plays an important role in the solutions in the human body. Specific pH values are vital to the roles of solutions such as saliva, stomach acid, and blood. The production of saliva in the mouth is known as the first step of digestion. Throughout the digestive tract, food must be broken down by acidic solutions. It is important that the pH of saliva should be between 6.5-7.5, slightly acidic, to begin this process. Later on, stomach acid functions in the digestive system as well. It is important that stomach acid has a very acidic pH, ranging from about 1.5 to 3.5, due to the secretion of HCl and the high concentration of hydrogen ions. This strongly acidic environment kicks digestion into high gear and begins to break down food particles in preparation for the excretion process. Healthy blood has a pH of 7.4. Hundreds of reactions occur in the bloodstream, such as enzymes, which require a specific pH. Blood with a higher or lower pH can result in negative symptoms. Acidosis is a symptom of a condition in which the pH value of blood is too low and alkalosis indicates blood with a pH value which is too high. Humans aren’t the only organisms that rely on appropriate pH levels. Some species only thrive in alkaline (basic) environments and would not be able to survive in neutral or acidic environments. Entire ecosystems revolve around pH.

Questions for Discussion

1. Why is it important that oceans keep a specific pH?
2. Name some common household items with an basic pH.
3. What is the pH of vinegar? Why?
4. If a patient suffers from acidosis, what are they suffering from?
5. How does pH play a role in the blood?

More with pH: Acid-Base Equilibrium, Titrations, Buffers, pKa, Equilibrium Constant, Neutralization, Conjugate Acids, Conjugate Base.

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Check out our other articles on Chemistry. You can also find thousands of practice questions on Albert.io. Albert.io lets you customize your learning experience to target practice where you need the most help. We’ll give you challenging practice questions to help you achieve mastery in Chemistry. Start practicing here. Are you a teacher or administrator interested in boosting Chemistry student outcomes? Learn more about our school licenses here. pH is a measure of how acidic or basic a substance is. In our everyday routine, we encounter and drink many liquids with different pH. Water is a neutral substance. Soda and coffee are often acidic. The pH is an important property, since it affects how substances interact with one another and with our bodies. In our lakes and oceans, pH determines what creatures are able to survive in the water. Read on to learn vocabulary associated with pH, how to use the pH formula, how to calculate pH, and why pH is an important measurement! Test tubes containing substances of different pH as indicated by the color of the solution.

What is pH?

pH is a measure of how basic or acidic a substance is. pH has a range of 0-14. A pH greater than 7 means the substance is basic. A pH less than 7 means the substance is acidic. When the pH is exactly 7 that indicates that the substance is neutral. An acidic substance is anything that will give up a proton. A basic substance will accept a proton. For more on acids and bases see this article. The pH formula is: pH = -log ([H⁺]) This formula for pH is discussed in more detail in a section below, including how to use the pH formula.

Related Links

• Strong Acids and Bases
• Properties of Acids and Bases
• Acid Base Chemistry
• Buffers
• Acid Base Theories: Arrhenius & Brønsted-Lowry Acids
• Acid Base Neutralization Reaction

What Is the pH Scale?

The pH scale starts from the number 0 and ends at the number 14. These numbers allow the classification of substances based on their pH; the most acidic substances will be close to 0, while the most basic or alkaline substances will be close to 14. The lower the pH, the more H⁺ ions will be present and the stronger the acid. The most basic or alkaline substances will have a classification between 7 and 14.

pH Formula

pH = −log ([H⁺]) The formula for pH is shown above. pH is defined as the negative log base 10 of the hydronium concentration. The pH is a logarithmic measure of the concentration of hydrogen ions in a solution. Because pH is on a log scale that means that increasing the pH by 1 corresponds to multiplying the concentration of H⁺ ions by 10! So even though the difference between pH 6 and pH 7 might sound small, it’s actually quite sizeable. For the pH equation, the concentration of hydrogen ions is always a molar concentration, that is, moles of H⁺ per liter.

How to calculate pH

If you know the concentration of hydrogen ions, then calculating the pH is just plugging in to the pH equation. Sometimes, a problem will tell you that an acid completely dissociates into ions in solution. In this, knowing the hydrogen ion concentration is straight forward. For example: Assume a 0.2 molar solution of HCl completely dissociates in solution. This means for each mole of hydrochloric acid, there is 1 mole of H⁺ ions. (HCl is a strong acid, so completely dissociates). So the concentration of hydrogen ions is 0.2 M. The pH formula tells us that the pH is the negative log of the hydrogen ion concentration – which is 0.2 molar. Then use a calculator to plug the 0.2 M into the pH formula. pH = -log ([H⁺]) = – log (0.1) = 0.699 Now if the acid does not completely dissociate, and they give you the K_a of the acid, you need to calculate pH from K_a, which is covered in this article.

pH Equation – Converting pH to H⁺

In some situations, we know the pH and need to convert to the molar concentration of H⁺ ions. For this, we need to invert the logarithm from the first equation, by raising 10 to the power of the negative pH. [H ⁺ ] = 10 ^-pH Using this equation, we find that if the pH of a solution is 7, then [H⁺] = 10^-7 M. If the pH of a solution is 0, then [H⁺] = 10^-0 M = 1M (a one molar solution), and if the pH of a solution is 14, then [H⁺] = 10^-14 M.

Measuring pH

There are several ways to measure the pH of a substance in the laboratory, at home, or in the field.

1. pH Probe: A pH probe is an instrument that has an arm with two small electrodes in it. The arm is placed in a substance and the electrodes respond to the pH of the solution. The pH is typically displayed on a small screen. There are both large and small portable models of this instrument.

2. Litmus paper: Litmus paper is a thin strip of paper that changes color based on the pH of the solution it is dipped in to. This pH indicator can be found at some pet supply stores to test the pH of aquariums. 3. Chemical Indicators: Chemical indicators are chemicals you can place into solution that will change the color of the solution as the pH changes. Some common ones are phenolphthalein, bromthymol blue, and litmus. Although theses won’t give you the exact pH they are a great way to tell about what pH a substance is. 4. Many Others: There are many other chemicals that can act as indicators in different ways. One easy one to test at home is using cabbage juice. Instructions for making your own cabbage juice indicator can be found here. Another great activity for exploring the pH of different substances can also be found here. pH indicator strips (Source: Wikimedia Commons)

pOH

The term pOH is similar to pH, but refers to alkalinity or basicity, that is, the concentration of hydroxide ion (OH-) in a solution. The two scales function identically, except that the scale is reversed. A neutral substance has both pH and pOH of 7. However in the pOH scale a basic substance will have a pOH of less than 7. The equation for pOH is the same as that for pH except using the concentration of hydroxide instead of H⁺: pOH = −log [OH^–]

Water and its pH

Pure water has a pH of 7 on the pH scale, meaning that it is neutral. In pure water, the concentration of hydrogen ions, and hydroxide ions, are both the same – 10^-7 M. However, dissolved acids, bases, or salts can make it either acidic or basic. For example, ocean water tends to have a pH of about 8. Even a glass of water won’t have a pH of exactly 7 due to the carbon dioxide from the atmosphere that dissolves into it. Pollutants can also change the pH of water, so water pH is frequently monitored in many situations for both safety and research.

K_w in Chemistry

K_w is the dissociation constant or ionization constant of water. When water ionizes, it splits into a hydrogen ion (H⁺) and a hydroxide ion (OH^–). H⁺ is unstable in water on its own and prefers to form a hydronium ion (H₃O⁺), but for convenience we usually still refer to it as H⁺. The ionization constant of water represents the degree to which it exists as ions versus together as a molecules and equals the concentration of H⁺ times the concentration of OH^–: K _w = [H ⁺ ][OH ^– ] pK _w = -log _Kw At room temperature, K_w = 10^-14 and pKw = 14. We can use this constant to convert between the pH and pOH via the equation below: pH + pOH = pK _w

Why is pH important?

Some chemical reactions only take place under certain pH conditions. Sometimes this is because H⁺ or OH^– acts as a reactant in the reaction. In other cases, acid or base can catalyze a reaction, meaning that they affect the rate of the reaction. Living organisms rely on a wide variety of biochemical reactions and processes, most of which require specific pH ranges. As a result, ecosystems like lakes and rivers thrive under the pH conditions that are favorable to the biochemistry of the local flora and fauna. Like an ecosystem, the human body has a certain pH that allows the proper functioning of the different tasks that our body performs. We require one value in our blood, and a totally different one (much more acidic) in our digestive fluids. Otherwise, normal biochemistry could break down, causing serious health issues. Luckily, humans and many other creatures have blood that is buffered, so that the pH cannot change easily. This is why if you drink a bottle of alkaline water or acidic soda, your blood will stay nearly the same, keeping you safe from the effects of imbalance! Another consequence of this safety net built into your blood is that the common health benefits associated with alkaline water are mostly made-up. Even if your blood were somehow too acidic, drinking some water with pH 8 would barely change it!

Fun Facts About the pH Scale

• Acids often have a sour taste to them, like lemons
• Bases tend to taste bitter
• Søren Peder Lauritz Sørensen devised the pH formula and pH scale in 1909. Sørensen was a Danish chemist working at the Carlsberg Laboratory.
• Many cleaners are very basic (pH > 10). Some examples include drain cleaner, bleach and ammonia solution. Always be careful working with very basic or very acidic substances.
• Due to the increase in carbon dioxide in the air, our oceans are currently decreasing in pH which means they are getting more acidic. The change in pH has cascading effects on the creatures in the ocean. For background and activities on this topic, see this article.

Further Reading

• What is specific heat?
• Acid Base Neutralization Reactions
• What is a reactant?
• What is a chemical bond?

pH is a measure of how acidic or basic a chemical solution is. The pH scale runs from 0 to 14—a value of seven is considered neutral, less than seven acidic, and greater than seven basic. pH is the negative base 10 logarithm («log» on a calculator) of the hydrogen ion concentration of a solution. To calculate it, take the log of a given hydrogen ion concentration and reverse the sign. See more information about the pH formula below. Here’s a more in-depth review of how to calculate pH and what pH means with respect to hydrogen ion concentration, acids, and bases.

Review of Acids and Bases

There are several ways to define acids and bases, but pH specifically only refers to hydrogen ion concentration and is applied to aqueous (water-based) solutions. When water dissociates, it yields a hydrogen ion and a hydroxide. See this chemical equation below.

H₂O ↔ H⁺ + OH^—

When calculating pH, remember that [ ] refers to molarity, M. Molarity is expressed in units of moles of solute per liter of solution. If you are given concentration in any other unit than moles (mass percent, molality, etc.), convert it to molarity in order to use the pH formula. The relationship between pH and molarity can be expressed as:

K_w = [H⁺][OH^—] = 1×10^-14 at 25°C
for pure water [H⁺] = [OH^—] = 1×10^-7

• K_w is the dissociation constant of water
• Acidic Solution: [H⁺] > 1×10^-7
• Basic Solution: [H⁺] < 1×10^-7

How to Calculate pH and [H+]

The equilibrium equation yields the following formula for pH:

pH = -log₁₀[H⁺]
[H⁺] = 10^-pH

In other words, pH is the negative log of the molar hydrogen ion concentration or the molar hydrogen ion concentration equals 10 to the power of the negative pH value. It’s easy to do this calculation on any scientific calculator because more often than not, these have a «log» button. This is not the same as the «ln» button, which refers to the natural logarithm.

pH and pOH

You can easily use a pH value to calculate pOH if you recall:

pH + pOH = 14

This is particularly useful if you’re asked to find the pH of a base since you’ll usually solve for pOH rather than pH.

Example Calculation Problems

Try these sample problems to test your knowledge of pH.

Example 1

Calculate the pH for a specific [H⁺]. Calculate pH given [H⁺] = 1.4 x 10^-5 M Answer: pH = -log₁₀[H⁺]
pH = -log₁₀(1.4 x 10^-5)
pH = 4.85

Example 2

Calculate [H⁺] from a known pH. Find [H⁺] if pH = 8.5 Answer: [H ⁺ ] = 10 ^-pH
[H ⁺ ] = 10 ^-8.5
[H ⁺ ] = 3.2 x 10 ^-9 M

Example 3

Find the pH if the H⁺ concentration is 0.0001 moles per liter. Here it helps to rewrite the concentration as 1.0 x 10^-4 M because this makes the formula: pH = -(-4) = 4. Or, you could just use a calculator to take the log. This gives you: Answer: pH = — log (0.0001) = 4 Usually, you aren’t given the hydrogen ion concentration in a problem but have to find it from a chemical reaction or acid concentration. The simplicity of this will depend on whether you have a strong acid or a weak acid. Most problems asking for pH are for strong acids because they completely dissociate into their ions in water. Weak acids, on the other hand, only partially dissociate, so at equilibrium, a solution contains both the weak acid and the ions into which it dissociates.

Example 4

Find the pH of a 0.03 M solution of hydrochloric acid, HCl. Remember, Hydrochloric acid is a strong acid that dissociates according to a 1:1 molar ratio into hydrogen cations and chloride anions. So, the concentration of hydrogen ions is exactly the same as the concentration of the acid solution. Answer: [H⁺ ]= 0.03 M pH = — log (0.03)
pH = 1.5

Check Your Work

When you’re performing pH calculations, always make sure your answers make sense. An acid should have a pH much less than seven (usually one to three) and a base should have a high pH value (usually around 11 to 13). While it’s theoretically possible to calculate a negative pH, pH values should be between 0 and 14 in practice. This means that a pH higher than 14 indicates an error either in setting up the calculation or the calculation itself.

Sources

• Covington, A. K.; Bates, R. G.; Durst, R. A. (1985). «Definitions of pH scales, standard reference values, measurement of pH, and related terminology». Pure Appl. Chem. 57 (3): 531–542. doi:10.1351/pac198557030531
• International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry (2nd ed.) Oxford: Blackwell Science. ISBN 0-632-03583-8.
• Mendham, J.; Denney, R. C.; Barnes, J. D.; Thomas, M. J. K. (2000). Vogel’s Quantitative Chemical Analysis (6th ed.). New York: Prentice Hall. ISBN 0-582-22628-7.

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