Medicine & HealthcareAcid-Base PhysiologyA-Level

pH Calculation

Calculates the acidity or alkalinity of a solution based on the logarithmic concentration of hydrogen ions ([H⁺]).

Understand the formulaSee the free derivationOpen the full walkthrough

p H = - lo g_{10} [H^{+}]

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

The pH scale provides a convenient way to express the concentration of hydrogen ions in a solution, as these values are often extremely small. Because the scale is logarithmic, a change of one pH unit represents a tenfold change in [H⁺] concentration. This relationship is critical in biological systems where even minor deviations in hydrogen ion concentration can have drastic effects on enzyme activity and cellular function.

When to use: Apply this when the concentration of hydrogen ions ([H⁺]) in mol/L is provided and you need to determine the acidity or alkalinity of a physiological fluid or chemical solution.

Why it matters: Maintaining precise pH levels is vital for human homeostasis; for example, blood pH must be kept within a very narrow range (7.35–7.45) to prevent acidosis or alkalosis.

Symbols

Variables

[H^+] = Hydrogen Ion Concentration (mol/L), pH = Potential of Hydrogen

[H^{+}]

Hydrogen Ion Concentration (mol/L)

V a r iab l e

p H

Potential of Hydrogen

V a r iab l e

Walkthrough

Derivation

Derivation of pH Calculation

The pH scale is derived from the molar concentration of hydrogen ions using a logarithmic function to manage the wide range of values found in aqueous solutions.

The solution is dilute, allowing concentration to be treated as equivalent to activity.
The temperature is standard at 25°C (298K).

Definition of Ion Product of Water

The auto-ionization of water establishes the relationship between hydrogen and hydroxide ions, providing a baseline for neutrality.

K_{w} = [H^{+}] [O H^{-}] = 1.0 \times 1 0^{- 14} mol^{2} dm^{- 6}

Note: Remember that Kw changes with temperature.

Introduction of Logarithmic Scale

Since [H+] values are often extremely small (e.g., 10^-7), taking the negative base-10 logarithm simplifies these values into a manageable scale.

- lo g_{10} (K_{w}) = - lo g_{10} ([H^{+}] [O H^{-}])

Note: The negative sign is used to ensure pH values are generally positive.

Application of Logarithm Laws

Applying the product rule for logarithms (log(ab) = log a + log b) separates the hydrogen ion component.

- lo g_{10} (K_{w}) = - lo g_{10} [H^{+}] - lo g_{10} [O H^{-}]

Note: This identity is fundamental to the relationship pH + pOH = 14.

Defining pH

The term -log[H+] is defined as the pH, representing the power of hydrogen in the solution.

pH = - lo g_{10} [H^{+}]

Note: Always ensure your concentration is in mol/dm³.

Result

pH = - lo g_{10} [H^{+}]

Source: AQA A-Level Chemistry Specification / Pearson Edexcel International A-Level Chemistry

Free formulas

Rearrangements

Solve for $p H$

Make pH the subject

p H = - lo g_{10} [H^{+}]

The formula is already provided with pH as the subject.

Difficulty: 1/5

Solve for $[H^{+}]$

Make [H⁺] the subject

[H^{+}] = 1 0^{- p H}

Rearrange the logarithmic equation by applying the base-10 exponential function to both sides.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Think of the pH scale as a 'magnifying glass' for a logarithmic zoom lens. Because hydrogen ion concentrations span many orders of magnitude (e.g., 0.1 to 0.0000001), the log scale acts like a compressed ruler that turns massive decimal gaps into simple, manageable whole numbers.

Potential of Hydrogen

A simple numerical label representing the 'acidity level' of a liquid, where lower numbers signal a more aggressive, acidic environment.

[H⁺]

Molar concentration of hydrogen ions

The 'density' or count of free protons swimming in a specific volume of solution; more protons mean a higher chance of chemical reactions.

Signs and relationships

-: Since hydrogen concentrations are typically tiny fractions (e.g., 10⁻⁷), the negative sign is used to flip the resulting negative exponent into a positive, user-friendly number.
log₁₀: This indicates a base-10 logarithmic scale; each whole number change in pH represents a tenfold difference in H⁺ concentration, mirroring how we perceive magnitude in chemistry.

One free problem

Practice Problem

Calculate the pH of a solution where the hydrogen ion concentration [H⁺] is 1.0 x 10⁻⁷ mol/L.

[H⁺]1e-7

Solve for: $p H$

Hint: The negative log of 10^-x is simply x.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the pH of arterial blood given a hydrogen ion concentration of 4.0 x 10⁻⁸ mol/L helps clinicians identify if a patient is experiencing respiratory or metabolic acidosis.

Study smarter

Tips

Remember that the log is base 10, not the natural logarithm (ln).
Higher [H⁺] values result in lower pH values due to the negative sign.
Ensure [H⁺] is expressed in moles per liter (M) before calculation.

Avoid these traps

Common Mistakes

Forgetting the negative sign in the calculation.
Confusing the base 10 log with the natural log (ln).

Common questions

Frequently Asked Questions

The pH scale is derived from the molar concentration of hydrogen ions using a logarithmic function to manage the wide range of values found in aqueous solutions.

Apply this when the concentration of hydrogen ions ([H⁺]) in mol/L is provided and you need to determine the acidity or alkalinity of a physiological fluid or chemical solution.

Maintaining precise pH levels is vital for human homeostasis; for example, blood pH must be kept within a very narrow range (7.35–7.45) to prevent acidosis or alkalosis.

Forgetting the negative sign in the calculation. Confusing the base 10 log with the natural log (ln).