Question

If the \(\mathrm{pH}\) of an \(\mathrm{HCl}\) solution is \(2.0\), what is the concentration of HCl?

Short answer

The concentration of HCl is \(0.01 M\).

Step-by-step solution

Understand the Relationship between pH and Concentration

First, understand that the pH of a solution is defined as the negative logarithm to base 10 of the activity of the hydronium ion (\(H_3O^+\)). Mathematically, \(pH = -log10[H_3O^+]\), where \([H_3O^+]\) is the concentration of the hydronium ions in moles per litre. Because HCl is a strong acid, it will completely dissociate in water, and one molecule of HCl will give one hydronium ion. Hence, the concentration of HCl can also be considered as \([H_3O^+]\). So, we can modify the equation as \(pH = -log10[HCl]\)

Convert pH to Concentration

To determine the concentration, we rearrange the formula to solve for \([HCl]\) which gets us \([HCl]=10^{-pH}\).

Calculate Concentration

Substitute the given pH value into the formula to get the concentration. \([HCl]=10^{-pH} = 10^{-2.0} = 0.01 M\).

Key concepts

pH and Hydronium Relationship

Understanding the relationship between pH and hydronium ions is crucial in many chemistry applications. pH is an index used to specify the acidity or basicity of an aqueous solution. Mathematically, pH is expressed as the negative logarithm of the hydronium ion concentration:
\( pH = -\text{log}([H_3O^+]) \).
Hydronium ions, represented by \( H_3O^+ \), are the form in which protons (H+) exist in water. For solutions of strong acids, such as hydrochloric acid (HCl), the assumption is that every acid molecule contributes one hydronium ion to the solution. Therefore, the concentration of hydronium ions is equal to the concentration of the acid itself. With this in mind, by knowing the pH, we can readily determine the hydronium ion concentration, which in the case of strong acids, is synonymous with the acid's concentration.

Acid Dissociation in Water

Acids dissociate in water to produce hydronium ions (H3O+) and their corresponding anions. The degree to which an acid dissociates is indicative of its strength. Strong acids like HCl fully dissociate in water, which means each molecule of the acid contributes one hydronium ion to the solution. In contrast, weak acids only partially dissociate, establishing an equilibrium between undissociated molecules and ions.
In the case of HCl, a strong acid, the dissociation equation is:
\( HCl \rightarrow H^+ + Cl^- \).
In aqueous solutions, the hydrogen ions (H+) associate with water molecules to form hydronium ions:\( H^+ + H_2O \rightarrow H_3O^+ \).
This complete dissociation makes calculating the concentration of HCl straightforward once the pH is known.

Logarithmic pH Scale

The pH scale is logarithmic, which means each whole pH value below 7 (the neutral pH of water) is ten times more acidic than the next higher value. This log scale compresses the range of hydrogen ion concentrations, which can vary over many orders of magnitude, into a manageable scale from 0 to 14 for most practical purposes.
A key aspect to understand is that a decrease in one pH unit reflects a tenfold increase in acidity. For instance, a solution with a pH of 2 is not just twice as acidic as a solution with a pH of 3; it is in fact ten times more acidic. This logarithmic nature allows for handling very small or very large numbers in a more convenient way and is pivotal in pH concentration calculations.

Chemistry Problem-Solving

Problem-solving in chemistry typically involves understanding concepts and applying mathematical relationships. In our exercise, the pH value gave us the necessary information to determine the concentration of HCl using the logarithmic relationship of pH.
Crucial steps in chemical problem-solving include:

• Identifying the relevant chemical principles or equations.
• Substituting the known values into the equations.
• Manipulating the equation to solve for the unknown.

With practice, students develop the ability to accurately apply these steps to a wide range of chemistry problems, from calculating concentrations to predicting reaction outcomes.