Question

Write the mathematical definition of \(\mathrm{pH}\), and explain what fundamental change there would be in the pH scale if you left the negative out.

Short answer

The mathematical definition of pH is given by the equation: \(pH = -\log_{10} [H^+]\), where \([H^+]\) represents the concentration of hydrogen ions in the solution, measured in moles per liter (M). If we were to leave out the negative sign in the pH definition, the scale would be reversed, with lower pH' values indicating higher basicity, and higher pH' values indicating higher acidity. The neutral point would shift to pH'=0, and it would make the scale less intuitive to interpret as it would no longer represent how many times more acidic or basic a solution is relative to the neutral point.

Step-by-step solution

Definition of pH

pH is a measure of the acidity or basicity of a solution. It is defined as the negative logarithm (base 10) of the concentration of hydrogen ions (\(H^+\)) in the solution. Mathematically, the pH is given by the following equation: \[pH = -\log_{10} [H^+]\] Where \([H^+]\) represents the concentration of hydrogen ions in the solution, measured in moles per liter (M).

Changes in the pH scale without the negative sign

If we were to leave out the negative sign in the pH definition, the equation would be: \[pH' = \log_{10} [H^+]\] The omission of the negative sign would lead to fundamental changes in the pH scale, as detailed below: 1. Reverse interpretation of acidity and basicity: In the original pH scale, a lower pH value (less than 7) corresponds to higher acidity, whereas a higher pH value (greater than 7) corresponds to higher basicity. By removing the negative sign, the scale would be reversed; lower pH' values would indicate higher basicity, and higher pH' values would indicate higher acidity. 2. Shift in the neutral point: The neutral point in the original pH scale is at pH=7, representing a solution with equal concentrations of \(H^+\) and \(OH^-\) ions. In the modified pH' scale, the neutral point would shift to pH'=0, as the logarithm of unity is zero, i.e., \(log_{10}(1)=0\). 3. Loss of intuitive interpretation: The original pH scale ranges from 0 to 14, with each whole number step representing a tenfold change in hydrogen ion concentration. The loss of the negative sign would make it more difficult to intuitively interpret and understand the scale, as it would no longer represent how many times more acidic or basic a solution is relative to the neutral point. In conclusion, leaving the negative sign out of the pH definition would fundamentally change the pH scale, altering the interpretations of acidity and basicity, shifting the neutral point, and making the scale less intuitive.

Key concepts

Acidity

Acidity refers to the level of acid in a substance, particularly an aqueous solution. When we talk about an acid, we're looking at a solution that has more hydrogen ions ( [H^+] ) present. In chemistry, acidity is directly related to the concentration of these hydrogen ions. The greater the concentration, the stronger the acid, and conversely, the lesser the concentration, the weaker the acid. Acidity affects how a solution behaves in reactions; for example, acids typically turn blue litmus paper red and can dissolve some metals and neutralize bases. Understanding acidity is crucial when examining solutions because this property will dictate how the solution interacts with other substances.

Logarithm

A logarithm is a mathematical function used to express how many times a number (called the base) must be multiplied by itself to reach another number. In the context of pH, the base is 10, and we use a logarithmic scale to express hydrogen ion concentration because this allows us to manage large variations in size more comfortably. The pH of a solution is defined as the negative logarithm (base 10) of its hydrogen ion concentration:\[ \text{pH} = -\log_{10} [H^+] \]This logarithmic scale means that each whole number step on the pH scale represents a tenfold change in [H^+] . It simplifies massive numerical differences and makes them easier to comprehend, ranging from highly acidic (pH 0) to highly basic (pH 14).

Hydrogen Ion Concentration

Hydrogen ion concentration, [H^+], measures how many hydrogen ions are present in a given volume of a solution. These ions are responsible for the acidic nature of a solution. In a typical scenario, the concentration is measured in moles per liter (M), representing the number of moles of hydrogen ions in one liter of the solution. As the concentration of hydrogen ions increases, the solution becomes more acidic, which corresponds to a lower pH value. Conversely, lower concentrations result in a higher pH, indicating a more basic solution. Understanding [H^+] is crucial for determining the pH, as it provides the necessary data to calculate and interpret the acidity or basicity of the solution using the pH formula.

Basicity

Basicity, often referred to as alkalinity, refers to the presence of base components in a solution, particularly those that can accept hydrogen ions or donate electron pairs. In the pH scale, a basic solution is one with a pH greater than 7, indicating fewer hydrogen ions present and more hydroxide ions (OH⁻) compared to an acidic solution. Basic substances, like sodium hydroxide (NaOH) and ammonia (NH₃), are characterized by their ability to neutralize acids and are typically slippery to the touch. They also turn red litmus paper blue. In terms of ion interactions, the converse of high hydrogen ion concentration, a high concentration of OH⁻ ions corresponds to a basic solution. By understanding basicity, one can predict how solutions will behave when combined with acids and design reactions and products accordingly.