What is the \(\mathrm{pH}\) of a solution whose hydronium ion concentration is \(2.2 \times 10^{-12} \mathrm{M} ?\)

Understanding the hydronium ion concentration is crucial when studying the acidity or basicity of a solution. Hydronium ions, denoted as \( H_3O^+ \), are the positive ions that result when water molecules gain an extra proton. The concentration of these ions in a solution, typically measured in moles per liter (M), directly determines the solution's acidity.

In the context of our exercise, a solution with a hydronium ion concentration of \( 2.2 \times 10^{-12} \mathrm{M} \) is considered. This notation, known as scientific notation, is a way to express very large or very small numbers in a compact form. The exponent (-12) indicates how many places to move the decimal point to the left. Therefore, in this example, the concentration is a very dilute acid solution signifying a very low number of hydronium ions present.

The precise quantification of \( H_3O^+ \) ions in this example serves as the foundation for calculating the pH, which we will explore next with the help of the pH formula.

The term 'logarithm' may seem daunting, but its concept is key to understanding pH calculations. A logarithm is the power to which a number, called the base, must be raised to produce another number. The negative logarithm, specifically, is used in pH calculations and it involves taking the logarithm of a number and then multiplying the result by -1.

In simple terms, if you have a positive number \(x\), the negative logarithm is just \( -\log(x) \). So, why negative? In the context of pH, we're often dealing with very small numbers (like the concentration of hydronium ions), which result in negative pH values. Since pH cannot be negative, we use the negative logarithm to transform these values into a positive pH scale that ranges from 0 to 14.

A logarithm with a base of 10 (logarithm base 10) is commonly used, and for our exercise, this is the type of logarithm we apply to the hydronium ion concentration to calculate the pH value.

The pH formula is the backbone of the pH calculation process. It is elegantly simple: \(\text{pH} = -\log [H_3O^+]\), where \(\text{pH}\) represents the power of hydrogen or the 'potential of Hydrogen' and \( [H_3O^+] \) represents the concentration of hydronium ions in the solution. By applying this formula, we can convert the concentration of hydronium ions into a value that represents the acidity or basicity of the solution.

To apply the formula, as done in the exercise, one plugs in the hydronium ion concentration in its scientific notation form into the logarithmic function and then applies the negative sign. This gives us a pH value, which in the case of our exercise, is a positive number indicating where the solution falls on the pH scale. It's a quick and effective way to assess the character of the solution, allowing scientists and students alike to categorize it as acidic, neutral, or basic.