Question

A beer has a pH of \(4.30 .\) What is the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) ? (a) \(3.0 \times 10^{-4}\) (b) \(2.0 \times 10^{-4}\) (c) \(2.0 \times 10^{-5}\) (d) \(\cdot 5.0 \times 10^{-5}\)

Short answer

The concentration of hydronium ions \(\left[\mathrm{H}_{3}\mathrm{O}^{+}\right]\) is \(5.0 \times 10^{-5} M\). Option (d) is correct.

Step-by-step solution

- Understanding pH

pH is a measure of the acidity or basicity of an aqueous solution. The pH scale ranges from 0 to 14 and is defined as the negative logarithm (base 10) of the hydrogen ion concentration. So, pH = -log10[H3O+].

- Convert pH to Hydrogen Ion Concentration

To find the hydrogen ion concentration [H3O+], we can reverse the pH definition. We calculate the antilog (inverse log) of the negative pH which is 10 raised to the power of the negative pH value: [H3O+] = 10^(-pH).

- Calculate the Hydrogen Ion Concentration

Given pH = 4.30, we calculate [H3O+] = 10^(-4.30). Using a calculator, this yields [H3O+] = 5.0 x 10^(-5) M.

Key concepts

pH scale

The pH scale is a chart that provides information about the acidity or basicity of a substance. It ranges from 0 to 14, with 7 being neutral. Substances with a pH lower than 7 are considered acidic, while those with a pH higher than 7 are basic (or alkaline). The scale is logarithmic, meaning each whole pH value below 7 is ten times more acidic than the next higher value, and conversely, each whole pH value above 7 is ten times less acidic (or more basic) than the one below it. For example, a substance with a pH of 3 is ten times more acidic than one with a pH of 4.
For students delving into the understanding of pH, it's fundamental to recognize that the scale is a measure of the concentration of hydrogen ions (\([\text{H}^+]\)) in a solution. Interestingly, pH stands for 'potentia hydrogenii' which means 'potential of hydrogen'. In the context of the exercise provided, a beer with a pH of 4.30 is categorized as acidic, as it is below 7 on the pH scale.

acidity and basicity

Acidity and basicity are two fundamental characteristics that describe the nature of substances. Acids are substances that can donate a hydrogen ion (\([\text{H}^+]\)), and they usually taste sour and can corrode metals. Examples include lemon juice and vinegar. On the other hand, bases are substances that can accept a hydrogen ion and often have a bitter taste and a slippery feel, like soap or baking soda. When an acid and a base react, they neutralize each other and form water and a salt in a chemical reaction known as neutralization.
When exploring these concepts, it's helpful to employ the Bronsted-Lowry definition, which expands on acids and bases as proton donors and acceptors, respectively. It's critical for students to grasp that the concentration of hydrogen ions is inversely related to pH: the higher the \([\text{H}^+]\) concentration, the lower the pH and greater the acidity. In the exercise scenario, the measured pH value (4.30) indicates that beer is acidic, although it is not a strong acid, as it does not have a pH that is extremely close to 0.

logarithmic calculations

Logarithmic calculations are essential for understanding the pH scale and are based on powers of 10. A logarithm answers the question: 'To what power must we raise 10 to get this number?'. In the context of pH, \(\text{pH} = -\text{log}_{10}[\text{H}_3\text{O}^+]\), the logarithm is negative because the hydrogen ion concentration in aqueous solutions is typically a small decimal number (less than 1). Therefore, taking the negative logarithm of the concentration results in a positive pH value.
To convert pH to hydrogen ion concentration, you use the antilog function, which is essentially raising 10 to the power of the negative pH value: \([\text{H}_3\text{O}^+] = 10^{-\text{pH}}\). In the sample exercise, to find the \([\text{H}_3\text{O}^+]\) based on a given pH of 4.30, we raise 10 to the power of -4.30, yielding \(5.0 \times 10^{-5}\) M. It's crucial for students to become comfortable with these calculations, as they pave the way for understanding more complex chemical concepts related to acid-base chemistry.