Question

Calculate \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)for each solution. (a) \(\mathrm{pH}=1.76\) (b) \(\mathrm{pH}=3.88\) (c) \(\mathrm{pH}=8.43\) (d) \(\mathrm{pH}=12.32\)

Short answer

\([H_3O^+]\) for the solutions are: (a) \(10^{-1.76}\), (b) \(10^{-3.88}\), (c) \(10^{-8.43}\), and (d) \(10^{-12.32}\).

Step-by-step solution

Understanding pH and Hydronium Concentration

pH is the negative logarithm base 10 of the hydronium ion concentration, \[H_3O^+\]. The formula to find \[H_3O^+\] from pH is \[H_3O^+ = 10^{-pH}\].

Calculating \[H_3O^+\] for Solution (a)

To find \[H_3O^+\] for pH=1.76, use the formula \[H_3O^+ = 10^{-1.76}\].

Calculating \[H_3O^+\] for Solution (b)

To find \[H_3O^+\] for pH=3.88, use the formula \[H_3O^+ = 10^{-3.88}\].

Calculating \[H_3O^+\] for Solution (c)

To find \[H_3O^+\] for pH=8.43, use the formula \[H_3O^+ = 10^{-8.43}\].

Calculating \[H_3O^+\] for Solution (d)

To find \[H_3O^+\] for pH=12.32, use the formula \[H_3O^+ = 10^{-12.32}\].

Key concepts

pH calculation

Understanding how to calculate pH is a fundamental skill in chemistry, as it provides a measure of the acidity or basicity of a solution. To calculate the pH, you need to know the concentration of hydronium ions, \(\left[\mathrm{H}_{3}O^{+}\right]\), in the solution. The pH is calculated using the formula \(\text{pH} = -\log_{10}\left[\mathrm{H}_{3}O^{+}\right]\). This equation expresses the pH as the negative logarithm of the hydronium ion concentration. The importance of this formula lies in its ability to convert the potentially complex concentrations of hydronium ions into a more manageable number on a simple scale.

For instance, a hydronium ion concentration of \(1 \times 10^{-7} M\) would result in a pH of 7, which is neutral on the pH scale. Solutions with a lower concentration of hydronium ions (and thus a higher pH) are basic, while those with a higher concentration of hydronium ions (and a lower pH) are acidic. To improve understanding, visualize the pH as a scale that inversely reflects the amount of acidity in a solution—the lower the pH, the higher the acidity.

hydronium ion

The hydronium ion, \(\mathrm{H}_{3}O^{+}\), plays a central role in the discussion of acids and bases in chemistry. It is formed when an acid donates a proton (\(H^{+}\)) to a water molecule (\(H_2O\)), resulting in \(\mathrm{H}_{3}O^{+}\). The concentration of hydronium ions in a solution determines its acidity and, consequently, its pH value. Acids increase the hydronium ion concentration in a solution, while bases reduce it.

To grasp the concept, consider that pure water at 25°C has a hydronium ion concentration of \(1 \times 10^{-7} M\), making it neutral. Any increase in this concentration makes the solution more acidic, and any decrease makes it more basic. When solving problems, it is crucial to express the hydronium ion concentration in molarity (moles per liter) to accurately determine the pH.

pH scale

The pH scale is a dimensionless range commonly used to express the acidity or basicity of a solution on a logarithmic scale. It typically ranges from 0 to 14, with 7 being neutral, below 7 acidic, and above 7 basic. One unit change in pH represents a tenfold change in the hydronium ion concentration. Understanding this scale allows one to quickly assess whether a substance is acidic or basic and to what degree.

For example, a substance with a pH of 4 is ten times more acidic than one with a pH of 5 and one hundred times more acidic than one with a pH of 6. This logarithmic nature makes the pH scale a very efficient way to express a wide range of concentrations. It's important for students to familiarize themselves with the scale by considering real-world examples, such as the pH of vinegar or baking soda, to see how everyday substances fit into the scale.

logarithmic calculations

Logarithmic calculations are crucial for understanding the pH scale as the pH is the negative logarithm of the hydronium ion concentration. Essentially, logarithms allow us to work with large or small numbers more conveniently. In the context of pH, the logarithmic scale condenses the wide range of hydronium ion concentrations into a manageable size.

Applying logarithmic calculations requires understanding certain properties such as the inverse relationship between the logarithms and exponents—considering \(\log_{10}(10^{x}) = x\). In pH calculations, taking the negative logarithm of \(10^{-x}\), which is often the case for hydronium ion concentration, simplifies to \(x\) because of the inverse properties mentioned. This concept assists in making accurate, quick converts from a hydronium ion concentration to pH and vice versa, which is a valuable tool when analyzing the chemical properties of substances.