Question

Calculate the hydrogen ion concentration, in moles per liter, for solutions with each of the following pH values. a. \(\mathrm{pH}=8.34\) b. \(\mathrm{pH}=5.90\) c. \(\mathrm{pH}=2.65\) d. \(\mathrm{pH}=12.6\)

Short answer

The hydrogen ion concentrations for each pH value are as follows: a. \([\mathrm{H^+}] \approx 4.57 \times 10^{-9}\ \mathrm{M}\) b. \([\mathrm{H^+}] \approx 1.26 \times 10^{-6}\ \mathrm{M}\) c. \([\mathrm{H^+}] \approx 2.24 \times 10^{-3}\ \mathrm{M}\) d. \([\mathrm{H^+}] \approx 2.51 \times 10^{-13}\ \mathrm{M}\)

Step-by-step solution

Write the formula for pH

Recall the formula for pH: \[\mathrm{pH} = -\log_{10}[\mathrm{H^+}]\]

Plug in the given pH value

We are given the pH value 8.34, so we plug it into the formula to solve for \([\mathrm{H^+}]\). \[8.34 = -\log_{10}[\mathrm{H^+}]\]

Solve for the hydrogen ion concentration

Now, we need to solve for \([\mathrm{H^+}]\). We can do this by using exponentiation to isolate the concentration term: \[[\mathrm{H^+}] = 10^{-8.34}\] Calculate \(10^{-8.34}\): \[[\mathrm{H^+}] \approx 4.57 \times 10^{-9}\ \mathrm{M}\] b. pH = 5.90

Write the formula for pH

Recall the formula for pH: \[\mathrm{pH} = -\log_{10}[\mathrm{H^+}]\]

Plug in the given pH value

We are given the pH value 5.90, so we plug it into the formula to solve for \([\mathrm{H^+}]\). \[5.90 = -\log_{10}[\mathrm{H^+}]\]

Solve for the hydrogen ion concentration

Now, we need to solve for \([\mathrm{H^+}]\). We can do this by using exponentiation to isolate the concentration term: \[[\mathrm{H^+}] = 10^{-5.90}\] Calculate \(10^{-5.90}\): \[[\mathrm{H^+}] \approx 1.26 \times 10^{-6}\ \mathrm{M}\] c. pH = 2.65

Write the formula for pH

Recall the formula for pH: \[\mathrm{pH} = -\log_{10}[\mathrm{H^+}]\]

Plug in the given pH value

We are given the pH value 2.65, so plug it into the formula to solve for \([\mathrm{H^+}]\). \[2.65 = -\log_{10}[\mathrm{H^+}]\]

Solve for the hydrogen ion concentration

Now, we need to solve for \([\mathrm{H^+}]\). We can do this by using exponentiation to isolate the concentration term: \[[\mathrm{H^+}] = 10^{-2.65}\] Calculate \(10^{-2.65}\): \[[\mathrm{H^+}] \approx 2.24 \times 10^{-3}\ \mathrm{M}\] d. pH = 12.6

Write the formula for pH

Recall the formula for pH: \[\mathrm{pH} = -\log_{10}[\mathrm{H^+}]\]

Plug in the given pH value

We are given the pH value 12.6, so plug it into the formula to solve for \([\mathrm{H^+}]\). \[12.6 = -\log_{10}[\mathrm{H^+}]\]

Solve for the hydrogen ion concentration

Now, we need to solve for \([\mathrm{H^+}]\). We can do this by using exponentiation to isolate the concentration term: \[[\mathrm{H^+}] = 10^{-12.6}\] Calculate \(10^{-12.6}\): \[[\mathrm{H^+}] \approx 2.51 \times 10^{-13}\ \mathrm{M}\]

Key concepts

Hydrogen Ion Concentration

Understanding the concept of hydrogen ion concentration is crucial in acid-base chemistry. When we mention \([\mathrm{H}^+]\), we're referring to the concentration of hydrogen ions in a solution, expressed in moles per liter. This measure helps us determine how acidic or basic a solution is. The scale on which hydrogen ion concentration is evaluated is called the pH scale.

The relationship between pH and hydrogen ion concentration is logarithmic, meaning small changes in pH correspond to large changes in \([\mathrm{H}^+]\). For instance, when the pH decreases by one unit, the hydrogen ion concentration increases tenfold. This exponential relationship is why accurate calculations are so vital in chemistry.

In practical terms:

• A high \([\mathrm{H}^+]\) indicates a low pH, resulting in an acidic solution.
• A low \([\mathrm{H}^+]\) indicates a high pH, resulting in a basic or alkaline solution.

Acid-Base Chemistry

Acid-base chemistry revolves around the concept of proton donors and acceptors. Acids are substances that donate protons (hydrogen ions \([\mathrm{H}^+]\)), while bases are substances that accept them. This simple principle is the foundation of many chemical reactions and processes.

The pH scale, ranging from 0 to 14, serves as a convenient tool to express the acidity or basicity of a solution:

• Neutral solutions: Have a pH of around 7, such as pure water.
• Acidic solutions: Have a pH less than 7, like lemon juice or vinegar.
• Basic solutions: Have a pH greater than 7, such as soap or baking soda.

Understanding this scale helps predict the behavior of substances when they interact. For example, the reaction between an acid and a base often results in the formation of water and a salt, a process known as neutralization.

Logarithmic Calculations

Logarithmic calculations are key to understanding how the pH scale works. The formula for pH, \(\mathrm{pH} = -\log_{10}[\mathrm{H}^+]\), highlights the logarithmic nature of the scale. This means that each unit change in pH represents a tenfold change in hydrogen ion concentration.

To calculate \([\mathrm{H}^+]\) from a given pH, you reverse the logarithmic function using exponentiation:

• Rewrite the equation as \([[\mathrm{H}^+]] = 10^{-\mathrm{pH}}\).
• Plug in the pH value to find the concentration.

This mathematical manipulation makes it easy to switch between pH and hydrogen ion concentration. It allows chemists to quickly assess solutions and predict how they might interact in different conditions. Getting comfortable with these calculations is essential for anyone studying chemistry or related fields.