Question

Calculate the \(\left[\mathrm{H}^{+}\right]\)in each of the following solutions: (a) \(\left[\mathrm{OH}^{-}\right]=4.5 \times 10^{-2}\) (b) \(\left[\mathrm{OH}^{-}\right]=5.2 \times 10^{-9}\)

Short answer

For part (a): \( \text{[H}^+] = 2.22 \times 10^{-13} \). For part (b): \( \text{[H}^+] = 1.92 \times 10^{-6} \).

Step-by-step solution

- Understand the relationship between \(\text{H}^+\) and \(\text{OH}^-\text{)\text{ concentrations}

The relationship between the hydrogen ion concentration \( \text{[H}^+\text{]} \) and the hydroxide ion concentration \( \text{[OH}^-] \) in water at 25°C can be given by the equation: \[ \text{[H}^+] \times \text{[OH}^-] = 1.0 \times 10^{-14} \]

- Rearrange the equation

To find \( \text{[H}^+]\), rearrange the equation: \[ \text{[H}^+] = \frac{1.0 \times 10^{-14}}{ \text{[OH}^-]} \]

- Calculate \( \text{[H}^+]\) for part (a)

Given \( \text{[OH}^-] = 4.5 \times 10^{-2}\), substitute this value into the equation: \[ \text{[H}^+] = \frac{1.0 \times 10^{-14}}{4.5 \times 10^{-2}} = 2.22 \times 10^{-13} \]

- Calculate \( \text{[H}^+]\) for part (b)

Given \( \text{[OH}^-] = 5.2 \times 10^{-9}\), substitute this value into the equation: \[ \text{[H}^+] = \frac{1.0 \times 10^{-14}}{5.2 \times 10^{-9}} = 1.92 \times 10^{-6} \]

Key concepts

Ion Concentration Calculation

One of the fundamental concepts in acid-base chemistry is calculating ion concentrations. Specifically, the concentrations of hydrogen ions \( \text{[H}^+]\text{]} \) and hydroxide ions \( \text{[OH}^{-}]\text{]} \) in solutions. This is vital for understanding the properties of acids and bases. In water at 25°C, the product of \( \text{[H}^+]\text{]} \) and \( \text{[OH}^-]\text{]} \) is a constant value, given by: \[ \text{[H}^+] \times \text{[OH}^-] = 1.0 \times 10^{-14} \] This relationship allows us to find one ion concentration if we know the other. For example, if you know the hydroxide ion concentration in a solution, you can find the hydrogen ion concentration using this formula: \[ \text{[H}^+] = \frac{1.0 \times 10^{-14}}{\text{[OH}^-]} \] Let’s practice this with two scenarios. In the first, we are given \[ \text{[OH}^{-}] = 4.5 \times 10^{-2} \] , we find \[ \text{[H}^+] = \frac{1.0 \times 10^{-14}}{4.5 \times 10^{-2}} = 2.22 \times 10^{-13} \] In the second, we are given: \[ \text{[OH}^{-}] = 5.2 \times 10^{-9} \], leading us to find: \[ \text{[H}^+] = \frac{1.0 \times 10^{-14}}{5.2 \times 10^{-9}} = 1.92 \times 10^{-6} \] By understanding this foundational relationship and knowing how to rearrange and apply the equation, you can consistently calculate ion concentrations in various chemical solutions.

pH and pOH Relationship

The pH and pOH scales are tools chemists use to describe the acidity or basicity of a solution. The pH scale ranges from 0-14 and measures the concentration of hydrogen ions \[ \text{[H}^+] \]. On the other hand, the pOH scale measures the concentration of hydroxide ions \[ \text{[OH}^-] \]. An important thing to remember is: \[ \text{pH} + \text{pOH} = 14 \] This is derived from the constant product relationship of \[ \text{[H}^+] \times \text{[OH}^-] = 1.0 \times 10^{-14} \]. If you know one value, you can easily calculate the other. \[ \text{pH} = -\text{log[H}^+] \] and \[ \text{pOH} = -\text{log[OH}^-] \] For instance, if you have \[ \text{[OH}^-] = 4.5 \times 10^{-2} \], you can calculate \[ \text{[H}^+] = 2.22 \times 10^{-13} \] using the earlier method. From here, use the pH formula: \[ \text{pH} = -\text{log}(2.22 \times 10^{-13}) \] Calculating this would give you the pH value. Similarly, with the value of \[ \text{[OH}^-] = 5.2 \times 10^{-9} \], which gives \[ \text{[H}^+] = 1.92 \times 10^{-6} \], you can find the pH value. Understanding the pH and pOH relationship is crucial in chemistry since it provides a measure of the solution's hydrogen ion and hydroxide ion concentration, helping classify solutions as acidic, neutral, or basic.

Acid-Base Chemistry

Acid-base chemistry is a key area in understanding chemical reactions and solutions. Acids are substances that release hydrogen ions \[ \text{[H}^+] \] when dissolved in water, while bases release hydroxide ions \[ \text{[OH}^-] \]. In water, these ions interact in a balance described by the ion product constant \[ K_w = \text{[H}^+] \times \text{[OH}^-] = 1.0 \times 10^{-14} \] at 25°C.
Here are a few important concepts:

• Strong acids and bases dissociate completely in water.
• Weak acids and bases do not fully dissociate.
• The pH scale quantifies acidity, with values below 7 indicating acidic solutions, above 7 indicating basic solutions, and around 7 being neutral.

To calculate and understand the nature of a solution, you often start with known concentrations of ions, applying the \[ K_w \] constant to find missing values. For instance, by understanding and calculating \[ \text{[H}^+] \] from \[ \text{[OH}^-] \], and vice versa, you can determine whether a solution is acidic or basic. Moreover, using the pH and pOH formulas helps classify and understand the properties of the solutions. By grasping these concepts and equations, you build a strong foundation in acid-base chemistry.