Problem 22 - Chemistry Matter and Change

Question

The concentration of either the \(\mathrm{H}^{+}\) ion or the \(\mathrm{OH}^{-}\) ion is given for four aqueous solutions at 298 \(\mathrm{K}\) . For each solution, calculate \(\left[\mathrm{H}^{+}\right]\) or \(\left[\mathrm{OH}^{-}\right] .\) State whether the solution is acidic, basic, or neutral. \begin{equation} \begin{array}{ll}{\text { a. }\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-13} M} & {\text { c. }\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-3} \mathrm{M}} \\\ {\text { b. }\left[\mathrm{OH}^{-}\right]=1.0 \times 10^{-7} M} & {\text { d. }\left[\mathrm{H}^{+}\right]=4.0 \times 10^{-5} \mathrm{M}}\end{array} \end{equation}

Step-by-Step Solution

Answer

a. Basic, b. Neutral, c. Basic, d. Acidic.

Step 1: Understanding Water Ion Product

At 298 K, the ion product of water \( K_w \) is \( 1.0 \times 10^{-14} \). The equations \( [\mathrm{H}^+] \times [\mathrm{OH}^-] = K_w \) can be used to find the missing ion concentrations.

Step 2: Calculate for Part (a)

Given \( [\mathrm{H}^+] = 1.0 \times 10^{-13} \). Calculate \( [\mathrm{OH}^-] = \frac{K_w}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-13}} = 1.0 \times 10^{-1} \). Since \( [\mathrm{H}^+] < [\mathrm{OH}^-] \), the solution is basic.

Step 3: Calculate for Part (b)

Given \( [\mathrm{OH}^-] = 1.0 \times 10^{-7} \). Calculate \( [\mathrm{H}^+] = \frac{K_w}{[\mathrm{OH}^-]} = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-7}} = 1.0 \times 10^{-7} \). Since \( [\mathrm{H}^+] = [\mathrm{OH}^-] \), the solution is neutral.

Step 4: Calculate for Part (c)

Given \( [\mathrm{OH}^-] = 1.0 \times 10^{-3} \). Calculate \( [\mathrm{H}^+] = \frac{K_w}{[\mathrm{OH}^-]} = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-3}} = 1.0 \times 10^{-11} \). Since \( [\mathrm{H}^+] < [\mathrm{OH}^-] \), the solution is basic.

Step 5: Calculate for Part (d)

Given \( [\mathrm{H}^+] = 4.0 \times 10^{-5} \). Calculate \( [\mathrm{OH}^-] = \frac{K_w}{[\mathrm{H}^+]} = \frac{1.0 \times 10^{-14}}{4.0 \times 10^{-5}} = 2.5 \times 10^{-10} \). Since \( [\mathrm{H}^+] > [\mathrm{OH}^-] \), the solution is acidic.

Key Concepts

Water Ion Product

The water ion product, often symbolized as \( K_w \), is a fundamental concept in chemistry, especially when dealing with acid-base reactions. At 298 K, the value of \( K_w \) is \( 1.0 \times 10^{-14} \). This constant is derived from the equilibrium between water molecules and their ions (\( \mathrm{H}^+ \) and \( \mathrm{OH}^- \)).
To understand how it works, remember that water can self-ionize, meaning it can act as both an acid and a base. This reaction is described by the equation:

• \[ \mathrm{H}_2\mathrm{O}(l) \rightleftharpoons \mathrm{H}^+(aq) + \mathrm{OH}^-(aq) \]

Since the ion product \( K_w \) is very small, pure water only contains \( 10^{-7} \) M of both \( \mathrm{H}^+ \) and \( \mathrm{OH}^- \) at 298 K, reflecting its neutrality.
Knowing \( K_w \), we can calculate the concentration of the hydrogen ion \( \mathrm{H}^+ \) if the hydroxide ion \( \mathrm{OH}^- \) is known and vice versa using the formula:

• \[ [\mathrm{H}^+] \times [\mathrm{OH}^-] = K_w \]

This relationship is crucial for determining the nature of solutions, whether they're basic, neutral, or acidic.

pH Calculation

Calculating the pH of a solution is a way to express its acidity or basicity. pH is a logarithmic measure and is defined as the negative base-10 logarithm of the hydrogen ion concentration, \([\mathrm{H}^+]\):

• \[ \text{pH} = -\log_{10}([\mathrm{H}^+]) \]

Understanding pH is fairly straightforward when you follow the logic of this formula. If the concentration of \( [\mathrm{H}^+] \) is known, you can plug it into the formula to find the pH.
For neutral solutions at 298 K, the \( [\mathrm{H}^+] \) is equal to \( 10^{-7} \), hence a pH of 7. Solutions with a pH lower than 7 are acidic, meaning they have a higher concentration of \( [\mathrm{H}^+] \). Conversely, solutions with a pH above 7 are basic, indicating a higher concentration of \( [\mathrm{OH}^-] \).
One thing to note is that because pH is a logarithmic scale, each whole number change represents a tenfold change in hydrogen ion concentration. This makes pH a very sensitive measure for changes in acidity or basicity.

Acidic and Basic Solutions

An acidic solution is defined by a higher concentration of hydrogen ions \( [\mathrm{H}^+] \) than hydroxide ions \( [\mathrm{OH}^-] \). Common examples include substances like lemon juice or vinegar.
To determine if a solution is acidic, we often see that its pH is less than 7 and the ratio of \( [\mathrm{H}^+] \) to \( [\mathrm{OH}^-] \) exceeds 1. You calculate \([\mathrm{OH}^-]\) from \([\mathrm{H}^+]\) using \( K_w \), if starting \([\mathrm{H}^+]\) is given.
Basic solutions, on the other hand, are characterized by a higher concentration of \( [\mathrm{OH}^-] \) ions than \( [\mathrm{H}^+] \). Substances like baking soda or soap represent basic solutions.
Here, the pH will be greater than 7, and the ratio of \( [\mathrm{H}^+] \) to \( [\mathrm{OH}^-] \) is less than 1. The strength of the acidity or basicity relies heavily on the magnitude of these concentrations relative to each other.