Question

For a given aqueous solution, if \(\left[\mathrm{H}^{+}\right]=1.0 \times 10^{-3} \mathrm{M},\) what is \(\left[\mathrm{OH}^{-}\right] ?\)

Short answer

\(\left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-11} \mathrm{M}\)

Step-by-step solution

Understand the Relationship

In aqueous solutions, the product of the concentrations of hydrogen ions (\(\left[\mathrm{H}^{+}\right]\)) and hydroxide ions (\(\left[\mathrm{OH}^{-}\right]\)) is constant at 25°C. This relationship is given by the water dissociation constant, \(K_w\), which is \(1.0 \times 10^{-14}\).

Use the Ion Product of Water

Given \(\left[\mathrm{H}^{+}\right] = 1.0 \times 10^{-3} \mathrm{M}\), use the equation for the ion product of water: \[ \left[\mathrm{H}^{+}\right] \times \left[\mathrm{OH}^{-}\right] = K_w = 1.0 \times 10^{-14} \]

Substitute and Solve for \(\left[\mathrm{OH}^{-}\right]\)

Substitute the known value \(\left[\mathrm{H}^{+}\right] = 1.0 \times 10^{-3} \mathrm{M}\) into the equation and solve for \(\left[\mathrm{OH}^{-}\right]\): \[ 1.0 \times 10^{-3} \times \left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-14} \] Divide both sides by \(1.0 \times 10^{-3}\): \[ \left[\mathrm{OH}^{-}\right] = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-3}} = 1.0 \times 10^{-11} \mathrm{M} \]

Verify Units and Finalize Answer

Ensure that the solution for \(\left[\mathrm{OH}^{-}\right]\) is dimensionally consistent and the units are correct (Molarity, M). Conclude that the concentration found is correct and represents the hydroxide ion concentration in the solution.

Key concepts

Hydrogen Ion Concentration

The hydrogen ion concentration, denoted as \([\mathrm{H}^+]\), is a crucial factor in determining the acidity of a solution. This concentration is derived from the dissociation of acids in an aqueous solution. \([\mathrm{H}^+]\) is measured in moles per liter (M), which represents the number of moles of hydrogen ions present per liter of solution.
The scale to measure how acidic a solution is involves the pH scale. It is formulated as: \[ \text{pH} = -\log_{10}[\mathrm{H}^+] \] In this formula, a lower \[\text{pH}\] value indicates a higher hydrogen ion concentration and therefore a more acidic solution.
In the given exercise, a concentration of \([\mathrm{H}^+] = 1.0 \times 10^{-3} \, \mathrm{M}\) translates to a relatively low pH, denoting an acidic solution. Understanding this indicator helps chemists and students comprehend the nature of the solutions they are working with, allowing for control and prediction of chemical reactions.

Hydroxide Ion Concentration

The hydroxide ion concentration, represented as \([\mathrm{OH}^-]\), plays a vital role in defining the basicity or alkalinity of a solution. Like hydrogen ions, hydroxide ions are expressed in moles per liter (M). When a base dissociates in water, it increases \([\mathrm{OH}^-]\) in the solution.
\([\mathrm{OH}^-]\) is inversely related to \([\mathrm{H}^+]\) through the water dissociation constant. A higher hydroxide concentration leads to a basic (or alkaline) solution.
For instance, if a solution’s \([\mathrm{H}^+] = 1.0 \times 10^{-3} \; \mathrm{M}\), the \([\mathrm{OH}^-]\) can be computed using the equation: \[ [\mathrm{H}^+] \times [\mathrm{OH}^-] = K_w = 1.0 \times 10^{-14} \] Knowing one ion concentration allows us to quickly find the other, granting insight into the solution's profile.

Water Dissociation Constant

An essential concept to understand in chemistry is the water dissociation constant, \[ K_w \]. This constant represents the relationship between hydrogen ions and hydroxide ions in pure water at a temperature of 25°C. The value of \[ K_w \] is \[ 1.0 \times 10^{-14} \], meaning: \[ [\mathrm{H}^+] \times [\mathrm{OH}^-] = 1.0 \times 10^{-14} \]
This equilibrium condition allows us to understand that the product of \([\mathrm{H}^+]\) and \([\mathrm{OH}^-]\) will always equal \[ K_w \] in an aqueous solution, provided it's at the stated condition.
\[ K_w \] is fundamental because it provides a vital link between acidity (crate[H+]) and basicity (crate[OH-]) under standard conditions. By using \[ K_w \], anyone can find the concentration of either of the ions if the other is known. It essentially serves as the "yardstick" for assessing how acidic or basic a solution is, independent of conditions directly added to the water.