Question

Write the mathematical expression that allows you to solve for the \(\mathrm{OH}^{-}\) concentration in water when you know only the \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration.

Short answer

The mathematical expression to solve for the OH⁻ concentration in water when only the H₃O⁺ concentration is known is [OH⁻] = \(\dfrac{1.0 \times 10^{-14}}{[H_{3}O^+]}\), using the ion product constant of water (Kw).

Step-by-step solution

Understand the ion product constant of water (Kw)

In pure water, a small amount of water molecules ionize to form H₃O⁺ and OH⁻ ions. The ion product constant of water (Kw) is a measure of this process and is given by the following expression: Kw = [H₃O⁺][OH⁻] For water at 25°C, the value of Kw is 1.0 x 10⁻¹⁴. Note that the concentrations are given in moles per liter (M).

Rearrange the formula for Kw to solve for the OH⁻ concentration

Our goal is to find an expression for the OH⁻ concentration when the H₃O⁺ concentration is known. We can do this by rearranging the formula for Kw: [OH⁻] = \(\dfrac{Kw}{[H_{3}O^+]}\)

Write the final expression for OH⁻ concentration

Now that we have derived the formula for the OH⁻ concentration, we can write it as a final expression: [OH⁻] = \(\dfrac{1.0 \times 10^{-14}}{[H_{3}O^+]}\) This is the mathematical expression we can use to solve for the OH⁻ concentration in water when only the H₃O⁺ concentration is known.

Key concepts

Kw (Ion Product of Water)

The ion product constant of water, often denoted as \( K_w \), is a fundamental concept in chemistry that helps us understand the behavior of water molecules. In pure water, a tiny fraction of the water molecules dissociate into hydronium ions \( \text{H}_3\text{O}^+ \) and hydroxide ions \( \text{OH}^- \). This self-ionization can be expressed by the equation:\[\text{H}_2\text{O(l)} \leftrightarrow \text{H}_3\text{O}^+(aq) + \text{OH}^-(aq)\]The ion product \( K_w \) is defined as the product of the molar concentrations of these ions:\[ K_w = [\text{H}_3\text{O}^+][\text{OH}^-] \]At a temperature of 25°C, the value of \( K_w \) is \( 1.0 \times 10^{-14} \) \( M^2 \), which is crucial for calculating either ion's concentration when the other is known. It's important to note that the concentrations are measured in moles per liter \( M \). This constant can change with temperature, reflecting how the environment can influence the degree of water's ionization.
Understanding \( K_w \) helps us make sense of acid-base chemistry, where water's self-ionization plays a pivotal role in maintaining equilibrium in middle school to advanced chemistry studies. When solving problems related to \( K_w \), it's useful to remember its dependence on temperature and its significance in both acidic and basic solutions.

H₃O⁺ concentration

The concentration of hydronium ions, \( \text{H}_3\text{O}^+ \), is a critical component when considering the acidity of a solution. It arises from the dissociation of acids in water or from water's autoionization. In pure water at 25°C, the concentration of \( \text{H}_3\text{O}^+ \) matches that of \( \text{OH}^- \), each being \( 1.0 \times 10^{-7} \) M. This balance maintains what's known as a neutral pH of 7. Any increase in \( \text{H}_3\text{O}^+ \) concentration signifies an increasingly acidic solution, while a decrease points to a basic solution.
Calculating \( \text{H}_3\text{O}^+ \) concentration is important in different chemical contexts, including determining pH. The pH is simply a logarithmic measure of hydronium ion concentration, given by:\[pH = -\log [\text{H}_3\text{O}^+]\]Thus, knowing \( \text{H}_3\text{O}^+ \) concentration enables us to find the pH, which further informs us about the solution's characteristics. In exercises requiring these calculations, it's crucial to remember that the logarithmic nature of pH means small changes in \( \text{H}_3\text{O}^+ \) concentration can result in significant pH shifts.

OH⁻ concentration

Understanding the concentration of hydroxide ions, \( \text{OH}^- \), is crucial in determining a solution's basicity. Hydroxide ions are produced either through the dissociation of bases or the ionization of water. The formula derived from the ion product constant of water (\( K_w \)) allows us to calculate \( \text{OH}^- \) concentration if the \( \text{H}_3\text{O}^+ \) concentration is known:\[[\text{OH}^-] = \dfrac{1.0 \times 10^{-14}}{[\text{H}_3\text{O}^+]}\]This expression shows an inverse relationship between \( [\text{OH}^-] \) and \( [\text{H}_3\text{O}^+] \): as one increases, the other decreases. In pure water at 25°C, both are \( 1.0 \times 10^{-7} \) M, reflecting neutrality. But as \( \text{OH}^- \) concentration increases, the solution becomes more basic, and the pH rises above 7. This relationship is pivotal for understanding equilibrium in aqueous solutions and is frequently applied in contexts such as buffer preparations, where maintaining specific ion concentrations is vital.
Whether tackling homework or real-world problems, being able to calculate \( \text{OH}^- \) concentration is a valuable tool for anyone studying chemistry.