Question

For a given aqueous solution, if \(\left[\mathrm{OH}^{-}\right]=3.77 \times 10^{-4} \mathrm{M},\) what is \(\left[\mathrm{H}^{+}\right] ?\)

Short answer

\([\mathrm{H}^{+}] = 2.65 \times 10^{-11} \text{ M}\).

Step-by-step solution

Recall the Ion Product of Water

In any aqueous solution at 25°C, the product of the concentrations of hydrogen ions (\[\left[\mathrm{H}^{+}\right]\]) and hydroxide ions (\[\left[\mathrm{OH}^{-}\right]\]) is constant and is known as the ion product of water (\[K_w\]). It is given by:\[K_w = \left[\mathrm{H}^{+}\right] \times \left[\mathrm{OH}^{-}\right] = 1.0 \times 10^{-14}\]

Apply the Ion Product Expression

We know \[\left[\mathrm{OH}^{-}\right] = 3.77 \times 10^{-4}\, \text{M}\] and we need to find \[\left[\mathrm{H}^{+}\right]\]. Using\[K_w = \left[\mathrm{H}^{+}\right] \times \left[\mathrm{OH}^{-}\right]\], substitute the known value:\[1.0 \times 10^{-14} = \left[\mathrm{H}^{+}\right] \times 3.77 \times 10^{-4}\]

Solve for [H⁺]

Rearrange the equation from Step 2 to solve for \[\left[\mathrm{H}^{+}\right]\]:\[\left[\mathrm{H}^{+}\right] = \frac{1.0 \times 10^{-14}}{3.77 \times 10^{-4}}\] Perform the division:\[\left[\mathrm{H}^{+}\right] \approx 2.65 \times 10^{-11} \text{ M}\]

Final Answer

Therefore, the concentration of hydrogen ions \(\left[\mathrm{H}^{+}\right]\) in the solution is approximately \(2.65 \times 10^{-11} \text{ M}\).

Key concepts

Hydrogen Ion Concentration

The hydrogen ion concentration, denoted by \( \left[\mathrm{H}^{+}\right] \), is a crucial measure in chemistry, particularly when dealing with aqueous solutions. This concentration indicates the number of hydrogen ions present in a given solution. Hydrogen ions are a measure of acidity in a solution.
A lower hydrogen ion concentration corresponds to a more basic (or less acidic) solution, while a higher concentration means the solution is more acidic. It's important to understand that even though we may refer to hydrogen ions, in water, they mostly take the form of hydronium ions \( \left(\mathrm{H}_3\mathrm{O}^{+}\right) \). However, \( \left[\mathrm{H}^{+}\right] \) is still the notation used for simplicity.

• In pure water, the hydrogen ion concentration is very low, around \( 1.0 \times 10^{-7} \text{ M} \) at 25°C.
• This corresponds to pH 7, a neutral pH, because water auto-dissociates into equal parts hydrogen and hydroxide ions.

Hydroxide Ion Concentration

Hydroxide ion concentration, represented by \( \left[\mathrm{OH}^{-}\right] \), is another fundamental aspect of aqueous solutions. It provides a measure of the basicity of a solution rather than its acidity. When a solution has a high \( \left[\mathrm{OH}^{-}\right] \), it is more basic.
The hydroxide ions play an essential role in balancing the hydrogen ion concentration in a solution. For any aqueous solution, the concentrations of hydrogen ions and hydroxide ions are interconnected through the ion product constant of water (\( K_w \)).

• For instance, in pure water, \( \left[\mathrm{OH}^{-}\right] \) also equals \( 1.0 \times 10^{-7} \text{ M} \) at 25°C, complementing \( \left[\mathrm{H}^{+}\right] \).
• In a basic solution, \( \left[\mathrm{OH}^{-}\right] \) is greater than \( 1.0 \times 10^{-7} \text{ M} \), indicating a decrease in \( \left[\mathrm{H}^{+}\right] \).

Aqueous Solutions

Aqueous solutions are those where water is the solvent, dissolving various substances (solutes). Water's polar nature makes it an excellent solvent, capable of dissolving many ionic and covalent compounds. The chemistry of aqueous solutions often involves the study of ion concentrations, such as hydrogen and hydroxide ions, to understand the solution's properties.
These solutions are omnipresent in chemical reactions and biological processes. The behavior of ions in aqueous solutions can significantly impact reactions.

• Water's amphoteric property allows it to act as both an acid and a base, which is fundamental to its ability to self-ionize into hydrogen and hydroxide ions.
• Aqueous solutions can be neutral, acidic, or basic, depending on the relative concentrations of \( \left[\mathrm{H}^{+}\right] \) and \( \left[\mathrm{OH}^{-}\right] \).

Ion Product Constant

The ion product constant of water, denoted as \( K_w \), is a vital concept in aqueous chemistry. It represents the product of the molar concentrations of hydrogen ions and hydroxide ions at 25°C in water. This value is always constant, hence its name. For pure water and neutral solutions, \( K_w = 1.0 \times 10^{-14} \).
The formula \( K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] \) helps us solve many problems involving pH, pOH, and the overall balance of acids and bases in a solution.
This constant is instrumental because it allows us to predict and calculate changes in concentration when acids or bases are added to a solution.

• Since \( K_w \) is constant, if you know one of the ion concentrations, you can always find the other using the formula.
• This relationship provides insight into the nature of the solution: whether it leans towards being acidic or basic.