Question

For a given aqueous solution, if \(\left[\mathrm{H}^{+}\right]=7.92 \times 10^{-5} \mathrm{M},\) what is \(\left[\mathrm{OH}^{-}\right] ?\)

Short answer

\([\mathrm{OH}^{-}] = 1.26 \times 10^{-10} \mathrm{M}\).

Step-by-step solution

Understand the water dissociation constant

In any aqueous solution, the concentration of hydrogen ions (H⁺) and hydroxide ions (OH⁻) is related by the ion-product constant for water, denoted as Kw. At 25°C, Kw is equal to \(1.0 \times 10^{-14}\). This means \([\mathrm{H}^+][\mathrm{OH}^-] = 1.0 \times 10^{-14}\).

Rearrange the formula

To find \([\mathrm{OH}^-]\), rearrange the formula for the ion-product constant: \([\mathrm{OH}^-] = \frac{K_w}{[\mathrm{H}^+]}\). This allows us to calculate the concentration of hydroxide ions if the concentration of hydrogen ions is known.

Plug in the known values

Substitute the given value of \([\mathrm{H}^+]\) and \(K_w\) into the rearranged formula. This gives \([\mathrm{OH}^-] = \frac{1.0 \times 10^{-14}}{7.92 \times 10^{-5}}\).

Perform the calculation

Calculate \([\mathrm{OH}^-]\) using the values substituted in the previous step:\[ [\mathrm{OH}^-] = \frac{1.0 \times 10^{-14}}{7.92 \times 10^{-5}} = 1.26 \times 10^{-10} \mathrm{M} \].

Key concepts

Water Dissociation Constant

The water dissociation constant, commonly represented as Kw, is a fundamental concept in chemistry that describes the relationship between hydrogen ions (H⁺) and hydroxide ions (OH⁻) in water.
In pure water, or any aqueous solution at 25°C, water molecules ionize to release these ions. This ionization reaches a state of dynamic equilibrium, where the concentration of the ions remains constant.

• The product of the concentrations of H⁺ and OH⁻ ions is always a constant, termed as the ion-product constant for water.
• This value, denoted by Kw, is essential for understanding the behavior of acids and bases in water.

Kw can be calculated or used to determine unknown ion concentrations in various solutions. Knowing how to work with this constant is crucial for solving problems related to the acidity or basicity of a solution.

Kw at 25°C

At a standard temperature of 25°C, the ion-product constant for water, Kw, is always \[ 1.0 \times 10^{-14} \].
This value is vital because it sets the baseline for calculations involving ion concentrations in water.

• The value of Kw is slightly dependent on temperature, meaning it changes if the temperature varies, but at 25°C, it is consistently \[ 1.0 \times 10^{-14} \].
• Every time you encounter a problem related to ion concentrations in aqueous solutions at this temperature, you can be confident using this specific value.

In practical applications, using Kw allows chemists to predict how substances behave in solution. This constant becomes the key to understanding how the balance between H⁺ and OH⁻ ions shifts under different conditions.

H⁺ and OH⁻ Concentration

The concentration of H⁺ and OH⁻ ions in water is interconnected and determined by the ion-product constant Kw.
If you know the concentration of one ion, you can always calculate the concentration of the other using the equation:
\[ \text{[H}^+\text{][OH}^-\text{]} = K_w \].For example, if you know \\( [H^+] = 7.92 \times 10^{-5} \text{ M} \), \then you can find \[ [OH^-] \] using\[ [OH^-] = \frac{K_w}{[H^+]} \].

• This relationship is essential for understanding the properties of acids and bases in solutions.
• In neutral solutions, the concentration of H⁺ is equal to the concentration of OH⁻, each at \( 1.0 \times 10^{-7} \text{ M} \).

The ability to interconvert H⁺ and OH⁻ concentrations helps predict if a solution is acidic or basic, where an abundance of H⁺ indicates an acidic solution, and OH⁻ suggests a basic solution.