The acidity or basicity of any water-based solution is measured using the pH scale. This scale is a straightforward way to express the concentration of hydrogen ions (H+) in a solution on a compressed, logarithmic scale. Calculating the concentration of hydroxide ions (OH-) from a known pH value requires a two-step mathematical conversion. This process translates the measure of acidity (pH) into the quantity of the opposing hydroxide ion (OH-).
Calculating pOH from the pH Value
Before determining the hydroxide ion concentration, the pH value must first be converted into the pOH value, which is the potential of hydroxide. Both pH and pOH are logarithmically related scales that together describe the full range of a solution’s acid-base properties. The value of pOH is a direct measure of the solution’s alkalinity, just as pH measures its acidity.
The relationship between these two values in any aqueous solution is fixed by the ion product constant of water, \(K_w\). At a standard temperature of 25°C, \(K_w\) is \(1.0 \times 10^{-14}\). This constant establishes that the product of the hydrogen ion concentration and the hydroxide ion concentration is always \(1.0 \times 10^{-14}\) in aqueous solutions at this temperature.
Because pH and pOH are derived using the negative logarithm of these concentrations, the constant \(1.0 \times 10^{-14}\) converts to a simple integer relationship. Applying the negative logarithm to the \(K_w\) constant results in the value \(14.00\), which is known as p\(K_w\). This means the sum of the pH and the pOH of any aqueous solution must equal \(14.00\) at 25°C.
Therefore, the first mathematical step to find the OH- concentration is to subtract the known pH value from \(14.00\), yielding the pOH value. Expressed as a formula, this relationship is pOH = \(14.00\) – pH. This initial calculation provides the necessary logarithmic measure that is directly tied to the concentration of the hydroxide ions.
Converting pOH to Hydroxide Ion Concentration
Once the pOH value is known, the final step involves converting this logarithmic measure back into a true concentration, expressed in Molarity (M). The pOH is defined as the negative logarithm of the hydroxide ion concentration, represented by the notation [OH-].
To reverse the negative logarithm function used to create the pOH scale, the anti-logarithm must be applied. This mathematical operation is represented by the formula [OH-] = \(10^{-\text{pOH}}\). The anti-log function, often found as \(10^x\) on a calculator, uses the calculated pOH value as a negative exponent to the base of ten.
Performing this operation results in a value that represents the molar concentration of hydroxide ions in the solution. This final concentration is a physical quantity, indicating the number of moles of hydroxide ions present per liter of solution.
Applying the Calculation Steps with an Example
Considering a hypothetical scenario where a solution has a measured pH of \(8.5\), the steps outlined above can be followed precisely to determine the hydroxide ion concentration. The pH of \(8.5\) indicates a slightly basic solution, as it is above the neutral point of \(7.0\). This value is the starting point for the two-stage conversion.
The first step is to calculate the pOH by subtracting the pH from the constant \(14.00\). This calculation is pOH = \(14.00 – 8.5\), which yields a pOH value of \(5.5\). This pOH value of \(5.5\) is the logarithmic measure of the solution’s alkalinity.
The second and final step is to apply the anti-logarithm function to this pOH value to find the molar concentration [OH-]. The calculation becomes [OH-] = \(10^{-5.5}\). Inputting this into a calculator provides the resulting concentration, which is approximately \(3.16 \times 10^{-6}\) M.
The resulting concentration is \(3.16 \times 10^{-6}\) M. This confirms that a single pH measurement is sufficient to determine the concentration of both hydrogen and hydroxide ions in an aqueous solution at 25°C.