The \(\text{pH}\) scale provides a standardized measurement for the acidity or basicity of an aqueous solution. This scale is logarithmic, meaning each whole number change represents a tenfold difference in chemical activity. The goal of this process is to reverse-engineer the starting molar concentration of the acid or base dissolved in the water by using the measured \(\text{pH}\) value. This chemical and mathematical conversion depends entirely on whether the dissolved substance is classified as a strong or a weak solution.
The Core Conversion: \(\text{pH}\) to Hydrogen Ion Concentration
The first step in calculating any concentration from \(\text{pH}\) involves determining the molar concentration of hydrogen ions, represented as \([\text{H}^+]\). The relationship between these two values is defined by the equation \(\text{pH} = -\log[\text{H}^+]\).
The negative sign is included because hydrogen ion concentrations are typically very small numbers. Taking the negative logarithm converts these small exponential values into the more manageable, positive numbers commonly seen on the \(\text{pH}\) scale.
To solve for the concentration, the formula is algebraically rearranged, resulting in the expression \([\text{H}^+] = 10^{-\text{pH}}\). This calculation directly yields the concentration of hydrogen ions, measured in moles per liter (\(\text{M}\)), present in the solution.
This initial concentration value, \([\text{H}^+]\), is the foundation for all subsequent concentration calculations. It does not yet represent the starting concentration of the substance that was dissolved.
Determining Concentration for Strong Solutions
Strong acids simplify concentration calculations because they undergo complete dissociation in water. For a strong acid, every molecule dissolved releases its maximum number of hydrogen ions. Therefore, the \([\text{H}^+]\) value calculated from the \(\text{pH}\) is directly equal to the original molar concentration of the strong acid.
For strong bases, the process requires an intermediate step because the \(\text{pH}\) scale measures acidity. Strong bases release hydroxide ions, \([\text{OH}^-]\), upon complete dissociation. The relationship between \(\text{pH}\) and \(\text{pOH}\) is \(\text{pH} + \text{pOH} = 14\). Once \(\text{pOH}\) is determined, the hydroxide ion concentration is calculated using the inverse logarithmic relationship: \([\text{OH}^-] = 10^{-\text{pOH}}\).
Since the strong base dissociates completely, the resulting \([\text{OH}^-]\) value is equal to the original molar concentration of the strong base. For bases that release more than one hydroxide ion per molecule, such as \(\text{Ca}(\text{OH})_2\), the final \([\text{OH}^-]\) concentration must be divided by the number of hydroxide ions released per molecule to find the original concentration.
Determining Concentration for Weak Solutions
Weak acids and weak bases introduce a complication because they only partially ionize in water. They establish a state of chemical equilibrium between the undissociated compound and its dissociated ions. Consequently, the \([\text{H}^+]\) value calculated from the \(\text{pH}\) is only a fraction of the original concentration of the weak acid.
To link the measured \([\text{H}^+]\) to the initial concentration, the acid dissociation constant (\(\text{Ka}\)) or base dissociation constant (\(\text{Kb}\)) is necessary. The mathematical expression for a weak acid (\(\text{HA}\)) in water is \(\text{Ka} = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\), where \([\text{HA}]\) represents the concentration of the undissociated acid remaining at equilibrium.
It is assumed that the concentration of the conjugate base \([\text{A}^-]\) is equal to the concentration of the hydrogen ions \([\text{H}^+]\). The initial concentration of the weak acid is the sum of the undissociated acid \([\text{HA}]\) and the dissociated hydrogen ions \([\text{H}^+]\). Therefore, the \(\text{Ka}\) expression can be rearranged to solve for the undissociated acid concentration at equilibrium: \([\text{HA}] = \frac{[\text{H}^+][\text{A}^-]}{\text{Ka}}\).
By substituting the calculated \([\text{H}^+]\) and the known \(\text{Ka}\) into this equation, the undissociated concentration \([\text{HA}]\) is found. The original, total concentration is determined by adding \([\text{HA}]\) to the dissociated hydrogen ion concentration \([\text{H}^+]\). A similar process is followed for weak bases, using the \(\text{Kb}\) constant and the \([\text{OH}^-]\) concentration derived from the \(\text{pH}\).
Step-by-Step Calculation Examples
Strong Acid Example
To find the original concentration of a strong acid like hydrochloric acid (\(\text{HCl}\)) with a measured \(\text{pH}\) of \(2.50\), the first step is to convert the \(\text{pH}\) to the hydrogen ion concentration. \([\text{H}^+] = 10^{-2.50}\), which yields a concentration of \(0.00316 \text{ M}\). Since \(\text{HCl}\) is a strong acid, it dissociates completely.
The initial concentration of the \(\text{HCl}\) solution is therefore \(0.00316 \text{ M}\), directly equaling the calculated \([\text{H}^+]\).
Weak Acid Example
Consider a weak acid, acetic acid (\(\text{CH}_3\text{COOH}\)), with a measured \(\text{pH}\) of \(3.50\) and a known \(\text{Ka}\) of \(1.8 \times 10^{-5}\). First, the hydrogen ion concentration is calculated: \([\text{H}^+] = 10^{-3.50}\), which is \(3.16 \times 10^{-4} \text{ M}\). This value is the concentration of \(\text{H}^+\) and \(\text{A}^-\) at equilibrium.
Next, this value is used with the \(\text{Ka}\) expression to find the concentration of the undissociated acid \([\text{HA}]\). The rearranged equation is \([\text{HA}] = \frac{(3.16 \times 10^{-4})^2}{1.8 \times 10^{-5}}\), which calculates to \(5.55 \times 10^{-3} \text{ M}\). The original, total concentration is the sum of the undissociated and dissociated portions: \(5.55 \times 10^{-3} \text{ M} + 3.16 \times 10^{-4} \text{ M}\), resulting in a starting concentration of \(5.87 \times 10^{-3} \text{ M}\).