Understanding the pH Scale and Formula
The measure of acidity or alkalinity in an aqueous solution is quantified by its potential of hydrogen (pH). This measurement is fundamental across various disciplines, from monitoring environmental health, such as in acid rain, to ensuring the stability of biological systems like blood, and maintaining household systems like swimming pools. Understanding how to calculate a change in pH, often denoted as \(Delta pH\), is a foundational skill for predicting chemical behavior.
The pH scale is based on the concentration of hydrogen ions, \([H^+]\), present in a solution. It is a logarithmic scale, meaning that a change of one whole pH unit represents a tenfold difference in acidity or alkalinity. For example, a solution with a pH of 3 is ten times more acidic than a solution with a pH of 4.
The scale is defined by the formula: \(pH = -log[H^+]\). The brackets around the hydrogen ion symbol indicate concentration, specifically measured in moles per liter (molarity). Because the formula uses a negative logarithm, a higher concentration of hydrogen ions results in a lower, more acidic pH value.
In any water-based solution, the concentrations of hydrogen ions (\([H^+]\)) and hydroxide ions (\([OH^-]\)) are inversely related. While pH measures the hydrogen ion concentration, a related value, pOH, measures the hydroxide ion concentration, using the formula \(pOH = -log[OH^-]\). These two measures are always linked by the simple relationship \(pH + pOH = 14\), providing a complete picture of the solution’s ionic balance.
Calculating Change in Unbuffered Solutions
Calculating the change in pH for an unbuffered solution, such as adding a strong acid or base to pure water, requires tracking the direct change in the molar concentration of hydrogen or hydroxide ions. The process begins by determining the initial hydrogen ion concentration (\([H^+]_1\)) from the starting pH, using the inverse of the logarithmic function: \([H^+]_1 = 10^{-pH_1}\). For pure water, the initial pH is 7, corresponding to an \([H^+]\) of \(1.0 times 10^{-7}\) moles per liter.
The next step involves calculating the amount of acid or base being introduced into the system in moles. Every mole of a strong acid added contributes one mole of \(H^+\) ions to the solution. This number of moles is then divided by the total volume of the final solution to find the change in molarity (\(Delta M\)).
By accounting for the initial ions already present, the final total concentration of hydrogen ions (\([H^+]_2\)) is determined. Since the initial concentration in water is extremely small, the added strong acid often dominates the final ion count. If a strong base is added, the added hydroxide ions (\(OH^-\)) react with and neutralize the existing \(H^+\) ions, often causing the calculation to shift to finding the final \([OH^-]\).
Once the final \([H^+]_2\) is established, the new pH (\(pH_2\)) is calculated using the standard formula: \(pH_2 = -log[H^+]_2\). The total change in pH is found by taking the difference: \(Delta pH = pH_2 – pH_1\). For example, adding just a small amount of strong acid to a liter of water can cause a rapid, large drop in \(Delta pH\) because water has no mechanism to resist the influx of new ions.
Calculating Change in Buffered Systems
When a strong acid or base is added to a buffered system, the calculation of the resulting pH change is fundamentally different because these solutions actively resist large fluctuations. A buffer solution is typically composed of a weak acid and its corresponding conjugate base, existing as a pair that can neutralize small amounts of added \(H^+\) or \(OH^-\) ions. This neutralizing capacity means that the addition of a strong acid or base will cause a much smaller \(Delta pH\) compared to an unbuffered system.
To calculate the new pH in a buffered solution, chemists use the Henderson-Hasselbalch equation: \(pH = pK_a + logfrac{[conjugate base]}{[weak acid]}\). The \(pK_a\) value is a constant specific to the weak acid used in the buffer and represents the pH at which the concentrations of the acid and its conjugate base are equal. This equation demonstrates that the pH of the system is dictated not by the total concentration of \(H^+\) ions, but by the ratio of the base component to the acid component.
When a strong acid is introduced, it reacts with and consumes a proportional amount of the conjugate base component, simultaneously creating more of the weak acid component. This neutralization changes the base-to-acid ratio in the logarithmic term of the Henderson-Hasselbalch equation.
The resulting calculation involves a two-step process where the moles of the strong acid or base added are used to recalculate the new molar ratio of the conjugate pair. The new ratio is then plugged back into the Henderson-Hasselbalch equation to determine the final \(pH_2\). Since the buffer components absorb the change, the resulting ratio shift is minor until the buffer’s capacity is exhausted, resulting in a significantly smaller \(Delta pH\) than in unbuffered water.