The concept of pH serves as a standardized measure for the acidity or alkalinity of an aqueous solution. Mathematically, pH is derived from the concentration of hydrogen ions (\([H^+]\)) using a logarithmic transformation. Because chemical measurements are never exact, every reported value must reflect the precision of the tools used to obtain it. Accurately determining the number of significant figures in a pH value is necessary for proper scientific reporting. This ensures the calculated result does not appear more precise than the original laboratory measurement allows.
Precision in Concentration Measurements
The first step in calculating pH is determining the hydrogen ion concentration, typically expressed in molarity. The precision of this concentration is established by the laboratory equipment used during the solution’s preparation. The number of significant figures in this initial concentration measurement limits the precision of the final pH value. The concentration is calculated by dividing the mass of the solute by the volume of the solution, meaning its significant figures are limited by the least precise measurement used. For example, if the concentration is determined to be \(2.56 \times 10^{-4}\) M, it has three significant figures, and this number dictates the precision of the resulting pH.
The Logarithm Rule for Significant Figures in pH
The mathematical operation used to calculate pH is taking the negative base-10 logarithm of the hydrogen ion concentration (\(pH = -\log[H^+]\)). This operation introduces a special rule for tracking precision. The number of significant figures in the original concentration must be conserved in the calculated pH value, but they are only counted in the decimal portion of the logarithm.
The digits in a logarithm are divided into two parts: the characteristic (the integer part) and the mantissa (the decimal part). The characteristic, which appears to the left of the decimal point, indicates the power of ten in the original concentration. Because it only serves to locate the decimal point, the characteristic is not considered a significant figure. The mantissa is the only part that conveys the measurement’s precision. Therefore, the rule for pH is that the number of significant figures in the original concentration measurement equals the number of decimal places in the calculated pH value. This convention ensures that the final result accurately reflects the precision of the initial measurement.
Calculating and Reporting pH Values
This rule is applied when converting any concentration value to a “p” function, such as pOH or pKa. If a hydrogen ion concentration is \(4.5 \times 10^{-2}\) M (two significant figures), calculating the negative log gives a pH of \(1.346787…\). This must be rounded to two decimal places to match the concentration’s precision. The correctly reported pH value is \(1.35\).
A concentration of \(1.000 \times 10^{-5}\) M has four significant figures. Although the calculated pH is exactly 5, it must be reported as \(5.0000\) (four decimal places) to maintain the required precision. The integer 5 is the non-significant characteristic, while the four zeros in the mantissa are the significant digits. Conversely, a concentration of \(0.0452\) M (three significant figures) yields a calculated pH of \(1.34486…\). This must be rounded to three decimal places, resulting in a final reported pH of \(1.345\).