Titration is a common laboratory method of quantitative chemical analysis used to determine the unknown concentration of a substance dissolved in a solution, known as the analyte. This technique involves the careful addition of a second solution, called the titrant, which has a precisely known concentration, until a complete chemical reaction occurs between the two substances. The calculation uses the precisely measured volume of the added titrant to determine the amount of the analyte present in the original sample. This process relies entirely on the principles of stoichiometry, which dictates the fixed ratios in which chemical substances react.
Essential Concepts for Calculation
Before beginning any calculation, it is necessary to understand the specific terms that represent the data points collected during the experiment. The substance whose concentration is being sought is the analyte, while the solution of known concentration used to react with it is the titrant. Both substances are measured using two fundamental chemical properties: concentration and volume.
Concentration is typically expressed as Molarity (\(M\)), which quantifies the amount of solute in moles per liter of solution. The Volume (\(V\)) refers to the exact quantity of the solution used for both the analyte and the titrant, measured precisely during the experiment. The measurement of volume is particularly important for the titrant, as its dispensed volume is directly used in the final calculation.
The reaction is considered complete when the Equivalence Point is reached. This is the theoretical point where the moles of the titrant added are chemically equivalent to the moles of the analyte present, based on the reaction’s stoichiometry. An indicator is often used to signal the Endpoint, which is the point where a color change occurs and closely approximates the equivalence point.
The Core Stoichiometric Formula
The mathematical calculation for titration results is founded on the principle that the total moles of titrant and analyte consumed at the equivalence point must satisfy the mole ratio defined by the balanced chemical equation. This principle relates concentration (\(C\)), volume (\(V\)), and the stoichiometric coefficient (\(n\)) for the two reacting species. Moles are calculated by multiplying Molarity by Volume (\(Moles = C \times V\)).
At the equivalence point, the moles of titrant (\(C_{titrant} \times V_{titrant}\)) and the moles of analyte (\(C_{analyte} \times V_{analyte}\)) are related by their respective stoichiometric coefficients (\(n\)) from the balanced reaction. For a general reaction, the stoichiometric relationship is expressed as: \((C_{analyte} \times V_{analyte}) / n_{analyte} = (C_{titrant} \times V_{titrant}) / n_{titrant}\). This equation ensures that the molar quantities are adjusted by the correct mole ratio, allowing for calculation of the unknown concentration. For reactions with a one-to-one (1:1) stoichiometric ratio, the \(n\) values cancel out, simplifying the formula to \(C_{analyte} V_{analyte} = C_{titrant} V_{titrant}\).
Step-by-Step Calculation Procedure
The first step in calculating the results of a titration is to establish the complete chemical context by writing and balancing the reaction equation between the titrant and the analyte. The coefficients in this balanced equation establish the mole ratio, which is the factor that links the quantity of the known substance to the unknown substance.
Next, all known experimental data must be identified and prepared for calculation. This includes the concentration (\(C\)) and the volume (\(V\)) of the titrant, and the initial volume (\(V\)) of the analyte. Since Molarity is defined in moles per liter, all volume measurements, which are typically recorded in milliliters (mL) during the experiment, must be converted to liters (L) by dividing by 1,000.
The calculation proceeds by first determining the actual number of moles of the titrant that were consumed to reach the equivalence point. This is achieved by using the titrant’s known concentration and its measured volume in liters: \(Moles_{titrant} = C_{titrant} \times V_{titrant}\).
The mole ratio from the balanced equation is then applied to convert the moles of titrant into the moles of analyte consumed in the reaction. If the mole ratio is \(n_{analyte}:n_{titrant}\), the equation becomes \(Moles_{analyte} = Moles_{titrant} \times \frac{n_{analyte}}{n_{titrant}}\). Once the moles of the analyte are calculated, the final step is to find its concentration by dividing the calculated moles by the initial volume of the analyte solution in liters: \(C_{analyte} = \frac{Moles_{analyte}}{V_{analyte}}\).
Applying the Calculation (A Worked Example)
Consider a simple acid-base titration where an unknown concentration of hydrochloric acid (\(\text{HCl}\)) is titrated with a known sodium hydroxide (\(\text{NaOH}\)) solution. The experiment begins with \(25.00 \text{ mL}\) of the \(\text{HCl}\) solution, and the titrant, \(\text{NaOH}\), has a known concentration of \(0.1050 \text{ M}\). The volume of \(\text{NaOH}\) required to reach the equivalence point is measured to be \(28.55 \text{ mL}\).
The balanced chemical equation for this reaction is \(\text{HCl} + \text{NaOH} \rightarrow \text{NaCl} + \text{H}_2\text{O}\), which shows a \(1:1\) stoichiometric ratio. The next step is to convert the volumes from milliliters to liters: \(V_{\text{HCl}} = 25.00 \text{ mL} / 1000 = 0.02500 \text{ L}\) and \(V_{\text{NaOH}} = 28.55 \text{ mL} / 1000 = 0.02855 \text{ L}\).
The moles of the known substance, the titrant (\(\text{NaOH}\)), are calculated using its Molarity and volume: \(Moles_{\text{NaOH}} = 0.1050 \text{ M} \times 0.02855 \text{ L}\), which equals \(0.002998 \text{ moles}\). Because the reaction ratio is \(1:1\), the moles of \(\text{HCl}\) must be equal to the moles of \(\text{NaOH}\) consumed, so \(Moles_{\text{HCl}} = 0.002998 \text{ moles}\).
The final concentration of the unknown \(\text{HCl}\) solution is then found by dividing the moles of \(\text{HCl}\) by its initial volume in liters: \(C_{\text{HCl}} = \frac{0.002998 \text{ moles}}{0.02500 \text{ L}}\). This calculation yields a final concentration of \(0.1199 \text{ M}\) for the unknown hydrochloric acid solution.