The technique known as titration is an analytical method commonly used in chemistry laboratories for determining the quantity or concentration of a substance in a solution. In a titration, an analyte -- the substance whose quantity or concentration is to be determined -- is reacted with a carefully controlled volume of solution of accurately-known concentration called a standard solution. The standard solution (also known as the titrant) is usually added to the solution containing the analyte by means of a buret, a piece of volumetric glassware capable of accurately measuring solution volumes.
There are many types of titrations in common use in the analytical chemistry laboratory. Each type uses a different kind of chemical reaction. Examples of titration types include
The most commonly studied type is the acid-base titration. For this project we focus our attention exclusively on this type, and we develop a mathematical model that describes the relationship between the volume of added titrant (a base) and changes in concentration of the analyte (an acid). The model presented would need only slight modifications to be applicable to other titration types listed above.
In a typical acid-base titration experiment, the solution containing the analyte (an acid of unknown identity and/or concentration) is placed into a container, and the titrant (a base of accurately-known concentration) is slowly added from the buret in small increments (see Figure 1).
After the addition of each increment of base, the volume of base is carefully read from the buret (measured to the nearest hundredth of a milliliter) and the hydrogen ion (H+) concentration of the solution is measured with a pH meter. The pH meter gives the hydrogen ion concentration in terms of pH, which is simply the negative logarithm (common log, i.e., base 10) of the hydrogen ion concentration:
where [H+] represents the concentration of H+.
The hydrogen ion concentration is directly related to the amount of acid present in the solution at any particular step in the titration according to the following chemical reaction:
The measurement of the hydrogen ion concentration (or pH) at each point in the titration allows us to find the location of the equivalence point, that volume of base which reacts completely with the unknown concentration of acid. It is at this equivalence point that the amount of base added is chemically equal to the amount of acid present. By chemically equal we mean that the number of molecules of base added is just enough to completely react with all of the molecules of acid originally present -- so that all of the acid molecules are, in a sense, used up. By knowing the volume of base at the equivalence point, as well as the concentration of the base, one can calculate (among other things) the initial concentration of the acid.
Figure 2.This is a typical titration graph plotting volume of titrant added versus pH of solution. Notice that only a small change in pH occurs as most of the titrant is added, then near the end of the titration there is a sudden, rapid change in pH. The sudden change occurs at the equivalence point. The location of the equivalence point is, therefore, determined by identifying the point of fastest increase.
The chemical reaction that occurs between the acid and the base allows one to calculate the initial concentration (or amount) of the acid. For example, in the titration of hydrochloric acid (HCl) with a base such as sodium hydroxide (NaOH), the chemical reaction between these two species would have to be known. The reaction is as follows:
This reaction states that 1 molecule of HCl will react with 1 molecule of NaOH to produce 1 molecule of the salt, sodium chloride (NaCl), and 1 molecule of water. The chemical equation allows us to calculate the concentration of a solution of HCl by titration with the base NaOH (where the concentration of NaOH is accurately known).
Let's suppose that our solution is 0.02500 L of an unknown concentration of the acid, HCl. We wish to find its concentration by titration with 0.1000 M NaOH. (M is the notation for the concentration unit called molarity, which is defined as the number of moles of a substance per liter of solution. A mole is equal to 6.022 x 1023 molecules.) By doing the titration and making a plot of the volume of NaOH added versus the resulting pH of the solution, we find that the equivalence point occurs at 0.04398 L of NaOH. (This is the point where the plot appears to increase most rapidly.) Recall that at the equivalence point the amount of base added is chemically equal to the amount of acid present in the solution. According to the chemical reaction (1) between HCl and NaOH, 1 molecule of HCl will react with exactly 1 molecule of NaOH. Therefore, the number of moles of base needed to react with all of the acid present is the same as the number of moles of acid present in the solution. So, at equivalence point,
We can calculate the number of moles of NaOH at the equivalence point by multiplying its concentration (in terms of molarity or moles/L) by the volume of NaOH added in order to reach the equivalence point:
This tells us that there were 0.004398 moles of HCl in our original 0.02500 L solution. Finally, we can calculate the unknown concentration of HCl:
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