Acid/Base Titration Curves

Introduction:

In the Bronsted-Lowry definition, an acid is a proton (H+) donor and a base is a proton acceptor. When it donates a proton, an acid produces a base, called its conjugate base. Likewise, when a base accepts a proton, it produces an acid, called its conjugate acid.  Strong acids, such as hydrochloric acid (HCl), are 100% dissociated in water, so the reaction is shown with a single arrow.  However, weak acids, such as acetic acid (HAc), dissociate only to a small degree and so an equilibrium is established:


Strong and weak bases show analogous behavior with the except that they form OH- ions rather than H3O+ ions.

Neutralization is a process in which an acid plus a base react to yield a salt and water:

The heart of this type of reaction is the combination of the proton and hydroxide ion to form water.  Therefore every acid base neutralization reaction involves acid base pairs.

Up to this point, we have only been interested in the total amount of acid or base that is present in a given solution.  For example, to determine the amount of acetic acid in a solution, we would titrate it with a known concentration of sodium hydroxide until the phenolphthalein indicator turned pink (endpoint).  Since we know the concentration of the base and how much we have added, we can calculate the number of moles of base.  From this we can calculate the number of moles of acetic acid in the solution which gives us the concentration.

While this 'endpoint' is useful, it is only an approximation of the true equivalence point.  To determine the actual equivalence point, we must acquire a titration curve.  Instead of determining a single point, you will determine the pH at hundreds of points as the neutralization reaction proceeds.  You can then generate a titration curve, which is a plot of pH versus the amount of titrant (NaOH in this case) added.  These titration curves have several interesting characteristics as shown in the figures below::

The most interesting points on a titration curve include:  the initial pH, the pH at the equivalence point, the pH at the point halfway between the initial and the equivalence point, and the pH at the end point. 

The first of these points is the initial pH of the solution.  The initial pH is solely determined by the original concentration of acid or base present.  If it is a strong acid or base, the [H+] or [OH-] is simply the original concentration of the acid or base.  For example, if you were titrating a 0.1 M HCl solution, the initial pH would equal the negative log of 0.1, which is 1.00.  If the solution is a weak acid or base, then the Ka or Kb and the initial concentration of the acid or base will be needed to calculate the [H+] or [OH- ].

As titrant is added, it begins to neutralize the solution and the pH changes.  If the solution is a strong acid (left hand figure above), you notice that the pH does not change appreciably until most of the acid has neutralized.  Then over a very short period, the pH rapidly changes from strongly acidic to strongly basic.  On the other hand, if the solution contains a weak acid (right hand figure above), you notice that the pH begins to change immediately and the titration curve shows a characteristic 'hump'.  Like the strong acid, the pH of a weak acid changes very rapidly when you near the equivalence point.

This 'hump' that is seen in the titration of weak acids and bases is the result of a buffer being formed during the titration.  Remember that a buffer is defined as a solution containing a relatively high concentration a weak acid or base and its conjugate.  As you titrate a weak acid, you are producing its conjugate.  So technically, you are dealing with a buffer solution from the time the first drop of titrant is added until a drop before the equivalence point.  Given that we are dealing with a buffer, we can use the Henderson-Hasselbalch equation to calculate the pH within the buffer regimen:

The second interesting point in a titration curve of a weak acid or weak base occurs at the point halfway between the initial point and the equivalence point.  When half of the weak acid has been titrated, the concentration of conjugate, [A-], equals the concentration of the remaining acid, [HA].  The ratio of their concentrations is 1, and the log of 1 is zero.  Therefore the pH equals the pKa.  Likewise, when half of the weak base has been titrated, the concentration of the conjugate, [BH+], equals the concentration of the remaining base, [B].  The ratio of their concentrations is 1, the log of  1 is zero, and therefore, the pOH equals the pKb.  This is also the point of maximum buffer capacity.  Often chemists find it much easier to determine pKas and pKbs experimentally from titration curves. 

The third interesting point in a titration curve is the equivalence point.  The actual equivalence point is the point where the number of moles of base or acid added as titrant is exactly equal to the number of moles of acid or base originally present in the solution.  If you know the concentration and volume of the titrant, you can calculate the number of moles added.  This must equal the moles of acid or base in the solution.  The pH at the equivalence point can be acidic, neutral, or basic depending on the solution being titrated and the titrant.  If both the solution and the titrate are strong, then the pH at the equivalence point will be 7 (neutral).  This is because the pH is being driven by the auto ionization of water.  However, if either the solution or the titrant is weak, then the situation is more complex because the conjugate of a weak acid or base, is a strong base or acid which will undergo a hydrolysis reaction with water.

The fourth interesting point in a titration curve is the end point, which is simply the pH at the end you decide to stop the titration.  If you are titrating with a strong acid or base, then the pH is determined by the excess [H +] or [OH-].  If you are titrating with a weak acid or weak base then you will have to use the Ka or Kb to calculate the concentration of [H +] or [OH-]. 

Many of the points discussed above can be seen in the following animation:

Purpose:

In a previous experiment, you calculated the Ka of a weak acid by determining the equilibrium concentrations of the weak acid, its conjugate, and the hydronium ion.  As mentioned above, it is often more convenient for chemists to measure pKas and pKbs by titrating the weak acid or base with strong base or acid.  In this experiment, you will again be determining the Ka for a weak acid or the K b for a weak base, however this time you will be using data from a titration curve that you will acquire.  Instead of just titrating to a fixed endpoint (change in an indicator), you will be using a pH probe to measure [H+] as you titrate your samples.

One of the most important aspects of this experiment is to accurately determine the equivalence point.  Unfortunately, this is not readily apparent from the titration curve itself.  To overcome this problem, you will employ some sophisticated mathematics (calculus) to very accurately determine the equivalence point.  Once the equivalence point is determined, it is easy to find the pH at the half way to equivalence point.  This is the point of maximum buffering for a weak acid or weak base system.  It is also the point where the pH equals the pKa or the pOH equals the pKb:

Method:

With the help of computer interfaced pH probes (PASCO), you will investigate the qualitative and quantitative aspects of acid base reactions. Such reactions are in a class known as neutralization reactions. Determining the molarities and/or volumes involved in a neutralization reaction involves the technique called titration (a titer refers to a known or fixed volume).

The following setup should look very familiar to you by now.  It is the same one we have used for the last several titrations.  The big difference in this setup is that instead of using a chemical indicator to signal the endpoint (which is NOT the same as the equivalence point), you will be using a pH probe to measure the [H+] as the titration proceeds.

For example, supposed that 10 mL of 0.1 M HCl was placed in the beaker with 20 mL of water.  The pH probe is submersed into the solution, the magnetic stirrer is started and an initial pH of 1.6 is determined (0.0 mL of NaOH titrant added).  Now 1.0 mL of  0.1 M NaOH is added from the buret, and the pH remains at 1.6.  This process of adding titrant and monitoring the pH is continued throughout the neutralization reaction and the following data is collected:  

mL NaOH
pH
mL NaOH
pH
mL NaOH
pH
0.0
1.6
9.3
2.2
10.5
11.8
1.0
1.6
9.4
2.2
10.6
11.8
2.0
1.6
9.5
2.3
10.7
11.9
3.0
1.6
9.6
2.5
10.8
12.0
4.0
1.6
9.7
2.6
10.9
12.0
5.0
1.6
9.8
2.7
11.0
12.1
6.0
1.7
9.9
3.0
12.0
12.2
7.0
1.7
10.0
7.2
13.0
12.2
8.0
1.8
10.1
11.0
14.0
12.3
9.0
2.1
10.2
11.3
15.0
12.3
9.1
2.1
10.3
11.6
16.0
12.3
9.2
2.1
10.4
11.7
  17.0  12.3

Note that during the addition of the first 9.0 mL of NaOH, each addition of 1.0 mL only produces only a small change in the pH.  This is consistent with what we would expect from our earlier discussion of strong acid, strong base systems.  After 9.0 mLs however, we notice that the pH is changing more rapidly and so the rate of NaOH addition is decreased from 1.0 mL per increment to 0.1 mL.  Since the pH is changing so rapidly, we must be approaching the equivalence point.  This reduced rate of NaOH addition is maintained until the rate of pH change becomes more gradual again.  These changes can best be viewed as graphs or titration curves:


Although you can clearly see that this is a strong acid, strong base titration, it is difficult to see the exact equivalence point.  To aid in the determination of the equivalence point, it is often necessary to take the first derivative of the titration curve.  This is easily accomplished by subtracting the pH at the nth point from the pH at the (n+1) point.  For example, subtract the pH at 0.0 mL from the pH at 1.0 mL (from the preceding table) and you obtain a difference of 0.0.  This process is repeated until the following table (the 1st Derivative of pH) is obtained:
 
mL NaOH
1st Derivative of pH
mL NaOH
1st Derivative of pH
mL NaOH
1st Derivative of pH
0.0
  
9.3
0.1
10.5
0.1
1.0
0.0
9.4
0.0
10.6
0.0
2.0
0.0
9.5
0.1
10.7
0.1
3.0
0.0
9.6
0.2
10.8
0.1
4.0
0.0
9.7
0.1
10.9
0.0
5.0
0.0
9.8
0.1
11.0
0.0
6.0
0.1
9.9
0.3
12.0
0.1
7.0
0.0
10.0
4.2
13.0
0.0
8.0
0.1
10.1
3.8
14.0
0.1
9.0
0.3
10.2
0.3
15.0
0.0
9.1
0.0
10.3
0.3
16.0
0.0
9.2
0.0
10.4
0.1
17.0
 

When this 1st derivative data is plotted, it is much easier to see the equivalence point.  From the expanded plot, we can see that the equivalence point is between 10.0 mL and 10.1 mL:

For many analyses this would be sufficient, however, for even greater accuracy, the second derivative of the titration curve can be calculated.  Again, this is easily accomplished by subtracting the pH at the nth point from the pH at point (n+1).  For example, subtract the change in pH at 2.0 mL from the change in pH at 1.0 mL (in the 1st Derivative table) and you obtain a difference of 0.0.  This process is repeated until the following table (the 2nd Derivative of pH) is obtained:
 
mL NaOH
2nd Derivative of pH
mL NaOH
2nd Derivative of pH
mL NaOH
2nd Derivative of pH
0.0
 
9.3
 0.1
10.5
0.0 
1.0
 
9.4
 -0.1
10.6
 -0.1
2.0
 0.0
9.5
 0.1
10.7
 0.1
3.0
 0.0
9.6
 0.1
10.8
 0.0
4.0
 0.0
9.7
 -0.1
10.9
 -0.1
5.0
 0.0
9.8
 0.0
11.0
 0.0
6.0
 0.1
9.9
 0.2
12.0
 0.1
7.0
 -0.1
10.0
 3.9
13.0
 -0.1
8.0
 0.1
10.1
 -0.4
14.0
 0.1
9.0
 0.2
10.2
 -3.5
15.0
 
9.1
 -.03
10.3
 0.0
16.0
 
9.2
 0.0
10.4
 -0.2
17.0
 

When the 2nd derivative data is plotted, an even more accurate picture of the equivalence point emerges.  Now we see a 'ringing' where the second derivative has a very rapid rise and then a very rapid decrease passing through zero.  It is this crossing at zero that represents the true equivalence point.  From the expanded plot, we can see that the equivalence point is reached at approximately 10.09 mL of titrant:

As you can see, we have had to use fairly sophisticated mathematics to obtain an accuracy of approximately 10 microliters (10 uL or 0.01 mL).  We have had to do this because each drop from our buret represents approximately 1/20th of a mL or 50 uL.  So we must use mathematics to artificially improve our analysis.  However, there are commercial titration instruments that are capable of accurately delivering titrate in qualities less than one microliter (0.001 mL).  As with most things in life:  you can get almost anything if you are willing to spend enough time and/or money.
 

Procedure:

Titration of Known Acids and Bases:
  1. The computer software will already be setup and your instructor will show you how to acquire pH data.
  2. Your instructor will show you how to assemble your titration station.  The pH probe will be in buffer solution so be sure to rinse it off thoroughly with distilled water before submerging it into you sample.  You will want to adjust the flow rate on the syringe (buret) to be approximately 1-2 drops per second.
  3. Add 10 mL of 0.1 M HCl and 50 mL of distilled water to a 250 mL beaker.
  4. Add a magnetic stir bar and set the stirrer speed as high as possible without splashing.
  5. Rinse out your 60 mL syringe with 0.1 M NaOH and refill.
  6. Clamp your pH probe so that the tip of the plastic probe is covered by solution but is not in the way of the stir bar.
  7. Start collecting data using the Science Workstation software (1 Hz).
  8. Once the titration is over, rinse the pH probe with distilled water and store it back in its buffer.
  9. You have now completed the titration of a strong acid (HCl) with a strong base (NaOH).
  10. Repeat this procedure for the following combinations.  Remember when you switch titrants, you must first rinse the syringe with the new titrant and then refilling it before starting the titration:
  11. Remember to determine the pH as both the equivalence point and at the halfway to equivalence point for each plot before erasing the data.
Titration of Unknown Acids and Bases:
  1. Obtain an unknown solution.  Be sure to record the unknown number.
  2. Using the pH probe, determine if you have an acid or a base.
  3. Clean and fill your buret with the appropriate titrant (0.1 M HCl or 0.1 M NaOH).
  4. Collect a titration curve as you did above.
  5. After completing all of the titrations, wash your buret with 'buret rinse' and then distilled water.


Calculations:

(Updated 6/5/07 by C.R. Snelling)