Buffers and the Henderson-Hasselbalch Equation:
Ten Examples

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As a reminder, here is the Henderson-Hasselbalch Equation:

  [base]
pH = pKa + log –––––
  [acid]

Be aware that this:

pH = pKa + log [base / acid]

is often the way you see it written on the Internet, for example, in the chemistry section of Yahoo Answers. A Q&A forum like YA lacks the ability to make a more typeset-appearing HH Equation.

Note: the bonus problem at the end of the file involves having to calculate how much of one of the buffer components is consumed and how much of the other is produced. There will be an unknown in the log portion of the Henderson-Hasselbalch. The third set of problems (#21 to 30) is devoted to this type.


Example #1: A buffer is prepared containing 1.00 M acetic acid and 1.00 M sodium acetate. What is its pH?

Solution:

1) To solve the above example, we must know the pKa of acetic acid. Often, the problem will provide the pKa. If the problem provides the Ka, you must convert it to the pKa (see below).

Comment: be aware, your teacher may create a test question where you must look up the Ka. Even in this era of fairly easy Internet access, try one of the appendices of your textbook. Tables of Ka values are also widespread on the Internet..

The Ka of acetic acid is 1.77 x 10¯5

2) Calculate the pKa:

pKa = -log Ka = -log 1.77 x 10¯5 = 4.752

3) Next, we simply insert the appropriate values into the HH equation:

  1.00
pH = 4.752 + log –––––
  1.00

Since the log of 1 is zero, we have pH = 4.752

Comment: 1.8 x 10¯5 is a commonly-seen value for the Ka of acetic acid.


Notice that the pH is greater than the solution of just the pure acid (4.752 as compared to 2.376). This is due to LeChatelier's Principle. Consider the dissociation equation for acetic acid:
HAc ⇌ H+ + Ac¯

Increasing the concentration of the acetate (Ac¯) will push the equiibrium back to the left, decreasing the concentration of H+. This makes the solution less acidic, making the pH of the buffer larger than the pure acid solution.


Example #2: A buffer is prepared containing 0.800 molar acetic acid and 1.00 molar sodium acetate. What is its pH?

Solution:

The Henderson-Hasselbalch Equation:

  1.00
pH = 4.752 + log –––––
  0.800

x = 4.752 + 0.097 = 4.849

Note how decreasing the amount of acid makes the buffer pH become more basic (compare to example #1).


Example #3: A buffer is prepared containing 1.00 molar acetic acid and 0.800 molar sodium acetate. What is its pH?

Solution:

The Henderson-Hasselbalch Equation (done in the Internet way):

pH = pKa + log [base / acid]

x = 4.752 + log (0.800 / 1.00)

x = 4.752 - 0.097 = 4.655

Note how decreasing the amount of base makes the buffer pH become more acidic (compare to example #1).


Example #4: (a) Calculate the pH of a 0.500 L buffer solution composed of 0.700 M formic acid (HCOOH, Ka = 1.77 x 10¯4) and 0.500 M sodium formate (HCOONa). (b) Calculate the pH after adding 50.0 mL of a 1.00 M NaOH solution.

Solution to (a):

We can use the given molarities in the Henderson-Hasselbalch Equation:

pH = pKa + log [base / acid]

pH = 3.752 + log [0.5 / 0.7]

pH = 3.752 + (-0.146)

pH = 3.606

Solution to (b):

1) We need to determine the moles of formic acid and sodium formate after the NaOH was added. We first calculate the amounts before the addition of the NaOH:

HCOOH ---> (0.700 mol/L) (0.500 L) = 0.350 mol
HCOONa ---> (0.500 mol/L) (0.500 L) = 0.250 mol

2) Now, determine the moles of NaOH:

NaOH ---> (1.00 mol/L) (0.0500 L) = 0.0500 mol

3) NaOH reacts in a 1:1 molar ratio with HCOOH:

HCOOH ---> 0.350 mol minus 0.0500 mol = 0.300 mol
HCOONa ---> 0.250 mol + 0.0500 mol = 0.300 mol

4) Calculate the new pH:

pH = 3.77 + log [0.300 / 0.300]

pH = 3.77 + log 1

pH = 3.77

Part (b) of the above question is a popular one to ask. Be sure to know how to calculate the pH of a buffer after some strong acid or base has been added. Note: one could add some weak acid or some weak base to a buffer. for example, adding ammonia (a weak base) to a buffer consisting on acetic acid and sodium acetate. The calculations for that type of situation are more complex and will not be addressed by the ChemTeam.


Example #5: 0.1 mole of CH3NH2 (Kb = 5 x 10¯4) is mixed with 0.08 mole of HCl and diluted to one liter. What will be the H+ concentration?

Solution:

1) The reaction is this:

CH3NH2 + H+ ---> CH3NH3+

2) When 0.1 mole of CH3NH2 and 0.08 mole of HCl react, this is what remains after the reaction:

0.02 mol CH3NH2
0.08 mol CH3NH3+

3) Since we now have a buffer, we will use the Henderson-Hasselbalch Equation:

First, however, we will use the Kb to get the pKa.

pKa + pKb = 14

pKb = -log 5 x 10¯4 = 3.30103

pKa = 14 - 3.30103 = 10.69897

4) Now, use the H-H Equation:

pH = 10.69897 + log (0.02 / 0.08) <--- since it's in 1 liter, these are the molarities

pH = 10.69897 + (-0.60206)

pH = 10.09691

5) Now, we antilog the pH to get the H+ conc

[H+] = 10¯pH = 10¯10.09691 = 8 x 10¯11 M

Example #6: Calculate the pH when 25.0 mL of 0.200 M acetic acid is mixed with 35.0 mL of 0.100 M NaOH.

Solution:

1) Determine moles of each substance:

(0.200 mol/L) (0.0250 L) = 0.00500 mol of acetic acid
(0.100 mol/L) (0.0350 L) = 0.00350 mol of NaOH

2) Acetic acid and NaOH react in a 1:1 molar ratio. Determine the moles remaining after reaction (acetic acid is in excess):

acetic acid ---> 0.00500 mol - 0.00350 mol = 0.00150 mol

The acetic acid that reacts with the NaOH produces sodium acetate. In the solution will be 0.00350 mol of acetate anion (we may ignore the sodium ion. It plays no role in the pH.)

3) Use the Henderson-Hasselbalch equation to determine the pH:

pH = 4.752 + log (0.00350 / 0.00150)

pH = 4.752 + 0.368

pH = 5.120

Note that we did not have a buffer to begin with. There was a solution of acetic acid and some strong base was added resulting in a solution of a weak acid and its salt. In other words, a buffer.


Example #7: Calculate the pH when 50.0 mL of 0.180 M NH3 is mixed with 5.00 mL of 0.360 M HBr. (The Kb of ammonia is 1.77 x 10-5.)

Solution:

1) Determine moles of each substance:

(0.180 mol/L) (0.0500 L) = 0.00900 mol of ammonia
(0.360 mol/L) (0.0050 L) = 0.00180 mol of HBr

2) Ammonia and HBr react in a 1:1 molar ratio. Determine the moles remaining after reaction (ammonia is in excess)

ammonia ---> 0.00900 mol - 0.00180 mol = 0.00720 mol

The ammonia that reacts with the HBr produces ammonium ion (we may ignore the bromide. It plays no role in the pH.) In the solution will be 0.00180 mole of ammonium ion.

3) Use the Henderson-Hasselbalch equation to determine the pH:

pH = 9.248 + log (0.00720 / 0.00180)

pH = 9.248 + 0.602

pH = 9.850

Notice that I used the pKa of the acetate ion in the Henderson-Hasselbalch equation.


Example #8: Determine the pH of a solution prepared by dissolving 0.35 mole of ammonium chloride in 1.0 L of 0.25 M aqueous ammonia. Kb for ammonia equals 1.77 x 10-5

Solution:

1) This is a buffer solution, with a weak base (the ammonia) and the salt of the weak base (the ammonium chloride) in solution at the same time. We must use the Henderson-Hasselbalch equation to solve this problem.

pH = pKa + log [base / acid]

2) We know the two concentrations:

pH = pKa + log [0.25 / 0.35]

3) Notice that the pKa is involved and we have been provided a Kb. What we need is the Ka (and then the pKa) for the ammonium ion. Here is the procedure for that:

Kw = KaKb

1.00 x 10-14 = (Ka) (1.77 x 10-5)

Ka = 5.65 x 10-10

pKa = - log 5.65 x 1010 = 9.248

4) Now, we can finish:

pH = 9.248 + log [0.25 / 0.35]

pH = 9.248 + (-0.146) = 9.10

Comment: Kw = KaKb is an important equation to know. It is a favorite trick of teachers to ask you something that requires either the Ka or the Kb, but only give you the other value. They want to know if you know to use Kw = KaKb to get the value you need.

The same thing was done in example #5, except there I used pKa + pKb = pKw.


Example #9: You have 0.500 liter of an acetic acid buffer (0.800 M total) at maximum buffering capacity. To it, you add 0.100 mole of HCl. What is the new pH?

Solution:

Note: since the problem is silent about volume change, we assume no volume change.

1) Moles present in solution:

moles HAc ---> 0.400 mol
moles acetate ---> 0.400 mol

Maximum buffering capacity occurs when the acid and its conjugate base are in a 1:1 molar ratio

2) The added HCl (being an acid) will react with the base (the acetate). It will do so in a 1:1 molar ratio. New amounts:

moles HAc ---> 0.400 + 0.100 = 0.500
moles acetate ---> 0.400 - 0.100 = 0.300

3) Use the Henderson-Hasselbalch:

pH = 4.752 + log (0.300 / 0.500)

pH = 4.752 + (-0.222)

pH = 4.530


Example #10: You have 0.500 liter of an acetic acid buffer (0.800 M total) at maximum buffering capacity. To it, you add 0.100 mole of salt. What is the new pH?

Solution: Note: this one is a bit sneaky because adding the salt does not change the amount of the acid. Also, we shall assume that salt means the salt of the weak acid (the acetate, say as sodium acetate). If we assumed salt meant NaCl, then this because a fairly trivial problem.

1) Starting amounts:

HAc ---> 0.400 mol
salt ---> 0.400 mol

2) Ending amounts:

HAc ---> 0.400 mol
salt ---> 0.400 + 0.100 = 0.500 mol

3) Use the Henderson-Hasselbalch:

pH = 4.752 + log (0.500 / 0.400)

pH = 4.752 + 0.097

pH = 4.849

This pH becomes more basic because we added a base (the salt) to the solution. In the first one, adding an acid caused the pH of the buffer to become more acidic. Notice that I did not bother to change moles to molarities. This step is not required since the volume is the same for the acid in solution as well as the base.


Bonus Problem: You need to prepare an acetate buffer of pH 6.420 from a 0.664 M acetic acid solution and a 2.50 M KOH solution. If you have 975 mL of the acetic acid solution, how many milliliters of the KOH solution do you need to add to make a buffer of pH 6.42? The pKa of acetic acid is 4.752.

Solution:

1) Moles of acetic acid:

(0.664 mol/L) (0.975 L) = 0.6474 mol

2) Henderson-Hasselbalch Equation:

pH = pKa + log [base / acid]

6.420 = 4.752 + log [x / (0.6474 - x)]

The x is the moles of acetate that must be present and the 0.6474 - x is the amount of acetic acid.

3) Algebra!

log [x / (0.6474 - x)] = 1.668

[x / (0.6474 - x)] = 46.5586

x = 30.142 - 46.5586x

47.5586x = 30.142

x = 0.63379 mol

Acetate and KOH are in a 1:1 stoichiometric ratio, so this is the required number of moles of KOH.

4) Volume of KOH needed:

0.63379 mol / 2.50 mol/L = 0.253516 L

254 mL seems like a reasonable answer

5) We can try our calculated values and see what happens:

pH = 4.752 + log (0.63379 / 0.01361) = 6.420

Buffer Problems 1-10     Buffer Problems 11-20     Buffer Problems 21-30     Buffer Problems 31-40
Intro. to the Henderson-Hasselbalch Equation     Return to the Acid Base menu