Maths for chemists
Tips for teaching maths skills, by MaNUS MonroE
Noelle Menesini, Nicole Litzie, Julieann Murella, Jan Elepano, Brook Sell, Rebecca Spector and Manus Monroe
In this issue: Estimating pH of ammonium solutions containing weak Brønsted-Lowry bases.
Teaching approaches
NH4+(aq)
NH3 (aq) + H+(aq)
(1)
CH3CO2−(aq) + H2O(l)
HCH3CO2 (aq) + OH–(aq)
(2)
H+(aq) + OH–(aq)
H2O(l)
(3)
NH3 (aq) + HCH3CO2 (aq) + H2O(l)
(4)
NH4
(aq) + CH3CO2 (aq) + H2O(l)
+
−
Three step technique
ShUTTERSTOCK
In secondary education and first year university students’
chemistry textbooks, the chapters regarding calculations
of pH for either a Brønsted-Lowry weak acid or weak
base solution are taught as separate topics.1–3
Students may be asked to calculate the pH of a 0.1 M
ammonium chloride solution and in another problem
calculate the pH of a 0.1 M sodium benzoate solution. In
one textbook2 there is a limited discussion, without
calculations, about the pH of a salt solution containing a
Brønsted-Lowry weak acid and weak base in which the
Ka and Kb are equivalent, eg ammonium acetate.
In essence, the pH is 7 because the acid and base
characteristics of the reactants are equal to each other, for
the conjugate acid and base Ka = Kb , and the production
of hydroxonium and hydroxide ions is equivalent, making
water via a neutralisation reaction (Equations 1–4).
Estimating pH
This technique is fairly accurate, for estimating the pH of
a solution where Ka and Kb are unequal. Ammonium salts
were selected due to their high solubility. Transition metal
ions with Brønsted-Lowry bases are insoluble to slightly
soluble and there is the issue of production of transition
metal hydroxide precipitates.
Ammonium oxalate is used as an example in this
discussion; the Ka is 5.6x10-10 M while the Kb is
6.5 x10-11 M.
Looking at Equations 5–7, we can point out to students
that calculating a final pH of a solution containing a
Brønsted-Lowry acid and base with unequal equilibrium
constants is virtually impossible. This is because there is a
disproportionate production of hydroxonium or
hydroxide ions that will react to produce water, leave an
excess unknown amount of either H+(aq) or OH–(aq) and, in
turn, shift interdependent equilibriums.
This three step technique for estimating pH is
empirically derived and requires the following
constraints:
1. The pKb of the anion is restricted to 8 ±2 when using
ammonium ions.
2. The anion can neither have any acidic hydrogen atoms
nor be a zwitterion.
Our research showed that pH readings for ammonium
salt solutions with concentrations less than 0.4 M varied
with the slightest temperature change from 25°C or with
minimal exposure to atmospheric gases, namely carbon
dioxide. We found that 0.4 M solutions at room
temperature had constant pH readings after a long
exposure to atmospheric gases and complied with
readings of 0.1 M solutions at 25°C and having a nitrogen
blanket. Our final research was conduction at 0.4 M,
a good concentration for student experimentation.
Practical notes
NH4+(aq) + C2O22-(aq) + H2O(l)
NH3 (aq) + HC2O2-(aq) + H+(aq) + OH–(aq)
(5)
where, either H+(aq) + OH–(aq)
H2O(l) + H+(aq)
(6)
or
H+
(aq)
+ OH–
(aq)
(excess)
H2O(l) + OH–(aq)
(excess)
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0511EiC - MathsChem.indd 156
(7)
1. Distilled water was degassed using helium.
A nitrogen blanket was used before making any solution
under 0.4 M.
2. When making ammonium salts, a mixture of
ammonium chloride and the sodium salt of a weak
acid was used.
3. When a sodium salt was not available, the weak acid
was stoichiometrically titrated with aqueous sodium
hydroxide.
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06/09/2011 13:22:58
Sample calculation using 0.4 M ammonium
oxalate, (NH4)2C2O4
The estimated pH of the solution is 6.69 while the measured
value is 6.44. Based upon the data in Table 2, the uncertainty
for estimated pH has been determined to be ±0.3 pH units.
Students would report their answer as 6.6 ±0.3. Table 3 lists
truncated data for estimated and measured pH values.
Data are truncated due to the impact of rounding logs.
Step one
Calculate a pH value for ammonium ions by computing
the acidity of a weak acid solution – assume ammonia’s
equilibrium is unaffected by the presence of oxalate ions:
NH4+(aq)
Table 1 – Calculations for adjusting pH and pOH values.
H+(aq) + NH3 (aq)
Ka = [H+][NH3]
[NH4+]
5.6x10-10 M =
Estimated
pH
4.67
Estimated
pOH
5.29
Sum
9.96
Difference
from 14.00
4.04
2.02
2.02
---
---
6.69
7.31
14.00
0
Half the
difference
Sum of
values
X2
0.800 M
X = [H+] = 2.1x10-5 M
pH = 4.67
Table 2 – Ammonium salts and pH values. The 0.4 M solutions are at 25°C
Or, using a general equation with logs, we may write:
Ka =
X2
[acidic cation]
X2 = Ka [acidic cation]
log X2 = log Ka + log [acidic cation]
log X = log Ka + log [acidic cation] = log[H+]
2
pH = pKa + (-log[acidic cation])
2
(8)
Ammonium salts
pKa of
anion4,5
Calculated pKb
of anion
Estimated pH
Measured pH
±Δ pH
Acetate
4.756
9.244
7.00
6.98
0.02
Benzoate
4.204
9.796
6.87
6.62
0.25
Citrate
4.76
9.24
6.92
7.22
0.30
Malonate
5.7
8.3
7.1
7.1
0
1-Naphthyacetate
4.236
9.764
6.88
7.00
0.12
Oxalate
3.81
10.19
6.69
6.44
0.25
Sulfite
7.2
6.8
7.5
7.8
0.3
DL-Tartrate
4.37
9.63
6.83
6.84
0.01
P-Toluate
4.37
9.63
6.83
7.08
0.25
Table 3 – Truncated pH values from Table 2.
and returning to our example,
pH = 9.25 + (-log 0.800 M)
2
pH = 4.67
Step two
Calculate a pOH value for oxalate ions
C2O22–(aq) + H2O(l)
HC2O2–(aq) + OH–(aq)
by using Equation 9, which is a modification of
Equation 8,
pOH = pKb + (-log[basic anion])
2
References
(9)
we may now write:
pOH = 10.19 + (-log 0.400 M) = 5.29
2
Step three
Add the pH and pOH values and compare the sum to
that of pKw at 25°C.
The sum, in this example, does not add to 14.00 and the
difference (14.00-9.96) is 4.04. In this technique, add one
half of the difference to the estimated pH value and the
other half to that of pOH – see Table 1.
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0511EiC - MathsChem.indd 157
1. J E Brady and J R Holum,
1st ed, Chemistry: the study
of matter and its changes,
p694–706. New York, US:
John Wiley & Sons, 1993
2. J C Kotz and P M
Treichel, Chemistry and
Chemical Reactivity, 5th ed,
p693–720. California, US:
Thomson Brooks/Cole,
2003
3. N J Tro, Chemistry: A
Molecular Approach, 1st ed,
p664–707. New Jersey, US:
Pearson Education, 2008
4. D R Lide (ed), Handbook
of Chemistry and Physics,
89th ed, p8.42–8.51 &
8.40–8.41. New York, US:
CRC Press, 2008–2009
5. J G Speight, Lange’s
Handbook, 16th ed,
p2.620–2.699. New York,
US: McGraw Hill, 2005
Ammonium salts
Truncated estimated
pH values
Truncated measured
pH
Acetate
7.0
6.9
Benzoate
6.8
6.6
Citrate
6.9
7.2
Malonate
7.1
7.1
1-Naphthyacetate
6.8
7.0
Oxalate
6.6
6.4
Sulfite
7.5
7.8
DL-Tartrate
6.8
6.8
P-Toluate
6.8
7.1
Conclusion
The application of this simple technique is limited in
application in a manner reminiscent to that of the Ideal
Gas Law. Nevertheless, this approach to pH calculations is
an introduction for students to the quite complex nature
of equilibrium between Brønsted-Lowry acids and bases
and also allows for a discussion into an area not taught in
secondary education and first year university chemistry.
Manus Monroe is a lecturer in chemistry at Sonoma State
University, California, US.
September 2011 | Education in Chemistry | 157
06/09/2011 13:24:22