Potentiometry example:
Assume two solutions, one containing 0.01 M
potassium dichromate, 1.0 x 10-6 M chromium III
from chromium III sulfate and 0.7 M hydrogen ion
and one containing 0.012 M iron II sulfate and 1.0 x
10-6 M iron III from iron III sulfate. Answer each
question below assuming a temperature of 25 °C and
that:
A. The two solutions are contained in separate
beakers joined by a salt bridge with an inert electrode
(such as platinum) immersed in each solution.
B. The two solutions are contained (no dilution) in a
single beaker with a saturated calomel electrode and
an inert electrode (e.g., Pt) immersed in the solution.
Relevant information:
Cr2O72-/Cr3+:
Fe3+/Fe2+:
For SCE:
E° = +1.33 V
E° = +0.77 V
Eref = +0.244 V
Potentiometry example:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Sketch the experimental setup for A.
Write the shorthand notation for A.
Write and balance the expected equation.
Identify the electrodes at which oxidation and reduction are
likely to occur.
Identify the anode and cathode (part A).
Calculate the potential of each half cell.
Calculate the cell potential (i.e., the cell potential that would
be measured by a high impedance voltmeter before any
reaction has occurred).
What is the cell potential when the primary reaction reaches
equilibrium?
Calculate the equilibrium constant for the redox reaction.
Calculate the iron (II) concentration when the primary
reaction is at equilibrium.
Calculate the difference between the initial and equilibrium
cell potentials.
Comment on the relative magnitudes of changes in the iron
(II) concentration and the cell potential from initial to
equilibrium conditions.
Why do small changes in cell potential frequently represent
very large changes in concentration? (hint: use Nernst
equation)
Comment on the relevance of points 11 and 12 to the
reliability of potentials required for direct measurement
applications of potentiometry.
Why are high-impedance (resistance) voltmeters used to
measure cell potentials? Why not use a low-impedance
voltmeter?
1. Sketch of experimental set-up (Part A):
2.
Shorthand notation for A:
Pt|Fe2+(0.012), Fe3+(1x10-6)||Cr2O72-(0.01), Cr3+(1 x 10-6), H+(0.7)|Pt
3. Write and balance the equation:
half reaction method:
(1)
(2)
Cr2O72- + 14H+ + 6e- → 2Cr3+ + 7H2O
-6(Fe3+ + e- → Fe2+)
______________________________________
(3) Cr2O72- + 6Fe2+ +14H+ → 2Cr3+ + 6Fe3+ + 7H2O
(1) - (2) = (3)
Write as a reduction to match E0, add H2O to balance
O, H+ to balance H, and e- to balance total charge.
Multiply each reaction by the appropriate number to
achieve equal numbers of transferred electrons for
each reaction.
Subtract half reaction with lesser Eo from reaction
with greater Eo to get the balanced equation for the
spontaneous reaction.
4.
Identify the electrodes:
oxidation
reduction
Fe2+→Fe3+ , left electrode
Cr2O72- → 2Cr3+, right electrode
5.
Identify the anode and cathode (part A).
oxidation takes place at the anode (left electrode)
Fe2+→Fe3+, left electrode
reduction takes place at the cathode (right electrode)
Cr2O72- → 2Cr3+, right electrode
6.
Calculate the potential of each half cell.
Cr2O72-/Cr3+:
E° = +1.33 V
3+
2+
Fe /Fe :
E° = +0.77 V
These voltages apply to unit activities of reactants
and products (not this situation). we turn to Nernst
anode E = E
E = 0.77 −
°
Fe 3 + / Fe 2 +
[
[
0.0592
Fe 2 +
−
log
1
Fe3+
0.0592
0.012
log
= 0.53V
−6
1
1x10
E = 1.33 −
[
]
3+ 2
0.0592
Cr
log
2−
+ 14
6
Cr2O7 H
[
(10 )
0.0592
E = 1.33 −
log
6
(10 )(0.7 )
cathode
]
]
−6 2
−2
14
][ ]
= 1.41V
7.
Calculate the cell potential (i.e., the cell potential
that would be measured by a high impedance
voltmeter before any reaction has occurred).
Ecell = Ecathode – Eanode = Eright - Eleft
= 1.41V – 0.53 V = 0.88 V
Reaction is spontaneous as written for the initial
conditions.
8.
What is the cell potential when the primary
reaction reaches equilibrium?
o
ΔG o = −nFEcell
= − RT ln K eq
E
o
cell
RT
=
ln K eq
nF
At equilibrium, E=0.
E = Eo −
RT
ln K = 0
nF
9.
Ecell
Calculate the equilibrium constant for the redox
reaction.
[
][
]
2
6
0.059
Cr 3+ Fe3+
o
o
= ECr − EFe −
log
6
14
6
Cr2O72 − Fe 2 + H +
[
o
o
0 = ECr
− EFe
−
][
][ ]
0.059
log K at equilibrium
6
o
o
− E Fe
6( ECr
)
log K =
0.059
K = 10
o
o
6 ( ECr
)
− E Fe
0.059
= 10
6 ( 0.56 )
0.059
= 8.9 x10 56
In general,
K = 10
n ( ΔE o )
0.059
( )
16.9 n ΔE °
= 10
Therefore, small changes in ΔE° lead to very large
changes in K.
10. Calculate the iron (II) concentration when the
primary reaction is at equilibrium.
Cr2O72- + 6Fe2+ +14H+
2Cr3+ + 6Fe3+ + 7H2O
[
][
2
]
6
0.059
Cr 3 + Fe3+
o
o
Ecell = 0 = ECr − EFe −
log
6
14
6
Cr2O72 − Fe 2 + H +
where all are equilibrium concentrations
[
K = 10
o
o
6 ( ECr
)
− E Fe
0.059
= 10
6 ( 0.56 )
0.059
][
][ ]
= 8.9 x10 56
initial concentrations:
Codichromate =
CoCr3+ =
CoH+ =
CoFe2+ =
CoFe3+ =
0.01M
1.0 x 10-6 M
0.7 M
0.012 M
1.0 x 10-6 M
Note that Cr2O72- is in stoichiometric excess relative
to Fe2+ and the K value is so large that we can
conclude that virtually all Fe2+ is converted to Fe3+.
Therefore,
[Fe3+]eq ≈ 1x10-6 M + 0.012 M
[Cr3+]eq ≈ 1x10-6 M + (2/6)(0.012 M)
[Cr2O72-]eq ≈ 0.01 M – (1/6)(0.012 M)
[H+]eq ≈ 0.7 M – (14/6)(0.012 M)
≈ 0.012 M
≈ 0.004 M
≈ 0.008 M
≈ 0.672 M
(0.004 ) (0.012 )
0.059
o
o
− EFe
−
0 = ECr
log
6
(0.008) Fe 2 + 6 (0.672 )14
2
[
6
]
solving for [Fe2+]eq = 3.5x10-12 M
11. Calculate the difference between the initial and
equilibrium cell potentials.
Ecell, initial = Ecathode – Eanode = Eright - Eleft
= 1.41V – 0.53 V = 0.88 V
Ecell, equil. = 0
Δ = 0.88 V
12. Comment on the relative magnitudes of changes
in the iron (II) concentration and the cell
potential from initial to equilibrium conditions.
CoFe2+ = 0.012 M ; [Fe2+]eq = 3.5x10-12 M
> 9 order of magnitude change in concentration with
0.88 V change in voltage. (wow)
13. Why do small changes in cell potential
frequently represent very large changes in
concentration? (hint: use Nernst equation)
o
o
Ecell = ECr
− EFe
−
Codichromate =
CoCr3+ =
CoH+ =
CoFe2+ =
CoFe3+ =
[
][
3+ 2
]
3+ 6
0.059
Cr
Fe
log
6
14
6
Cr2O72 − Fe 2 + H +
[
][
][ ]
0.01M
1.0 x 10-6 M
0.7 M
0.012 M
1.0 x 10-6 M
[Fe3+]eq ≈ 1x10-6 M + 0.012 M
≈ 0.012 M
[Cr3+]eq ≈ 1x10-6 M + (2/6)(0.012 M)
≈ 0.004 M
[Cr2O72-]eq ≈ 0.01 M – (1/6)(0.012 M) ≈ 0.008 M
[H+]eq ≈ 0.7 M – (14/6)(0.012 M)
≈ 0.672 M
2+
[Fe ]eq
= 3.5x10-12 M
Near equilibrium, all species except Fe2+ are present
at relatively high concentrations. Small absolute
changes in these concentrations will give rise to small
relative concentration changes. Small absolute
changes in Fe2+, however, can give rise to large
relative changes….
Therefore,
[Cr ] [Fe ]
[Cr O ][H ]
3+ 2
3+ 6
2−
7
+ 14
2
can be regarded as a constant, C, near
equilibrium
0.059
1
log C
6
6
Fe 2 + 1
0.059
1
o
o
= ECr
− EFe
−
log C
2+ 6
6
Fe 2
o
o
Ecell ,1 = ECr
− EFe
−
[
]
Ecell , 2
[
]
Ecell ,1 − Ecell .2
[
[
]
]
[
[
]
]
2+
⎛
⎞
Fe
2 ⎟
⎜
= ΔE = −0.059 log
2+
⎜
⎟
Fe
1 ⎠
⎝
ΔE
⎛ Fe 2 + 2 ⎞
−
⎜
⎟ = 10 0.059
⎜ Fe 2 + ⎟
1 ⎠
⎝
The concentration ratio is an exponential function of the
change in potential. Therefore, small changes in potential
correspond to large changes in concentration.
e.g., if ΔE = 0.1 V, [Fe2+]2/[Fe2+]1 = 49
14. Comment on the relevance of points 11 and 12 to
the reliability of potentials required for direct
measurement applications of potentiometry.
Small voltage error → large concentration error
(However, there are very large relative changes in
relative concentrations of reactants and products near
the equivalence point in redox titration, which allows
for the identification of equivalence points using an
indicator or by monitoring voltage. See redox and
potentiometric titrations.)
15. Why are high-impedance (resistance) voltmeters
used to measure cell potentials? Why not use a
low-impedance voltmeter?
Once current can flow, the reactants will proceed
toward equilibrium thereby changing the voltage
between the electrodes.
V=IR or I = V/R, current flow is minimized with a
high resistance meter.
A low impedance voltmeter would allow for
significant current flow.
Let’s look at Part B:
B. The two solutions are contained (no dilution) in a
single beaker with a saturated calomel electrode and
an inert electrode (e.g., Pt) immersed in the solution.
Relevant information:
Cr2O72-/Cr3+:
Fe3+/Fe2+:
For SCE:
E° = +1.33 V
E° = +0.77 V
Eref = +0.244 V
Cr2O72- + 6Fe2+ +14H+
2Cr3+ + 6Fe3+ + 7H2O
The reaction quickly reaches equilibrium. What is
the potential of this cell?
We know the potential of the SCE, we need to
determine the potential at the Pt electrode where at
equilibrium, ECr and EFe are equal.
We can use the Nernst equation to calculate either ECr
or EFe. Both calculations will give the same result.
However, the [Fe2+]eq is very small (3.5x10-12 M) and
is subject to relatively large uncertainty. Therefore, it
is more convenient to use the Cr couple.
Cr2O72-/Cr3+:
E° = +1.33 V
[Cr3+]eq ≈ 1x10-6 M + (2/6)(0.012 M)
[Cr2O72-]eq ≈ 0.01 M – (1/6)(0.012 M)
[H+]eq ≈ 0.7 M – (14/6)(0.012 M)
[
≈ 0.004 M
≈ 0.008 M
≈ 0.672 M
]
2
0.0592
Cr 3+
E = 1.33 −
log
14
6
Cr2O72 − H +
[
][ ]
(0.004 )
0.0592
log
E = 1.33 −
= 1.333V
14
6
(0.008)(0.672 )
(Coincidentally very close to E°)
2
Ecell = ECr-ESCE = 1.333 – 0.244 = 1.086 V