CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Nernst Equation
Module Author: Michael D. Gross
Author Affiliation: Bucknell University
Course: Thermodynamics
Text Reference: Sandler (2006), Section 14.6; Larminie and Dicks (2003), Section 2.5
Concepts: Nernst Equation
Problem Motivation:
Fuel cells are a promising alternative energy conversion technology. Most fuel cells use
hydrogen as a fuel which reacts with oxygen to produce electricity. Different types of
fuel cells are distinguished by the nature of the electrolyte. For example, a solid oxide
fuel cell (SOFC) contains a solid oxide electrolyte that transports oxygen ions as shown
in Figure 1.
The SOFC reactions are:
Anode:
Cathode:
Overall:
H2 + O2-  H2O + 2e1/2O2 + 2e-  O2H2 + 1/2O2  H2O
For each mole of hydrogen consumed, two moles of electrons are passed through the
electric load. Faraday’s constant is commonly used to relate number of electrons and
charge ( F  96,485 coulombs/mole of electrons). The objective of a fuel cell is to deliver
power to the load:
P=E·I
Power = Voltage · Current
watt 
joule
coulomb
 volt 
s
s
Clearly the voltage of a fuel cell directly affects power output. The maximum voltage,
commonly referred to as open circuit voltage (OCV), of a fuel cell can be calculated with
the Nernst Equation. The Nernst Equation takes into account the Gibbs free energy of
reaction and the partial pressures of reactants and products. In this module, we will
explore how the OCV of a fuel cell changes under various fuel cell operating conditions.
1st Draft
2nd Draft
M.D. Gross
Page 1
April 19, 2008
January 13, 2009
Electron
Flow
(Current)
-
-
e
e
N2
H2
H2O
O2
N2
H2
H2
O2
O2-
H2O
H2
H2 H2O
O2O2-
Air
In
Anode
Gas
Chamber
Cathode
Gas
Chamber
Fuel Cell
N2
H2O
Cell Voltage
H2
In
O2
O2-
Electric Load
O2
H2 &
H2O
Out
O2
Anode
Cathode
Electrolyte
Figure 1: Reactions within SOFC
Air
Out
Figure 2: Flow Diagram for SOFC
Problem Information
The following relates Gibbs free energy, partial pressure of reactants and products, and
the Nernst Equation.
H2 + ½ O2 → H2O
 a H  aO 1 / 2 
2

G  G  RT ln 2


a
H 2O


ΔGº is the Gibbs free energy of reaction at the fuel cell operating temperature and
standard pressure, and the logarithm is a ratio of component activities. The Gibbs free
energy can easily be converted to voltage, E.

 G
nF
E is the OCV, n is the moles of electrons released for every mole of hydrogen reacted (2
mol e-/ 1 mol H2), and F is Faraday’s constant. With this relationship between ΔG and E,
an Equation describing OCV can be written.
E
1/ 2
RT  a H 2  aO2 
EE 
ln

nF  a H 2O


Eº is the OCV at fuel cell operating temperature and standard pressure (1 atm). Assuming
an ideal gas, activities can be replaced with partial pressures.
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2nd Draft
M.D. Gross
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April 19, 2008
January 13, 2009
1/ 2
RT  PH 2  PO2 
EE 
ln

nF  PH 2O


Two common ways the partial pressure of reactants and products can change in a fuel cell
are by varying fuel utilization and pressurizing the anode or cathode. Fuel utilization
indicates the percent of fuel that reacts in the fuel cell. For example, a fuel utilization of
50% indicates half of the H2 oxidized to H2O.
Example Problem Statement:
You are operating a SOFC at 800ºC with H2 as the fuel. If the anode pressure is held
constant at 1.5 atm and the O2 pressure in the cathode is held constant at 1.5 atm,
generate a plot of the open-circuit voltage, E, as a function of fuel utilization.
The OCV at operating temperature and standard pressure (1 atm) could be determined by
calculating ΔGº. For convenience, Eº is given in the table below.
Form of water product
Gas
Gas
Gas
Gas
Gas
Gas
Temperature / oC
100
200
400
600
800
1000
Eo / V
1.17
1.14
1.09
1.04
0.98
0.92
Example Problem Solution:
The OCV, E, is calculated with the Nernst Equation as follows:
1/ 2
RT  PH 2  PO2 
EE 
ln

nF  PH 2O


Eº is given in the table as 0.98V and n is 2 (2 moles of e- released per 1 mole of H2
reacted).
An example calculation for 80% fuel utilization is shown below.
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2nd Draft
M.D. Gross
Page 3
April 19, 2008
January 13, 2009
Step 1) Calculate the partial pressure of gases in the anode ( PH 2 and PH 2O ).
To calculate the partial pressures of the H2 and H2O, fuel utilization and total
pressure in the anode must be considered.
For 80% fuel utilization: PH 2 = 1.5 atm · (1-fuel util.) = 1.5 atm · (1-0.8) = 0.3
atm
PH 2O = 1.5 atm · fuel util. = 1.5 atm · 0.8 = 1.2 atm
Step 2) Calculate the partial pressure of oxygen in the cathode ( PO2 ).
In the cathode O2 is consumed and a product gas is not generated in the cathode.
Therefore, we can assume the pressure of O2 in the cathode given as 1.5 atm is
constant.
PO2 = 1.5 atm
Step 3) The partial pressures can now be inserted into the Nernst Equation to calculate the
OCV, E.
E  E 
RT  PH 2 PO12/ 2 

ln 
nF  PH 2O 
J
 1073K
 0.3(1.5)1 / 2
mol  K

 0.98V 

ln
1.2
mol e C

2
 96485
mol H 2
mol e
 0.925 V
8.314



Note: 1 V = 1 J / C (energy/charge)
Step 4) Calculate the OCV as a function of fuel utilization ranging from 0.01% to
99.99%. Follow the example provided for 80% fuel utilization.
1st Draft
M.D. Gross
April 19, 2008
2nd Draft
January 13, 2009
Page 4
1st Draft
2nd Draft
M.D. Gross
Page 5
April 19, 2008
January 13, 2009
1.5
Eo = 0.98V
PO2 = 1.5 atm
1.4
1.3
PH2 / atm
1.500
1.485
1.350
1.200
1.050
0.900
0.750
0.600
0.450
0.300
0.150
0.015
0.000
PH2O / atm
0.000
0.015
0.150
0.300
0.450
0.600
0.750
0.900
1.050
1.200
1.350
1.485
1.500
E/V
1.415
1.202
1.091
1.053
1.029
1.008
0.989
0.971
0.950
0.925
0.888
0.777
0.564
1.2
1.1
E/V
% fuel utilization
0.01
1
10
20
30
40
50
60
70
80
90
99
99.99
1.0
0.9
0.8
0.7
0.6
0.5
0
10
20
30
40
50
60
70
80
% fuel utilization
Fuel cells are typically operated above 90% fuel utilization. While the OCV is lower at
fuel utilizations greater than 90%, it is desirable to operate at these high conversions so
that fuel is not wasted.
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2nd Draft
M.D. Gross
Page 6
April 19, 2008
January 13, 2009
90
100
Home Problem Statement:
In addition to fuel utilization, the potential of a fuel cell can change by pressurizing the
cathode or the anode. This problem will provide insight into the importance of these
parameters on potential. The effect of the gas pressure on the maximum voltage of a
SOFC at 800oC with hydrogen as fuel is given by the Nernst Equation.
RT  PH 2 PO12/ 2 

E  Eo 
ln 
2 F  PH 2O 
a. Calculate E as a function of O2 pressure sent into the cathode (from 1 atm to 10
atm) for a fuel utilization of 80% and a constant anode pressure of 1.5 atm.
b. Calculate E as a function of H2 pressure sent into the anode (from 1 atm to 10
atm) for a fuel utilization of 80% and a constant cathode pressure of 1.5 atm.
c. Based upon these results, compare the effects of anode and cathode pressure on E.
The maximum voltage at operating temperature and standard pressure (1 atm) could be
determined by calculating ΔGº. For convenience, Eº is given in the table below.
Form of water product
Gas
Gas
Gas
Gas
Gas
Gas
1st Draft
2nd Draft
Temperature / oC
100
200
400
600
800
1000
M.D. Gross
Page 7
Eo / V
1.17
1.14
1.09
1.04
0.98
0.92
April 19, 2008
January 13, 2009