Chemical Thermodynamics Of Sofcs

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Chemical Thermodynamics Analysis of Fuel Cells Analysing the electrochemical reactions taking place in a SOFC at equilibrium allows determination of the electric potential between the anode and cathode gas phases directly from cell conditions and reactant properties. The expression thus obtained for the potential is identical to that obtained by considering ‘Steady Flow’ operation of a SOFC under ideal conditions. This agreement is expected as ideal SOFC operation assumes quasi-equilibrium (that is a steady flow, but one involving negligible departure from equilibrium) characteristics. The advantage of the chemical analysis is that it can readily be extended to take into account non-equilibrium characteristics – namely activation polarisation. Theory Before considering SOFC’s it is useful to clarify the theory and notation which will be used in the electrochemical analysis. Consider a generalised chemical reaction at equilibrium: aA + bB + ... ⇔ kK + lL + ...

(1)

Where, all species are assumed to be uncharged. Using the concept of chemical potential [2] this reaction can be represented as:

∑υ M i

i

=0

(2)

i

Where, the υi are the stoichiometric coefficients, positive for RHS and negative for LHS (an arbitrary convention) and Mi the mass of the reactants and products. Thermodynamic theory shows that the condition for equilibrium of this reaction is:

∑υ µ i

i

=0

(3)

i

Where, µ i is the chemical potential per mole of species i. Now, if some of the species are charged, the condition for equilibrium is changed and it can be shown to be [2]:

∑υ (µ i

i

)

+ zi FΦ = 0

(4)

i

Where, F is Faradays constant (electronic charge * Avogadros Number), zi the number of charges on the species and Φ the electric potential of the phase in which ~

the species resides. The term µ i = µ i + zi FΦ is the electro-chemical potential per mole (notation is very varied).

For perfect gases, the chemical potential takes the form:

µ (T , p ) = µ 0 (T ) + RT ln p p  o 

(5)

By convention p0=1 bar. Hence µ 0 (T ) is the chemical potential per mole of a perfect gas at a pressure of 1 bar. It is a function of temperature only. In mixtures of perfect gases, the chemical potential of each species is independent and its pressure dependency governed by its partial pressure, that is:

µ i (T , pi ) = µ io (T ) + RT ln pi p  o  µ i (T , pi ) = µ io (T ) + RT ln p p  + RT ln ( xi ) o 

(6a) (6b)

Where, p = ∑ pi is the mixture pressure; xi is the mol fraction of component i, and i

µ (T ) is the chemical potential per mol of pure component i at 1 bar pressure. o i

Application to SOFC at Equilibrium At equilibrium there is no net current flow across the electrolyte hence: 1. Mol fraction of O2 gas is constant through the cathode. 2. Concentration of O2- ions is constant through the electrolyte. 3. Mol fraction of H2 and H2O gases are constant through the anode. At the cathode-electrolyte interface the half reaction, 1 O2 + 2e − ⇔ O 2− 2

(7)

is in equilibrium. Hence:

(

) (

1 µ O + 2 µ e − − FΦ C = µ O 2− − 2 FΦ E 2 2

)

(8)

And rearranging gives: ΦC − Φ E =

1 2F

(

1 2

µ O + 2µ e − µ O 2

−

2−

)

(9)

At the anode-electrolyte interface the half reaction,

H 2 + O 2− ⇔ H 2O + 2e −

(10)

is at equilibrium. Hence:

(

)

(

µ H + µ O − 2 FΦ E = µ H O + 2 µ e − FΦ A 2

2−

2

−

)

(11)

And rearranging gives: ΦA − ΦE =

(

1 µ H 2 O + 2 µ e − − µ H 2 − µ O 2− 2F

)

(12)

The electric potential difference between the cathode and anode is, therefore: ΦC − Φ A =

(

1 µ H 2 + 12 µ O2 − µ H 2O 2F

)

(13)

Assuming all gases behave as ideal gases, each of the chemical potentials terms in equation (13) can be written in the form of equation (6b) giving:

(

)

o o o RT  xH 2 xO22 1 1 ΦC − Φ A = µ H 2 + 2 µ O2 − µ H 2O + ln 2F 2 F  xH 2O 1

 RT  p  + ln   4 F  po  

(14)

Now introducing the Gibbs Function for a chemical reaction, which is defined as the change in chemical potential from reactants to products; and taking advantage of the t

incomplete

References: [1] J.B. Young, Thermodynamics and Kinetics of Fuel Cells (Notes), Cambridge University Engineering Dept., 17/11/2001. [2] Sonntag van Wylen…  Ben Todd 2002 Last Updated: December 3, 2002