Electrochemical oxygen gas sensors
For a stoichiometric solid the equilibrium pressure of X2(p2)is a monotonic function of temperature.
2.2. Effect of temperature The variation of EMF with sensor temperature will now be derived in terms of standard thermodynamic functions. This analysis is based upon a previous discussion by Heyne (1 978). The reader is referred to any standard text on physical chemistry
for the background, e.g. Moore (1972). The EMF of an electrochemical cell is given by
nFE =  AG.
(6)
AG is the free energy change for the passage of n faradays of charge through the cell. The GibbsHelmoltz relation may be written
d AG/dT=AS
(7)
where A S is the consequent change in the entropy of the system. Thus
dE/dT=

ASjnF.
(8)
This indicates that the temperature coefficient of the EMF is proportional to the entropy change. Where a gaseous reference is employed (assuming ideal behaviour)
and
A S = R ln(pl/p2)
(9)
dE/dT= (R/nF) In(pl /pi),
(10)
which could, of course, have been derived directly from equation (4). Equation (IO) reveals that the sensor has a temperature coefficient close to 0.2/n mV K  I per decade ratio between the sample and reference gas pressures. In an oxygen sensor using
the
commonly
employed
air
reference
(p2= 2 1 kPa),
temperature coefficients
of
approximately
0.5 and
1.O mV K
I
are anticipated at lo' and lo'' in the sample gas respectively.
Pa partial pressure of oxygen
Consider now the following cell incorporating the solid reference M,X2, that dissociates according to equation (5)
M, M.yX2., X, Z ( P ~ )PtlsElPt, X?(PI).
( 1 1)
It is readily shown by summing the contributions of the two cells
M. M,yX2y,X2(p2),Pt/sE;Pt,X2(10' Pa)
(12)
and
X2(10' Pa). Pt sEIPt, X2(pl). that the temperature differential of cell (1 1) is given by
(13)
dE/dT=

   S E l n F i " Y

l o  S M r x 2 ,  S i 2  R I n p i Y
where S orepresents the standard entropy of the species shown as subscript. The relative magnitudes of the terms in equation
(14)
for a number of solid oxides (Kubaschewski and Alcock
1979)
are shown in table 1. ( ? C / ~ )and~ (l/y)S~,~r are of
similar magnitude and tend to cancel. The term S& is large (205 J K' mol') and amounts to approximately 0.5 mV K': it is this term that results in the inevitably high temperature co efficient of cells containing a solid reference (Maskell and Steele 1986). The final term in equation (14), R I n p , , is the same as that found in the cell with a gaseous reference (equation 10) and contributes 0.05 mV KI per decade ratio between p : and 10' Pa pressure. The signs of S t 2 and R Inp, are the same (forp, < 10' Pa) and hence the contributions are additive.
This analysis has shown that the temperature coefficient of a cell with a gaseous reference is not large and can, if practicable,
Bi203
6.1
0.46
coo
 10.9
0.4 1
Cr203
 5.4
0.47
cui0
 12.8
0.39
Fe0
 15.1
0.37
IrO,
5.0
0.48
NiO
 3.9
0.49
os02
4.6
0.48
PbO
 0.6
0.52
PdO
0.7
0.52
Reo2
(2.5)
(0.56)
Ru02
 0.9
0.52
Ta20j
 5.7
0.47
ZnO
 1.0
0.52
Table 1. Thermodynamic data for some solid oxides at 298 K.
AS,+
AS2/4F'
(cal K  ' mol')§
(mV)
M Oxide , O 2
,,
be minimised by choosing p2 to be similar to p I: the coefficient
of a cell with solid by 0.5 mV KI
latter
type require
with
a gaseous
reference, on the other hand, is always higher (for p , < 10' Pa). This means that cells of the more precise temperature control than those
reference
for
comparable
precision
of
measurement.
2.3.
Response time
2.3.1.
General principles. The EMF of a potentiometric cell is
given by
E =(R TInF) 1^{r"P1 }
t,,, d In p.
(3)
.'P2
If t o , varies through the electrolyte then the observed EMF will depend upon the profile oft,,, against z where z is the distance from one interface. This effect has been considered in detail by Heyne and den Engelsen (1977): they showed that in general the effect was important with regard to the longterm drift of the EMF but had little influence on the shortterm response time. Also, by careful preparation of the solid electrolyte. in particular eliminating as far as possible second phases with ionic transport numbers different from the main bulk phase, drift can be minimised.
Variations in ionic transport number within the ceramic arise as follows. Electronic defects are generated at high and intermediate gas pressures in an oxygenion conductor by the following process (shown in KrogerVink notation)
Clearly, the hole (h') concentration is related to the oxygen partial pressure. A high hole concentration results in a significant electronic transport number so that t,,, is depressed below unity.
An important point arises from the above paper with regard to rate of response. Consider a system initially at equilibrium: it has an established t,,, against z profile. Let the sample gas pressure be changed rapidly to a different value. Equation (3) indicates that the EMF of the cell should change instantaneously to a new value because E is only governed by p I and p 2 and by the t,,, against z profile. The profile will change only slowly (for
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