Thermodynamics and Chemistry

CHAPTER 11 REACTIONS AND OTHER CHEMICAL PROCESSES

11.9 EFFECTS OFTEMPERATURE ANDPRESSURE ONEQUILIBRIUMPOSITION 356

11.9 Effects of Temperature and Pressure on Equilibrium Position

The advancementof a chemical reaction in a closed system describes the changes in the
amounts of the reactants and products from specified initial values of these amounts. We
have seen that if the system is maintained at constant temperature and pressure,changes
spontaneously in the direction that decreases the Gibbs energy. The change continues until
the system reaches a state of reaction equilibrium at the minimum ofG. The value of
the advancement in this equilibrium state will be denotedeq, as shown in Fig.11.15on
page 345. The value ofeqdepends in general on the values ofT andp. Thus when we
change the temperature or pressure of a closed system that is at equilibrium,equsually
changes also and the reaction spontaneouslyshiftsto a new equilibrium position.
To investigate this effect, we write the total differential ofGwithT,p, andas inde-
pendent variables
dGDSdTCVdpCÅrGd (11.9.1)

and obtain the reciprocity relations

@ÅrG
@T



p;

D



@S

@



T;p



@ÅrG

@p



T;

D



@V

@



T;p

(11.9.2)

We recognize the partial derivative on the right side of each of these relations as a molar
differential reaction quantity:

@ÅrG
@T



p;

DÅrS



@ÅrG

@p



T;

DÅrV (11.9.3)

We use these expressions for two of the coefficients in an expression for the total differential
ofÅrG:

dÅrGDÅrSdTCÅrVdpC



@ÅrG

@



T;p

d (11.9.4)

(closed system)

SinceÅrGis the partial derivative ofGwith respect toat constantTandp, the coefficient
.@ÅrG=@/T;pis the partialsecondderivative ofGwith respect to:

@ÅrG
@



T;p

D



@^2 G

@^2



T;p

(11.9.5)

We know that at a fixedTandp, a plot ofGversushas a slope at each point equal toÅrG
and a minimum at the position of reaction equilibrium whereiseq. At the minimum of
the plotted curve, the slopeÅrGis zero and the second derivative is positive (see Fig.11.15
on page 345 ). By settingÅrGequal to zero in the general relationÅrGDÅrHTÅrS,
we obtain the equationÅrSDÅrH=T which is valid only at reaction equilibrium where
equalseq. Making this substitution in Eq.11.9.4, and setting dÅrGequal to zero and d
equal to deq, we obtain

0 D

ÅrH

T

dTCÅrVdpC



@^2 G

@^2



T;p

deq (11.9.6)

(closed system)