Page 42 - Advanced thermodynamics for engineers

2.11 GIBBS ENERGY 25

2.11.1 THE USE AND SIGNIFICANCE OF THE HELMHOLTZ AND GIBBS ENERGIES
It should be noted that the definitions of Helmholtz and Gibbs energies, Eqns (2.26) and (2.29), have
been obtained for systems of invariant composition. The more general form of these basic thermo-
dynamic relationships, in differential form, is
P
dU ¼ TdS pdV þ m dn i
i
P
dH ¼ TdS þ Vdp þ m dn i
i
(2.30)
P
dF ¼ SdT pdV þ m dn i
i
P
dG ¼ SdT þ Vdp þ m dn i
i
The additional term, P m dn i , is the product of the chemical potential of component, i, and the
i
change of the amount of substance (measured in moles) of component i. (The chemical potential of a
substance has the same numerical value as the specific Gibbs energy of the substance, and is intro-
duced in Section 12.2 when dissociation is discussed. It is used extensively in the later chapters where
it can be seen to be the driving force of chemical reactions). Obviously if the amount of substance of
the constituents does not change then this term is zero. However, if there is a reaction between the
components of a mixture then this term will be nonzero and must be taken into account.
2.11.2 HELMHOLTZ ENERGY
i. The change in Helmholtz energy is the maximum work that can be obtained from a closed
system undergoing a reversible process whilst remaining in temperature equilibrium with its
surroundings.
ii. A decrease in Helmholtz energy corresponds to an increase in entropy, hence the minimum
value of the function signifies the equilibrium condition.
iii. A decrease in entropy corresponds to an increase in F; hence the criterion dFÞ > 0 is that for
T
stability. This criterion corresponds to work being done on the system.
iv. For a constant volume system in which dW ¼ 0; dF ¼ 0.
v. For reversible processes F 1 ¼ F 2 ; for all other processes there is a decrease in Helmholtz energy.
vi. The minimum value of Helmholtz energy corresponds to the equilibrium condition.

2.11.3 GIBBS ENERGY
i. The change in Gibbs energy is the maximum useful work that can be obtained from a system
undergoing a reversible process whilst remaining in pressure and temperature equilibrium
with its surroundings;
ii. The equilibrium condition for the constraints of constant pressure and temperature can be
defined as
a. dGÞ p;T < 0 spontaneous change
b. dGÞ p;T ¼ 0 equilibrium
c. dGÞ p;T > 0 criterion of stability;
iii. The minimum value of Gibbs energy corresponds to the equilibrium condition.