5.2 Thermodynamic equilibrium
A state of equilibrium is denned as the state to which a
thermodynamic system tends under given external conditions. If the external
conditions do not change, the state of equilibrium is retained in a system any
length of time. We must distinguish between the state of stable equilibrium,
the state of unstable (or labile) equilibrium, and the state of metastable
equilibrium.
The state of stable equilibrium (the stable state) is
characterized by that if the system considered is brought out of the state of
stable equilibrium by some external disturbance, then when this disturbance is
removed, the system will return into the initial state of equilibrium by
itself.
If a system is in a state of unstable equilibrium (the labile state),
the system will not return into the initial state and will pass into a new
state, the state of stable equilibrium.
Even the slightest external disturbance is sufficient to put a system
out of the state of labile (unstable) equilibrium.
Finally, the state of relatively stable equilibrium (the metastable
state) is defined as the state in which a system remains for a long period
of time, and any slight disturbance causing the system to deviate from the
metastable state does not result in the system passing into another state; as
soon as the external disturbance is removed, the system will return into the
initial metastable state. On the other hand, a sufficiently strong disturbance
will put the system out of the metastable state, and the system will pass into
a new state of stable equilibrium. Thus, the metastable state is intermediate
in relation to the stable and labile states.
Let us consider now in greater detail the state of stable equilibrium
for pure substances (single-component systems).
Since, as we already know, all spontaneous processes developing in a
system are directed toward the state of equilibrium, it is very important to
determine for each individual case the conditions for the state of equilibrium,
so as to determine the direction of any possible spontaneous process, which is
of paramount importance in analyzing various chemical reactions.
In Chapter 3 an important criterion for the equilibrium of an isolated
thermodynamic system was defined. It will be recalled that by an isolated
system we mean a system which exchanges neither heat nor mechanical work with
the surroundings. Hence, for such a system the internal energy and volume are
constant, i.e. U = const and V = const.
As was revealed in Chapter 3, in accordance with the second law of thermodynamics
the entropy of an isolated system tends to a maximum, and in the state of
equilibrium the entropy of an isolated system is the maximum possible for the
given system, i.e. for an equilibrium isolated system dS = 0. In fact,
for an isolated system dU = 0 and dV = 0, and from Eq. (3.157),
i.e.
(where
the inequality sign corresponds to irreversible processes, and the equality
sign to reversible processes) we obtain:
This is the condition of equilibrium for an isolated system. Here the
inequality sign corresponds to a non-equilibrium state, and the equality sign
indicates an already attained equilibrium state. Thus, for an isolated system
in equilibrium,
the
last relationship shows that in a state of equilibrium the value of entropy is
the maximum possible.
But if a system can interact with the surroundings, the conditions for
equilibrium will differ from those given in Eq. (3.149). The equilibrium will
depend on the conditions governing the interaction between the system and the
surroundings.
Four types of interaction conditions (sometimes referred to as
conjugation conditions) for a system tending to equilibrium with the surroundings
are of the greatest importance.
(1)
The volume of the system remains constant,
but there may be heat transfer between the system and the surroundings, with
the system's entropy remaining constant:
(2)
There may he heat and mechanical work
transfer between the system and the surroundings, provided that the system's
pressure and entropy remain constant:
(3)
The system's volume remains constant, but
there may be heat transfer between
the system and the surroundings, with the temperature remaining constant:
(4)
There may be heat and mechanical work
transfer between the system and the surroundings, provided that the system's
pressure and temperature remain constant:
We shall find the criterion of equilibrium for each of these four
possible cases of interaction between a system and its surroundings.
(1)
The interaction conditions are V =
const and S = const, i.e. dV =
0 and dS = 0. From relationship (3.157), expressed as
it
follows that the criterion of equilibrium for a system with V = const
and S = const is
in
other words, when a system approaches a state of equilibrium, its internal
energy diminishes, reaching its minimum value in the state of equilibrium.
Thus, in a state of equilibrium,
(2)
The interaction conditions are: p = const and S = const, i.e. dp =
0 and dS = 0. Inasmuch as
it is
clear that
and,
consequently, Eq. (3.157) reduces to
from
which it follows that the criterion of equilibrium for a system with p =
const and S = const is
that
is, as a system approaches a state of equilibrium, its enthalpy diminishes,
reaching a minimum in the state of equilibrium. Thus, in a state of equilibrium
(3) Interaction
conditions: V = const and T = const, i.e. dV = 0 and dT
= 0. Inasmuch as
Eq. (3.157) takes the following form:
It is clear that
The thermodynamic function (U
- TS) is called the Helmholtz free energy, or the isochoric-isothermal
potential. Denote it
Thus,
It follows from Eq. (5.11) that the equilibrium criterion for a system
with V = const and T = const (an isochoric-isothermal system) is
In other words, when a system approaches a state of equilibrium, its
isochoric-isothermal potential diminishes, reaching its minimum value in the
state of equilibrium.
Thus, in a state of equilibrium,
(4)
The interaction conditions are: p =
const and T = const, i.e. dp = 0 and dT = 0. Combining Eqs. (5.3)
and (5.8), we get:
The thermodynamic function (I -
TS) is called the Gibbs
free energy, or the isobaric-isothermal
potential. Denote it
Thus,
It follows from Eq. (5.16) that the equilibrium criterion for a system
with p = const and T =
const (an isobaric-isothermal system) is
In other words, when a system approaches a state of equilibrium, its
isobaric-isothermal potential diminishes, reaching its minimum value in the
state of equilibrium. Thus, in a state of equilibrium
These conditions
of equilibrium were derived from the assumption that the only kind of work done
by the system while it interacts with the surroundings is the work of
expansion. If we consider a case when, in addition to the work of expansion, a
system is capable of performing other kinds of work (for instance, work in a
magnetic, electric, or gravitational field, work expended in increasing a
surface, etc.), and make use of Eq. (3.153),
where
with dL' being the differential of
all kinds of work except expansion work, we shall obtain somewhat different
equilibrium criteria for the four types of interaction conditions considered
above:
(1) V = const, S = const:
and in a state of
equilibrium
(2) p = const,
S = const:
and in a state of
equilibrium
(3) V = const, T = const:
and in a state of
equilibrium
(4) p = const, T = const:
and in a state of
equilibrium
The quantities F and Φ are formed from the functions of
state H, U and S, which implies that these quantities
are also functions of state.
The internal energy U, enthalpy
H, free energy F and the
isobaric-isothermal potential Φ, which define the conditions of
equilibrium for a thermodynamic system under various conditions of interaction
with the surroundings, are called the characteristic junctions. Besides
of being the criteria of equilibrium for thermodynamic systems, the characteristic
functions possess one more important property: if one of the characteristic
functions, expressed in terms of its proper variables, is known, we can
calculate any thermodynamic quantity.
So, if internal energy U is represented as a function of volume V
and entropy S, it is
easy to show that the other basic thermodynamic quantities are determined as
and
Knowing U, V, S,
p and T, we
can easily calculate the enthalpy H,
free energy F, the isobaric-isothermal potential Φ and other
quantities. If enthalpy H is known as
a function of pressure p and entropy S,
and
If H, p, S, T and V are
known, it is easy to determine the internal energy U, free energy F, the
isobaric-isothermal potential Φ, etc. Making use of the expression
relating free energy F with volume V and temperature T, we
find:
and
Having available the values of F, V, T, p and S, we can determine the internal
energy U, enthalpy H, the isobaric-isothermal potential Φ
and other thermodynamic quantities.
Finally, if the dependence of the isobaric-isothermal potential Φ
on pressure p and temperature T is known,
and
Knowing the values of Φ, p,
T, V and S, it is
easy to calculate the internal energy U, enthalpy H, free
energy F and other quantities.
It can easily be assured that entropy does not possess the property of a
characteristic function.
The quantities F and Φ have not been given the name of
potentials by chance. Not only the quantities F and Φ, but also the
internal energy and enthalpy are usually referred to as thermodynamic
potentials, and for the following reason.
As was already mentioned, if in addition to the work of expansion a
system is capable of performing other kinds of work, the total work done by the
system is the sum of the work of expansion and of the other kinds of work.
It follows from Eqs. (5.21), (5.23), (5.25) and (5.27) that the work U
that can be performed by a system under given conditions of conjugation with
the surroundings is equal to the decrease in a corresponding thermodynamic
function. In consequence, the characteristic functions are called potentials by
analogy with mechanics.
Of all the thermodynamic potentials the free energy F can be
distinguished as being important not only as an isochoric-isothermal potential
but also as an isothermal potential. Consider a system, that obeys only one
condition T = const.
From the relationship
for an isothermal
system we obtain
and
since
consequently,
Thus, the total work (including the work of expansion) which can be done
by a thermodynamic system undergoing a reversible process at T = const
is equal to the decrease in free energy.
Let us say a few words about the origin of the term "free
energy". The expression for the internal energy of a system can be written
as
We already know that in an isochoric-isothermal system work can be
performed only at the expense of the free energy F. Thus, in such a
system it is not all of the internal energy that can be converted into work but
only its "free" fraction F. As concerns the product TS, often
referred to as bound energy, it cannot be converted into work.
Based on similar grounds, the isobaric-isothermal potential Φ is
sometimes called the free enthalpy; in an isobaric-isothermal system work can
be done only at the expense of Φ, which is but a fraction of the system's
enthalpy:\
It follows from Eqs. (5.15), (5.10) and (2.39) that the quantities Φ
and F are interrelated as follows:
Completing the study of the thermodynamic potentials, let us also become
acquainted with the concept of the chemical potential. The chemical
potential φ of a
substance is the name given to the specific (per unit mass) isobaric-isothermal
potential:
where h
and s are the specific enthalpy
and entropy. But why does the specific isobaric-isothermal potential, a
quantity that appears similar to the specific free energy, specific enthalpy
and specific internal energy, occupies such a special position? To give an
answer to this question, let us examine how the characteristic functions of a
system change in relation to the amount of substance in the system.
Considering so far the equilibrium criteria for various thermodynamic
systems, the amount of substance in a system, G, was assumed to be
constant. However, in solving a number of problems (for instance, in analyzing
the conditions for phase equilibrium) it is helpful to find the change in the
system's potential resulting from the removal of a certain quantity of
substance dG, or upon
the addition of some quantity of substance dG, to the system. In other
words, it is necessary to find the quantities
In illustration, consider an isobaric-isothermal system (V = const, T = const).
Since the characteristic functions are additive quantities, the free
energy of this system is
where f is the specific free energy,
From Eq. (5.44) we obtain:
It also follows from Eq. (5.45) that
in a
particular system capable of performing only work of expansion
Consequently, Eq. (5.46) reduces to
It is clear that
and
since by virtue of the additivity of volume,
we have
Substituting this quantity into Eq. (5.49), we obtain:
or, which
is the same,
Inasmuch as in the isochoric-isothermal system considered V = const
and T = const, i.e. dV = 0 and dT = 0, we have
It follows from Eq. (5.42) that
Hence, Eq. (5.55) yields
This result is somewhat surprising. In fact, we found that with a change
in the quantity of substance in an isochoric-isothermal system, the free energy
F of the system as a whole changes in proportion to the specific
isobaric-isothermal potential φ and not in proportion to f, as would seem to be more natural.
It will be emphasized that the partial derivative present in Eq. (5.57)
was taken at V = const,
i.e. on the condition that the volume of the system is constant. But if the
derivative of F with respect to G is taken at v = const,
i.e. on the condition that the specific volume of the system is constant, the
derivative will be different. In fact, if v = const, then, inasmuch as
it is
clear that
Substituting this value into Eq. (5.54) we find that for the system for
which v = const
and T = const,
from
which it follows that
The difference between the values of the derivatives expressed by (5.57)
and (5.60) is traced to the fact that the derivative (5.57) is calculated
assuming V = const,
i.e. for the case when the amount of substance in the system changes but its
volume remains constant; as regards the derivative (5.60), the condition v =
const holds for the case when the system's volume changes in proportion to the
change of substance in the system.
Making similar transformations for systems with p = const and T
= const, p = const and S = const, or V = const and S
= const, we can make
sure that for each of the enumerated cases the derivative of the corresponding
characteristic function with respect to the quantity of substance in the system
equals φ, i.e.
Thus, the specific isobaric-isothermal potential φ possesses a
striking property: it permits calculation of the change in a characteristic
function of any system following a change in the quantity of substance in the
system. It is precisely because of this that the quantity φ was given the
name of chemical potential. As it will be seen below, the chemical potential is
of great importance in analyzing processes that involve changes of phase when a
substance changes one phase into another.
The concept of the chemical potential proves to be of paramount
importance in chemical thermodynamics, when studying chemical reactions.
In conclusion, one more detail must be noted. As is known, the quantity
characterizing the state of equilibrium of an isolated system (V = const, U = const) is entropy, which, as
was already mentioned above, is not a characteristic function. It is, however,
interesting to note that the derivative with respect to G of the entropy
of a system is also related to the chemical potential φ; it is easy to
show that