CHEMICAL THERMODYNAMICS CHAPTER ONE
10/16/2010
CHEMICAL THERMODYNAMICS.
Dedicated to the memory of Professor Odd Hassel
Professor of Physical Chemistry
University of Oslo, Norway
Nobel Laureate 1969
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CHEMICAL THERMODYNAMICS CHAPTER ONE
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CHAPTER ONE
1.
INTRODUCTION:
Converting Heat into Work.
1.1 Preamble:
Thermodynamics was once described to me as “a substantial
applied mathematical edifice that permeates all scientific fields and has
a major role to play in most fields of engineering”. Much of its early
activities were devoted to the study of the conversion of heat into work,
as exemplified by concern with the efficiency of steam engines.
From a chemist’s point of view, that edifice provides a rigorous
basis for monitoring the transfer of energy to or from macroscopic
systems that are under observation while they are undergoing
chemical or physical change.
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From the first law of thermodynamics (a statement concerning the
conservation of energy), it is possible to construct an elaborate
network of rigorous equations that link together experimentally
accessible quantities with each other and also with some inaccessible
properties that might possibly be of theoretical or practical importance.
There is a second law of thermodynamics that provides us with the
important ability of predicting the conditions under which spontaneous
physical and chemical changes can occur.
That thermodynamics is based upon these two laws means that it
is an essentially empirical science. We have, of necessity, embellished
it
from
time
to
time
with
theoretical
extensions
(molecular
thermodynamics) and adopted simplifying assumptions (such as the
ideal gas law).
Since thermodynamics is, in essence, a collection of rigorous
mathematical relationships, it calls for a precise vocabulary and
symbolism. A survey of the current literature and the numerous
textbooks dealing with chemical thermodynamics often reveals an
unfortunate
lack
of
uniformity
and
the
occasional
lack
of
understanding. We shall endeavor to be internally consistent and in
accord with the views expressed by the majority of practicing
thermodynamicists.
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We shall proceed with the task of building a basic vocabulary that
will, hopefully, be of some help in exploring many of the numerous
avenues of thermo-chemical activity.
1.2 Systems:
The term system is used over the complete range of physico –
chemical activities. We shall assume that it refers to any sample of
matter that is being subjected to investigation. On occasion, one might
wish to qualify the system by identifying its list of components.
It is helpful to partition the systems that we shall be dealing with
into three categories, based upon their composition:
a)
Pure substances.
b)
Mixtures with fixed composition (static systems).
c)
Mixtures with variable composition ( dynamic systems).
The terms static and dynamic are not commonly used but they
will provide us with a simple means of distinguishing between the two
types of multi - component systems
We must also recognize that we are likely to be dealing with
systems that exist in a single phase (homogeneous) or are partitioned
among two or more phases (heterogeneous).
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1.2.1 Isolated Systems:
Thermodynamics is concerned with the energy bookkeeping of the
physical and chemical changes that take place within a system. A
system is said to be isolated if there is no possibility for either matter
or energy to leave or to enter it. That is to say that the total mass and
the energy of an isolated sytem are both conserved.
i.e.
∆U
= 0
and ∆m = 0
where ∆U represents the change in energy and ∆m that of mass.
It should be emphasized that the fact that a system is isolated does not
rule out the possibility of chemical and physical changes occurring within it.
Whatever energy is produced on account of changes within the system has to
be absorbed by internal adjustments.
1.2.2Closed Systems:
We define a system as being closed if it is possible for energy to
enter or leave it, while there is no change in the total amount of matter
that it possesses. This has important implications for the experimental
determination of energy production. We frequently wish to consider the
case of an isolated super – system consisting of a closed sub –
system (the system of interest) and a closed monitoring sub –
system (sometimes referred to as the surroundings).
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i.e.
∆U
not = 0
and ∆m = 0
Isolated Super System A + B
Closed – Sub-system A
Closed – Sub system B
System of Interest
Monitoring
equipment
<-------- energy-------Æ
1.2.3
Open Systems:
An open system is one that is free to exchange both matter and
energy with other open systems.
i.e.
∆U
not = 0
and ∆m not = 0
Two phases, which are in contact, can be treated as being open
sub - systems. Together they may make up either a closed or an
isolated system.
The reactants and products of a chemical reaction may also be
treated as a pair of open sub – systems, despite the fact that they
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occupy the same space. Again, they may form a closed or an isolated
super - system.
Deleted: 1
Open – Sub-system A Closed – Sub
Monitor system
Deleted:
Deleted: system B¶
¶
Open - Sub system A 2
Å-------- energy-------Æ
1.2.3. States of a System:
The state of a system is defined by specifying its composition and
the conditions under which it exists. Unless we have specified the state
of a system that we are studying, any measured properties are of
relatively little value
Deleted: ¶
1.2.4 Composition:
The composition of a homogeneous mixture can be expressed in
terms of the numbers of moles of each of its components; the numbers
of moles are better suited to our needs than the numbers of grams.
Thus: nA, nB, nC etc.
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For many purposes, it is more useful to express the composition in
terms of the relative as opposed to absolute amounts of the
components, by determining the mole fractions: xA, xB, xC etc.
where the mole fraction, xA, is given by: xA = nA / nTOT
and
nTOT = nA + nB + nC etc
When an obvious solute – solvent relationship exists, we may
choose to express the composition in terms of the concentrations of the
solute species. For reasons that will become evident, we will often show
a preference for molalities (moles of solute per kilogram of solvent)
over molarities (mole of solute per liter of solution). We note, in
passing, that it is now considered to be more politically correct to replace
the term molarity by molar concentration. One is tempted to enquire as
to whether morality is also to be banned from chemistry.
1.2.5.
The Conditions:
The conditions, under which a system exists, are generally defined
by the combination of its temperature and its pressure. There are good
reasons to prefer the pressure to the volume, in general, but when
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dealing with gases, it is a fairly common practice to choose the
temperature and volume.
We should recognise that the molar volume or specific volume
(that per gram of sample) of a homogeneous system is uniquely defined
by the temperature and pressure.
Equilibria:
Conventional Thermodynamics is only concerned with the
properties of systems that are in states of equilibrium.
Provided that all of the sub – systems and each part of each
subsystem have the same uniform temperature, a state of thermal
equilibrium exists.
Provided that all of the subsystems are subject to the same
pressure, a state of mechanical equilibrium exists.
When two or more phases are in direct contact, matter may pass
freely from one to the other. At some point the quantities of each of the
components in each of the phases reach a constant level and a phase
equilibrium exists.
The reactants and products of a reaction mixture are free to
undergo chemical change. A point will be reached where the quantities
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of each of the species reach constant levels; a chemical equilibrium
then exists.
Both phase and chemical reaction equilibria are dynamic, meaning
that there is constant activity. In both instances, there are two opposing
processes occurring at identical rates.
The particles that make up a macroscopic sample are distributed
among
possible
energy
levels
and
structural
sites
(molecular
environments). The specific distribution that corresponds to a minimum
potential energy represents a microscopic equilibrium. Molecular
scale considerations are not directly addressed by thermodynamics, but
it would be unwise to disregard them entirely. When a system in
equilibrium is subjected to a stress, the equilibrium will be temporarily
destroyed; the subsequent relaxation to equilibrium may be virtually
instantaneous or take long enough to be monitored.
1.3
State Propertiess
A state property is one that possesses a unique reproducible value
for each state of the system, no matter how that state was attained. Of
particular utility is the fact that the change in a state property, resulting
from a shift from one specific state to another, is independent of the path
followed (sequence of intermediate states). It is significant that neither
heat nor work is a state property, since they are both path - dependent.
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2
INTERNAL ENERGY:
2.1
The Total (Internal) Energy of the System.
The first law of thermodynamics states that the total energy of
an isolated system is constant. Energy may be transferred between
sub – systems of an isolated super - system, but the total is fixed.
In classical thermodynamics, this total energy of a system is
referred to as its internal energy. It is usually represented by the
symbol U. In a sense, the absolute internal energy of a system is an
abstraction since there is no conceivable method of measuring all of it.
We note, for future reference, that statistical (molecular)
thermodynamics entertains the idea of an accessible total energy, E, in
which one selects a suitable energy baseline such as the combined
energies of the individual nuclei and electrons, or the ground state
isolated atomic energies.
2.2
Changes in the Internal Energy.
We may not be able to measure the absolute internal energy of a
system but there is generally no difficulty in obtaining reliable estimates
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of the changes, ∆U, that result from, or lead to, chemical and physical
transformation.
2.2.1
Closed system + monitor.
We envisage a situation where energy is free to flow between the
system of interest and a monitoring system (the thermodynamic
surroundings). The two make up an isolated super - system so that
nothing escapes the detection apparatus of the monitoring system.
The various ways in which energy can pass into or out of a system
of interest can be treated as contributing to either heat transfer or work
performed. Thus we write:
∆U = q + w
(1.1)
where q represents the heat entering the system and w
represents the work performed upon the system. (There was a period
of time during which engineers, notably those concerned with the
efficiency of heat engines, used a convention in which w was the work
done by the system)
q and w are not state properties since they are both path
dependent. That requires that we obtain ∆U by integrating incremental
changes along a prescribed path (sequence of changes in condition and
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composition). Thus we rewrite equation 1.1 in the differential form that is
appropriate for static systems.
dU =
dq + dw
(1.2)
2.2.2 Work:
I find it to be helpful to distinguish between two types of work: I
shall refer to them as device work and system configuration work.
Device Work.
We may envisage situations in which the system is linked to a
mechanical or electrical device. This kind of work performed by or upon
the system is often referred to as “useful” work.
The device may produce energy that is used to promote a
chemical or physical change within the system. Electrolysis, such as the
production of the more active metals, promoted by an external source of
electrical work, is a good example.
Alternatively, the devices may derive the energy, necessary for
them to function, from spontaneous chemical or physical change within
the system. The internal combustion engine and the flashlight battery
are obvious examples.
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We shall generally restrict our interest to cases where there are no
devices attached to the system. As chemists, we are not usually
interested in machines but we shall find it to be appropriate to venture
into the field of electrochemistry.
System Configuration Work.
Work is performed either upon or by the system, when its size and
shape undergo change. Contributions may be separated into linear,
surface and expansion work.
Linear work.
Work is required to stretch a molecule of a polymeric fibrous
material or a coiled protein. This type of work lies outside the scope of
these notes. It is not often a matter of concern but obviously could, in
certain circumstances, be of practical interest.
Surface Work,
Work needs to be performed to expand a surface area. This is
Comment [mid1]:
certainly the case when we are dealing with a liquid – air interface.
Comment [mid2R1]:
Comment [mid3]:
Comment [mid4]:
We can write the following expression for surface work:
dw(surface) = Γ dA
Deleted: gas i
(1.3)
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where dA is an incremental increase in the surface area and Γ is the
surface (interfacial) tension.
Surface work is generally of negligible magnitude, in the context of
chemical reactions. Those instances where surface work is significant,
as when dealing with emulsions or colloids, are generally treated in the
context of a separate surface - thermodynamics discipline.
Expansion Work.
Work is performed upon a system when it is compressed and by a
system when it expands. The energy expended on these volume
changes is proportional to the pressure that opposes the change in
volume. There is thus no work performed when a gas expands against a
vacuum.
We write the following equation:
dw (volume) = - popp dV
(1.4)
When there is a pressure difference between two systems that are
separated by a movable barrier (like a piston), the system with the
higher pressure expands while the other contracts. During the expansion
process, the higher pressure falls and the lower increases until such
time as the two are equal and a state of mechanical equilibrium is
attained.
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A
B
A
PA
A PA
B PB
PB
VA
VB
work
If PA > PB
VA will increase and VB will contract
PA will fall and PB will rise until PA = PB
The expansion work performed by the system of higher pressure is
said to be irreversible. That is due to the fact that the work that is
required to return the system to its initial state is greater than that which
it spontaneously performed.
This arises because the spontaneous
expansion is against pressures lower than the equilibrium value while
the return journey is against higher pressures.
Conventional
Chemical
Thermodynamics
is
limited
to
considerations of reversible expansion work. Reversibility requires that
the pressure of the system and its surroundings are always identical.
That might be accomplished in either of two ways. (a) If the pressure
within the system varies sufficiently slowly, that of the surroundings is
capable of being simultaneously adjusted to match it. (b) If the pressure
of the surroundings is constrained to be constant, such as the
atmospheric pressure or if it is maintained by some type of manostat
(pressure controller), both system and surroundings will stay at that
same constant level.
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In this equilibrated situation we no longer need to differentiate
between the two pressures. Limiting ourselves to reversible expansion
work leads us to write:
dwrev
=
- p dV
dq -
p dV
(1.5)
and thus :
dU =
(1.6)
2.2.3 Heat
We seek an analog to equation 1.5 for an incremental heat input
dq.
Equation 1.5 identifies pressure as being the potential for
expansion work. It is the temperature that serves as the potential for
heat transfer.
If two systems that are in thermal contact have different
temperatures, heat will naturally flow from the hotter to the colder. The
higher temperature will fall and the lower temperature will rise until the
two are equal and thermal equilibrium is attained.
A
PA
A
TA
A PA
B
B PB
TB
Heat (TA > TB)
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The cyclic process of spontaneous thermal equilibration and
reversion to the original temperature difference is irreversible since it
calls for a net heat transfer. This arises from the fact that an increment
dq required for a given dT increases with the temperature of the heat
sink.
Reversible heat transfer is only possible when the system of
interest and its surroundings continuously maintain thermal equilibrium.
This can occur when the transfer is essentially slow enough for the
system and surroundings to change their temperatures synchronously or
when there is adequate thermostating (temperature control).
If the temperature is the counterpart to the pressure of equation
1.5, the property that serves in the analogous role to the volume is the
entropy S:
dqrev
= T dS
(1.7)
This equation can be regarded as the classical thermodynamics
definition of entropy. It means that at constant temperature:
∆ST = qrev / T
We can now write the following fundamental equation:
dU = T dS -
p dV
(1.8)
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Equation 1.8 is the starting point for setting up the thermodynamic
network for static systems.
We see that U (for a system with constant composition) is being
expressed as a function of the two extensive variables V and S. We may
thus write:
dU(S.V) = (δU/δS)V dS
(δU/δV)S dV
+
(1.9)
Comparing equations 1.8 and 1.9 leads us to:
(δU/δS)V
(δU/δV)S
=
T
(1.10a)
= -p
(1.10b)
We can also differentiate equation 1.8 with respect to temperature
at constant volume giving:
(δU/δT)V = T(δS/δT)V = (δq/δT)V
= CV
(1.11)
where CV is called the constant volume (isochoric) heat capacity.
We may envisage some interesting scenarios. The primary system
is closed, in both instances. It and the monitoring (surroundings) system
form an isolated super - system.
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a) There are neither chemical nor physical changes occurring in
the primary system. In that case the energy gained by the primary
system is equal to that lost by the monitoring system. The volume
changes in the two subsystems are of equal magnitude and opposite
sign, as are the entropy changes.
b) A chemical change takes place in the primary system. We
need to add, to the right hand side of equation 1.8, a term accounting
for the impact on the internal energy of changing the numbers of moles
of the reactants and products
dU = T dS -
p dV + Σι (δU/δni)S.V.n(j) dni
(1.12)
The term pdV is the only kind of work being considered here. TdS,
however, includes not only entropy changes arising from heat
transfer but also from the changes in chemical identities.
By some means or other we might manage to maintain both
constant volume and constant entropy in our primary system. That leads
us to write:
dUS.V = Σι (δU/δni)S.V.n(j) dni
(1.13)
Chemical change will spontaneously take place in the direction
where:
dUS.V < 0
(1.14)
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The inequality 1.14 is one of a number of different, but
equivalent, ways of stating the second law of thermodynamics.
Since we are dealing with a closed but not an isolated system,
there is an energy output from the primary system that could be used
for device work (so – called useful work).
c)
If we isolate the primary system, we exclude any work
performance or heat transfer. There will still be the possibility of an
entropy change within the system due to the chemical reaction.
Thus:
dU = 0 = T dSrxn + Σι (δU/δni)S.V.n(j)
if
Then
Σι (δU/δni)S.V.n(j)
T dSrxn
<
>
(1.15)
0
0
Thus the second law of thermodynamics can be restated as: for a
spontaneous process, occurring in an isolated system, dS > 0. This is,
in fact, the more usual way of expressing the second law.
d) There is one further consideration, before we leave the internal
energy, and that is the case where our system is coupled to some type
of mechanical or electrical device. In the event that there is no change
in volume and the overall system is maintained at constant entropy, we
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can equate the spontaneous decrease in internal energy with the work
of the device.
dUS.V
= dwsystem = - dwdevice
(1.16)
Since dUS.V < 0 holds true for spontaneous processes, the
energy released may be used to perform useful (device) work.
The internal energy of a system serves as the potential for
spontaneous chemical and physical change when the two extensive
properties of entropy and volume are constrained to have constant
values. In a sense, neither of these constraints is natural.
The condition of constant volume is easily maintained for gaseous
samples but not for condensed phases. There is no practical means
available for restricting contraction of a liquid or solid sample. What
proved to be an extremely useful alternative was to replace the
constraint of constant volume by that of constant pressure. The potential
for spontaneous change under the combined constraints of constant
entropy and pressure is the enthalpy.
3
ENTHALPY:
The enthalpy, H, of a system is defined by the equation:
H
=
U
+
pV
(1.17)
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We can no more deal with absolute enthalpies than we could with
absolute internal energies so we again resort to working with the
changes in that state property.
∆H
∆U
=
+ ∆ (p V)
(1.18)
Again, we need to be prepared to follow incremental changes
along specific paths and write:
dH
=
dU
+ p dV
+ V dp
(1.19)
Combining this with equation 1.8 produces, for a static system:
dH
=
T dS
+
V dp
(1.20)
Thus the enthalpy is a function of the entropy and the pressure of
the system.
That provides us with the following analogs to equations 1.10a and
1.10b:
and
(δH/δS)p
=
T
(1.21a)
(δH/δp)S
=
V
(1.21b)
Further we can obtain:
dHp
and:
=
∆Hp
dqp
=
= T dSp
qp
(1.22)
(1.23)
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The majority of measurements, of the heat transfer arising from
chemical and physical changes, are carried out under the condition of
constant pressure. For that reason, terms like heats of reaction or heats
of solution usually refer to the enthalpy changes.
(δH/dT)p = T(δS/dT)p = (δq/dT)p = Cp
(1.24)
Cp is the isobaric (constant pressure) heat capacity of the
system.
For dynamic systems, we write the following equations:
dH
and
=
T dS
+
dHS.p
V dp + Σi (δH/δni)S.p.n(j) dni
= Σi (δH/δni)S.p.n(j) dni
(1.25)
(1.26)
Chemical change, under the constraints of constant entropy and
pressure will spontaneously take place in the direction for which:
dHS.p < 0
(1.27)
Thus far we have only dealt with isentropic processes. An
important alternative is the maintenance of constant temperature, which
is a far more likely condition to be encountered, or at least
approximated, both inside and outside the laboratory. The switch from
isentropic to isothermal processes calls for the introduction of two more
energy functions. These are frequently called the free energy functions
or simply free energies.
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4. THE HELMHOLTZ (FREE) ENERGY
The Hemholtz energy (A), often referred to as the Helmholtz free
energy, is defined by the equation:
A
=
U
-
TS
Hermann von Helmholtz
(1.28)
H
Before going any further, we should draw attention to a potential
conflict in symbolism. We shall adhere to the practice of using the letter
A to represent the Helmholtz energy while many Europeans use F. To
add to the confusion, F was also used in some of the earlier literature to
represent the Gibbs energy.
From equation 1.28, we can generate the following equations for
static systems:
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∆A
and
∆U
=
dA
=
-
- S dT
∆ (TS)
(1.29)
p dV
(1.30)
-
which allows us to write:
(δA/δT)V
=
-
S
(1.31)
(δA/δV)T
=
-
p
(1.32)
For dynamic systems, we can write:
dA
p dV + Σi (δA/δni)T.V.n(j) dni
= - S dT and
dAT.V
= Σi (δA/δni)T.V.n(j) dni
(1.33)
(1.34)
Chemical change, for systems with constant volume and
temperature, such as might be encountered in a homogeneous gasphase reaction, will spontaneously take place in the direction where:
dAT.V < 0
(1.35)
By far the most important pair of constraints, from a chemist’s or a
biologist’s standpoint, is that of constant temperature and pressure. In
that context, we encounter the fourth of the four energy functions; the
Gibbs energy.
THE GIBBS (FREE) ENERGY
The Gibbs energy is defined by the equation:
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G
=
U +
PV
- TS
=
H
-
TS
(1.36)
Josiah Willard Gibbs
We are most frequently interested in chemical and physical
changes that take place under the combined conditions of constant
temperature and pressure. It is under those constraints that the Gibbs
free energy becomes all - important. We may reasonably describe those
constraints as representing fixed conditions so that the changes of state
we deal with involve variations of composition only. It should be
mentioned that, in the past, some authors used the term Free Enthalpy
for the Gibbs free energy.
We can write, in general, that:
∆G
=
∆H
-
∆ (TS)
(1.37)
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And we note that, at constant temperature
:
∆ GT
∆H
=
- T∆S
(1.38)
S∆T
(1.39)
and under isentropic conditions
∆GS
dG
Further
∆H
=
=
-
- S dT +
V dp
(1.40)
implies that:
(δG/δT)p
=
-S
(1.41)
and
(δG/δp)T
=
V
(1.42)
For dynamic systems, adding the term representing the effects of
chemical change gives us:
dG
=
- S dT
+
V dp + Σi (δG/δni)T.p.n(j) dni
(1.43)
In this context, we encounter the quantity that is most frequently
implied when we use the term chemical potential:
µi.T.p = (δG/δni)T.p.n(j)
(1.44)
The chemical potential of a substance, in a mixture, is its partial
molar Gibbs free energy. For a pure substance, the chemical potential is
the same as the molar Gibbs free energy.
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An incremental change in the Gibbs free energy, under the
constraints of constant temperature and pressure, is given by:
dGT.p
= Σi µi.T.p dni
(1.45)
A spontaneous chemical or physical change under the same
constraints, takes place in the direction of:
dGT.p
<
0
(1.46)
The condition:
dGT.p
=
0
(1.47)
implies the existence of an equilibrium.
If two phases, α and β, are free to exchange matter, equilibrium
exists when:
µ(α)
=
µ(β)
(1.48)
For a chemical, or physical, change that takes place under the
constraints of constant pressure and temperature, we can write:
∆GT.p
=
∆HT.p
-
T∆ST.p
(1.49)
We recognize that, in this situation:
∆HT.p
=
qp
(1.50)
If a chemical equilibrium exists between the two open subsystems
that are involved in the change then we can write:
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∆GT.p
∆HT.p
=
=
0
T ∆ ST.p
(1.51)
(1.52)
When the subsystems are respectively the reactants and products
of a chemical reaction, the condition for equilibrium may be expressed
as either:
dGrxn =Σprod µprod.T.p dnprod - Σreact µreact.T.p dnreact = 0 (1.53)
or
Σprod µprod.T.p νprod = Σreact µreact.T.p νreact
(1.54)
where the chemical potentials, µI,, of the reactants and products have
values that are specific to the equilibrium state of the reaction mixture.
The quantities νprod and νreact represent the stoichiometric coefficients of
the various products and reactants.
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