Download Gibbs Free Energy and the Chemical Potential

Robert Gennis - University of Ill Urbana
Chapter 2: Gibbs Free Energy and the Chemical Potential
2.1 Free Energy Functions and Maximal Work Capacity
2.11 The special case of an isolated system.
2.12 The special case of a system held at constant temperature: introducing
Helmholz Free Energy.
2.13 The special case of a system held at constant temperature and pressure:
Gibbs Free Energy.
2.14 Gibbs Free Energy and the condition for a spontaneous change of a
system at constant T and P.
2.2 Thermodynamic Driving Forces: Temperature, Pressure and Chemical
Potential
2.3 The chemical potential of a pure substance is the molar Gibbs Free Energy.
2.31 The chemical potential of a component in a mixture is the partial molar
Gibbs Free Energy.
2.4 The chemical potential is dependent on log(concentration) of a substance.
2.5 The change in Gibbs Free Energy upon mixing two ideal gases.
2.6 The chemical potential of a substance in aqueous solution.
2.61 Selection of the standard state and the use of molarity units of
concentration
2.7 Gibbs Free Energy of formation
2.8 Summary.
Box 2.1: Deviations from ideality
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Chapter 2: Gibbs Free Energy and the Chemical Potential
2.1 Free Energy Functions and Maximal Work Capacity
We now know that the criterion for any spontaneous process in an isolated system
is that the entropy will increase. However, biological and chemical processes don’’t take
place in isolated systems and, in any event we don’’t know how to directly measure
entropy. Whereas classical thermodynamics was concerned with determining the
maximal work one could get out of a heat engine doing mechanical or PV work, we are
mostly concerned with the work required to synthesize proteins, make ATP, transport
ions or molecules across membranes, etc. In this section we will derive other state
functions that can be applied to these problems to determine the direction and driving
force for spontaneous changes as well as define the equilibrium conditions.
As we saw in Chapter 1, the thermodynamic definition of the change in entropy
of a system is related to the amount of heat needed to change the state of the system in a
reversible process. For a process that involves mechanical work, the maximal work is
obtained from a reversible pathway, occurring in small equilibrated steps. The reversible
pathway maximizes the work done on the environment by the system, and minimizes the
work we need to do on the system to change its state. Note that the heat change (dqrev)
refers to the change in heat of the system, and does not include the surroundings.
dS sys
dqrev
Tsys
(2.1)
2
From this, it follows that for any irreversible pathway between the two states, G qsys will
be less than TdS sys , i.e., G qsys will have a smaller negative value than dqrev for heat
removed from the system or a larger positive value for heat added to the system (Figure
2.1).
TdS sys t G qsys
(2.2)
where the equality applies only to a reversible process. For clarity, the subscript ““sys”” is
added to remind us that we are referring to heat added to or removed from the system.
Equation (2.2) is called the Clausius inequality.
Figure 2.1: Schematic showing heat and work increasing or decreasing the internal
energy of a system as it undergoes a transition between State 1 and State 2. In this
Figure, the energy change difference between the two states is considered to be
small, so differential notation is used (įq, įw, etc). The reversible process gets the
most work out of the transition where energy is decreased (State 1 to State 2)and
takes the least amount of work when energy is increased (State 2 to State 1). In
going from State 2 to State 1, the increase in energy of the system will be
spontaneous (irreversible) only when the work done on the system is greater than
the minimally necessary reversible work.
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We are using G qsys to specify a differential change in heat that is not necessarily
reversible and, hence, not necessarily a state function. For a change of state in which
heat is removed from the system, G qsys is negative and the maximum value (i.e., the
smallest negative value) is TdS sys .
Equation (2.2) is actually a restatement of the second law of thermodynamics (see
Section 1.18). To see this, consider that the surroundings consists of a ““heat reservoir””
which exchanges heat with the system, and a ““work reservoir”” on which the system can
either do work or which can do work on the system (Figure 2.2).
Figure 2.2: Schematic of a system which can exchange heat and work with the
surroundings, which we divide into a heat reservoir, equivalent to a thermal bath,
and a work reservoir, which can be a device which can either add to the energy of
the system be doing work on it, or can have work done on it by the system, which
will decrease the internal energy of the system. Differential notation is used to
indicate small amounts of heat and work.
For any given change of state of the system, there may be many combinations of work
and heat exchanges that are possible, as illustrated in Figure 1.24, for example. For any
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pathway, the heat added to the system is simply the negative of the heat removed from
the surroundings
G qsurr
G qsys
(2.3)
The heat reservoir can only change its internal energy by heat transfer. For a particular
pathway, once the change in the internal energy of the heat reservoir is defined for any
particular process, the amount of heat transferred is also defined, and will be the same
regardless of whether the process is reversible or irreversible. Hence, the entropy change
of the heat reservoir can be calculated using the value of G qsurr , regardless of whether the
process is reversible or irreversible. This is the same reasoning used in discussing the
heat lost to the room as a glass of hot water cools (see Section 1.18)
dS surr
G qsurr
Tsurr
(2.4)
When calculating the entropy change of the system, however, change in the internal
energy is determined by the sum of the work plus heat, and the entropy change must be
calculated using the heat for a reversible pathway only. Combining equations (2.3), (2.4)
and (2.2), we can now write
dS sys dS surr t 0
(2.5)
which is the criterion for any spontaneous process. This demonstrates that equation (2.2)
is equivalent to the statement that the total change in entropy increases for any
spontaneous process and is zero for a reversible process.
2.11 The special case of an isolated system.
Since G qsys
dU sys G wsys , we can substitute into equation (2.2) to yield
dU sys TdS sys d G wsys
(2.6)
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For the special case of an isolated system (Figure 2.3 B), dU sys
G wsys
G qsys
0 , so we
conclude that for any spontaneous process in an isolated system TdS sys t 0 , as we saw in
Chapter 1. For any spontaneous process in an isolated system, the entropy of the system
must increase.
2.12 The special case of a system held at constant temperature: introducing
Helmholz Free Energy.
Now let us consider another special case, in which the system is not isolated but
which can equilibrate with the surroundings by an exchange of heat and work in a way
that temperature of the system is held constant. This is pictured schematically in (Figure
2.3C) which shows that the system can gain or lose internal energy either though the
transfer of heat in a ““heat reservoir””, or by PV-work (expansion or compression), or by
nonPV-work (e.g., electrical, chemistry). In this case, we specify that the temperature of
the heat reservoir is fixed (Tres), as with a large thermal bath, and this fixes the
temperature of the system.
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Figure 2.3: Schematic showing three special cases we are considering. In
Panel B is the isolated system, with no energy exchange with the environment. In
Panel C is a system that can exchange energy in the form of work and heat with the
surroundings, with the constraint that the temperature of the system is maintained
at a constant value, determined by the temperature of the heat reservoir. In this
case, the maximal work capacity is given by the change in Helmholz Free Energy. In
Panel A is a system which can also exchange energy as both work and heat, but in
this case, equilibration with the ““PV work reservoir””, which can simply mean it is
open to the atmosphere maintains constant pressure. Temperature is also
maintained constant through the equilibration with the thermal bath. In this case,
the change of Gibbs Free Energy is equal to the maximal nonPV work capacity of
the system as the state of the system changes.
For convenience, we now define a new state function, U sys TS sys . We are now
considering the internal energy and entropy of the system itself, not the surroundings.
d (U sys TS sys )
dU sys TdS sys S sys dT
(2.7)
d (U sys TS sys )
dU sys TdS sys when dT
0
This new state function is called the Helmholz Free Energy, A
A U TS
(2.8)
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Where we have dropped the ““sys”” subscript but and it is assumed that the state functions
refer only to the system of interest and do not include the surroundings. The meaning of
this new function follows from equation (2.6).
dA d (U TS ) (dU TdS ) d G w (=G w PV +G w nonPV )at constant temperature (2.9)
The change in the Helmholz Free Energy (ǻA) between two states of a system at constant
temperature is the limiting value of the amount of work that can be obtained by any
pathway. The equal sign in equation (2.9) applies only for the reversible process. If the
internal energy of the system decreases, the work done by the system is negative, and its
magnitude is limited by the value of the Helmholtz Free Energy (Figure 2.4). For any
spontaneous process in which a system evolves from one state to another at constant
temperature, a portion of the decrease in the internal energy (TdS) cannot be used to do
work because of the requirement that the total entropy must increase.
Figure 2.4: Schematic energy diagram for the transition of a system from State 1 to
State 2. The panel on the left specifies that the temperature is held constant by
equilibration with a thermal bath or heat reservoir (see Figure 2.2). The change in
the Helmholz Free Energy (ǻA) is the limiting amount of work that can be obtained.
On the left is another system that is held at constant temperature and constant
pressure (see Figure 2.2). In this case the change in the Gibbs Free Energy is the
limiting amount of work, other than PV work than can be done by the system. Since
ǻG is the limiting amount of nonPV work, it is always the case that
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('G wnonPV ) d 0 . If the process occurs under conditions were no nonPV work is
actually accomplished, then 'G d 0 .
The Helmholz Free Energy is also called the ““work function””. For any process at
constant temperature,
'A d w
or
(2.10)
dA d G w
2.13 The special case of a system held at constant temperature and pressure: Gibbs
Free Energy.
Biological processes generally occur at constant temperature and, furthermore, are
open to the atmosphere. Any work that is done against the atmospheric pressure by a
change in volume is PV work, expressed as ––PǻV, where the negative sign signifies work
done on the surroundings, reducing the internal energy of the system. In an open system,
PV work is wasted and not of interest to us. By subtracting the PV work from the work
function (ǻA), we are left with an expression of the maximal amount of work that can be
obtained, other than PV work against the atmosphere. This is called the Gibbs Free
Energy (ǻG). We can rewrite equation (2.6) and explicitly divide the work into a
component done by a change in volume against the atmosphere (G wPV
PdV ) and all
work other than PV work (G wnonPV ) . This is pictorially shown in Figure 2.3. In
biological systems it is precisely this nonPV work that is of interest. This will include,
for example, the making and breaking of chemical bonds or transporting material across a
membrane.
Recall that equation (2.6) is equivalent to the second law of thermodynamics and
is a statement of the criterion that the total entropy (system plus surroundings) must
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increase for any spontaneous process, as we showed in deriving equation (2.5).
Explicitly dividing work into PV and nonPV components, we now have
dU TdS sys d G wPV G wnonPV
PdV G wnonPV
dU TdS sys PdV d G wnonPV
(2.11)
(2.12)
For convenience, we define another state function, the Gibbs Free Energy,
U sys TS sys PVsys
Gsys
(2.13)
from which it follows that the following is true for any small change in the Gibbs Free
Energy.
dGsys
dU sys TdS sys S sys dT PdVsys Vsys dP
At constant temperature and pressure, dT
0 and dP
0, this simplifies to
dU TdS sys PdV
dG
(2.14)
(2.15)
where the subscript ““sys”” has been dropped and it is assumed that all the state functions
refer to the system and do not include the surroundings. From the definition of enthalpy
in equation (1.26) , we can now write
dG
dH TdS sys
(2.16)
By defining the Gibbs Free Energy function as we have, we re-write equation (2.12)
dG d G wnonPV
(2.17)
The change in Gibbs Free Energy is the maximum amount of work, apart from PV work,
that can be obtained from any process performed at constant temperature and pressure
(see Figure 2.4). Note again that work done by the system on the surroundings (G wnonPV )
is negative, so expression (2.17) means that dG has a more negative value, and the
magnitude of dG is the limit of the magnitude of (G wnonPV ) .
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2.14 Gibbs Free Energy and the condition for a spontaneous change of a system at
constant T and P.
There are two reasons why the Gibbs Free Energy plays such a prominent role in
describing biological systems. The first reason, described in the previous section, is that
this function tells us the maximum amount of useful work that can be obtained from any
process that occurs spontaneously at constant temperature and pressure, which is the
condition under which most of biology occurs. The second reason is that the sign and
magnitude of the change in Gibbs Free Energy determines whether a process will occur
spontaneously at all and, if so, what the driving force will be for the process.
Furthermore, since the interactions between the system and the surroundings are limited
to maintaining constant temperature and pressure, these variables are sufficient to take
into account the thermodynamic effect of the changes to the surroundings during a
change in the state of the system.
Let’’s consider a spontaneous process in a system at constant T and P (Figure 2.4).
We can rewrite expression (2.18)
(dG G wnonPV ) d 0
(2.19)
The equal sign applies only to the reversible process which is at equilibrium throughout
the entire process.
(dG G wnonPV ) 0 for reversible processes only.
(2.20)
For irreversible processes, meaning those that proceed spontaneously and irreversibly
towards equilibrium, the inequality must hold.
(dG G wnonPV ) 0
(2.21)
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For the case in which there is no nonPV work done by the system on the surroundings (or
by the surroundings on the system), G wnonPV
0 , the following must hold for any
irreversible (spontaneous) process.
dG 0
(2.22)
Equation (2.22) is the thermodynamic criterion for any spontaneous change for a system
in which the initial and final states are at the same pressure and temperature and where no
work other than PV work is performed by the system on the surroundings (or vice versa).
For any spontaneous process in an open system that is maintained at constant pressure
and temperature in which G wnonPV
0 , the Gibbs Free Energy will decrease until the
equilibrium condition is reached, defined by
dG
0, at equilibrium (assuming G wnonPV =0).
(2.23)
In other words, the system will evolve until the minimum value of the Gibbs Free Energy
is reached. This is another extremum principle, such as we saw in Section 1.3, defining
equilibrium for a mechanical system by minimizing energy (U) and in Section 1.18,
defining the principle of maximizing entropy as a condition of any spontaneous process.
It is most important to realize that the principle of minimizing the Gibbs Free Energy is
really a re-statement of the principle of maximizing the total entropy. By maintaining
constant T and P through interactions with the surroundings, the entropy changes of the
surroundings are accounted for within the Gibbs Free Energy function by setting the
values of T and P. The great utility of the Gibbs Free Energy function is that it
incorporates within it the internal energy and entropy of the system plus surroundings ,
but it can be defined in terms of measurable parameters- temperature, pressure and
composition (T, P, and N).
12
The Gibbs Free Energy is the function we will be developing in much more detail
as we apply thermodynamics to problems in biochemistry and biology. If we want to
know the maximal amount of work that can be obtained by the hydrolysis of ATP, for
example under any defined circumstances, we need to know the change in the Gibbs Free
Energy. It is this function that will allow us to consider active transport, electrical work,
mechanical work and chemical reactions using a common vocabulary and to equate their
values using the same units of work potential. Furthermore, by knowing the change in
Gibbs Free Energy for any biochemical process, we can determine whether that process
will occur spontaneously. In terms of metabolic reactions taking place within cells,
knowledge of the changes of the Gibbs Free Energy during a particular reaction allows
one to predict in which direction the reaction will spontaneously proceed. We will discuss
these applications in the next Chapter.
2.2 Thermodynamic Driving Forces: Temperature, Pressure and Chemical
Potential
The systems we have been discussing have been defined as having a fixed
composition. We are primarily interested in biochemical reactions and material transport
in biological systems, so we need to allow the composition of the system to vary. If we
include chemical reactants in the system at constant temperature and pressure, for
example, they will react to form products. As a result of the chemical reaction, the
number of moles of the reactants will decrease and the products will increase until
equilibrium is reached. At this point the Gibbs Free Energy (G) of the system will be at
its minimal value. Since G U TS PV we can write this in differential form
dG
dU TdS SdT PdV VdP
(2.24)
13
For a system in which the composition is fixed, dU
PdV TdS , so we simplify
equation (2.25) to a compact form
dG
SdT VdP
(2.26)
Based on this compact form of the expression, T and P are referred to as the natural
variables for the Gibbs Free Energy: G(T, P). At fixed composition, we can also define
the differential expression of the free energy in the following form.
dG
(
wG
wG
) P ,ni dT ( )T ,ni dP
wT
wP
Comparing equations (2.28) and(2.29), we have the following.
wG
( ) P , ni S
wT
(
wG
)T ,n
wP i
V
(2.27)
(2.30)
(2.31)
We will return to these expressions later when we need to evaluate the temperature and
pressure dependence of the Gibbs Free Energy.
Equation (2.27) tells us that the Gibbs Free Energy of a system at fixed
composition and constant temperature and pressure (dT = dP = 0) is itself unchanging
(dG = 0). The Gibbs Free Energy is defined to conveniently deal with systems underoing
chemistry or transport. Now we will explicitly include the composition as an independent
variable. We can represent this by ““ni””, to denote the molar amount of each component of
the system.
G
G (T , P, ni )
(2.32)
We will define the initial composition of the system in terms of the number of moles of
each component- n1, n2, n3 etc. The differential expression for dG can now be written as
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dG
(
wG
wG
wG
) P ,ni dT ( )T ,ni dP ¦ (
)T , P ,niz j dn j
wT
wP
wn j
j
(2.33)
The change of Gibbs Free Energy with respect to an infinitesimal change in the amount
of component nj, at constant temperature and pressure and with the composition of all the
remaining components fixed is defined as the chemical potential.
wG
)T , P ,niz j
wn j
(2.34)
SdT VdP ¦ Pi dni
(2.35)
Pn
We can now write
dG
j
(
i
Equation (2.35) points out the relationships between pairs of extensive and
intensive thermodynamic variables (see Figure 2.5): S/T, V/P and µ/n. These are called
conjugate pairs. When a hot and cold object are brought together, heat is transferred from
the hot object to the cold object until the temperatures are equal. If heat is the only mode
of adding energy to the system, we can equate the heat transfer with a transfer of entropy,
increasing in the cold object and decreasing in the hot object. Temperature (an intensive
property) and entropy (an extensive property) are linked. A difference in pressure
between two chambers with a movable barrier results in increasing the volume of the side
with higher pressure and shrinking the volume on the low pressure side of the barrier. At
equilibrium, the pressures are the same on either side of the barrier. Hence, pressure (an
intensive property) and volume (an extensive property) are also linked.
15
Figure 2.5: The driving forces provided by a difference in temperature, pressure or
chemical potential are illustrated. Bringing hot and cold objects into contact results
in heat transfer, related to a change in entropy. The driving force to equalize the
temperature is proportional to the difference in temperature. Similarly a difference
in pressure provides a driving force to equalize pressure by adjusting volumes. The
chemical potential plays an equivalent role and the thermodynamic drive is towards
equalizing the chemical potential of each component in the system by the increase or
decrease in the amount of material. In this example, material is allowed to diffuse
between different chambers once they are in contact. Material flows from a region
of high chemical potential to low chemical potential.
Similarly, if matter can flow, it will move from a region of high chemical potential to
low chemical potential, until the chemical potentials are equal throughout the system.
Chemical potential (an intensive property) and the amount of material (an extensive
property) are linked.
Table 2.1
EXTENSIVE VARIABLE
INTENSIVE VARIBLE
S
T
V
P
ni
Pi
Q
Ȍ
16
Under conditions of constant temperature and pressure, applicable to most biological
problems, expression (2.35) simplifies to
dG
¦ P dn
i
i
at constant T and P.
(2.36)
i
In a mixture, each component has a chemical potential that is dependent on the
composition, temperature and pressure. For example, if we have a solution containing a
protein, ATP and water this will be characterized by P ATP , P prot and P water . We will see in
the Chapter 3 how to handle the complication that ATP actually exists in several distinct
states of ionization due to deprotonation of the phosphate groups.
Let’’s consider an example where we bring together two solutions A and
B,separated by a permeable membrane, where system A contains a 1 mM solution of
ATP and system B contains a solution of a protein (1 mg/ml) plus 0.1 mM ATP (Figure
2.6). The boundary allows the passage of all the molecular species and we observe the
system evolving towards equilibrium. Both the ATP and the protein will have a chemical
A
B
A
B
, P ATP
, P prot
and P prot
potential in each side of the combined system, designated P ATP
. We
know from experience that ATP will diffuse across from side A to side B, and that the
protein will diffuse from side B to side A, until equilibrium is reached.
17
Figure 2.6: Two chambers are brought together, separated by a permeable
membrane that allows both the protein and ATP to pass through. The system
evolves to a new equilibrium where the chemical potentials of each component in
the equilibrated chambers are equal.
The equilibrium condition, in the absence of nonPV work, is that dG = 0 (equation (2.23)
and the Gibbs Free Energy of the system will evolve to its minimum value. What happens
if we transfer an infinitesimal quantity of ATP from side A to side B at equilibrium?
dG
dG
A
B
( dn) P ATP
dnP ATP
0
(2.37)
B
A
dn( P ATP
P ATP
) 0
(2.38)
B
P ATP
A
P ATP
at equilibrium.
The same argument also holds for each component in the system, which tells us that at
B
equilibrium P ATP
A
B
P ATP
and P prot
A
P prot
. Furthermore, the same must apply to the water,
which can also pass through the membrane separating the two chambers, so at
18
B
equilibrium P water
A
. If we had used a membrane separating to two chambers which
P water
did not allow the protein to pass, then the chemical potential of the protein on each side
of the membrane would not be constrained to be equal.
We conclude that in a system where no nonPV work is done, each component will
be distributed so that its chemical potential is equal throughout the system if access is
allowed. We know from experience that we expect the concentrations of the equilibrated
substances will be equal, so it is clear that there must be a relationship between the
chemical potential, defined in equation (2.34), and concentration. We will obtain the
exact relationship in Section 2.4.
For any spontaneous change, the free energy of the system must decrease, dG < 0
(equation (2.22). In this example (Figure 2.6), the chemical potentials of each
component (ATP, protein and water) are effectively independent, so we expect dG < 0 for
spontaneous changes in the concentration of each component. Considering the change in
the free energy due to the transfer of an infinitesimal amount of ATP across the
membrane as the system equilibrates. From the definition of the chemical potential,
equation (2.34)
dG
B
A
dn( P ATP
P ATP
)0
(2.39)
A
B
Hence, if P ATP
, it follows that dn > 0, meaning that material will spontaneously
! P ATP
flow from a location of high chemical potential to a location of lower chemical potential.
The diffusion of each component is always from a region of high chemical potential to a
region of low chemical potential, until equilibrium is reached, at which point the
chemical potential of each component is constant throughout the system.
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The chemical potential can be thought of as an escape tendency. The magnitude
of the driving force for ATP to escape from the region where the chemical potential is
B
A
high to where the chemical potential is lower is given simply by ( P ATP
P ATP
).
Figure 2.7 shows the analogy between a mechanical and thermodynamic
equilibria. In the example of a mechanical equilibrium described in Section 1.2, an object
under the influence of a gravitational potential (V) responds to a force f
dV
which
dx
tends to move the object to the point of lowest potential energy.
Figure 2.7: Analogy between mechanical and thermodynamic systems. In the case of
the mechanical system (a ball rolling in a well), the equilibrium position is at the
minimal of potential energy. The force is the negative slope of potential energy as a
function of position, which is zero at the minimal value of V. The change in potential
energy from the starting point to the position of equilibrium is the work capacity.
In the thermodynamic system, exemplified by diffusion of ATP (Figure 2.6), the
thermodynamic driving force is also the negative slope of Gibbs Free Energy of the
system (both sides) as a function of the number of moles of ATP in chamber A. This
A
B
is equal to ( P ATP
P ATP
) , a positive number, which will decrease as the system gets
closer to equilibrium, at the minimal value of the Gibbs Free Energy. The capacity
for nonPV work is given by ǻG, measured from the starting point to the equilibrium
distribution.
20
The driving force is zero at the minimum in potential energy. We have a similar situation
with a charge (Q) in an electric potential, (Ȍ). Restricting ourselves to only one
dimension (x), the force experienced by the charge is the result of differences in the
potential at different locations, f el
E
d<
Q
dx
EQ , where the electric field is defined as
d<
. In three dimensions, E and fel are vector quantities with magnitude and
dx
direction. The charge reaches equilibrium when the force acting on it is zero, which will
be an extremum in the electrical potential (most positive for a negative charge; most
negative for a positive charge).
In the case of a thermodynamic system at constant temperature and pressure and
where wnonPV = 0, the components diffuse until the system reaches a minimum of the
Gibbs Free Energy. The driving force is given by (
wG
)T , P ,niz j which, in this example, is
wn j
A
B
( P ATP
P ATP
) . This is the change in the Gibbs Free Energy caused by a small
displacement of the system, analogous to a physical force being defined by the change in
energy due to a small displacement in position. As more ATP diffuses from A to B
A
B
(Figure 2.6) the difference in chemical potentials ( P ATP
P ATP
) decreases and is zero at
the point where the free energy of the system is minimal. Once this minimal value of the
Gibbs Free Energy is reached, the driving force will oppose any further net transfer of
ATP to either side. Diffusion of ATP towards equilibrium is spontaneous and irreversible
in the absence of any other processes. Diffusion will not spontaneously take the system
away from equilibrium.
21
2.3 The chemical potential of a pure substance is the molar Gibbs Free Energy.
For a pure substance ““i””, the free energy is simply equal to the product of the
molar free energy (Gm,i) and the number of moles, ni: G
ni Gm ,i . This is because the
Gibbs Free Energy is an extensive function and proportional to the size of the system.
Therefore,
Pi
(
wG
)T , P
wni
§ w ª ni Gm ,i º¼ ·
¨ ¬
¸
¨ wni
¸
©
¹ P ,T
Gm ,i
(2.40)
Hence, for a pure substance, such as water or solid ATP, the chemical potential is simply
the Gibbs Free Energy per mole.
2.31 The chemical potential of a component in a mixture is the partial molar Gibbs
Free Energy.
If we have a mixture of components, the chemical potential depends not only on
the amount of the substance itself, but also can depend on the concentrations of every
other component. This is why the expression for the chemical potential specifies that the
concentrations of every other component are defined, along with T and P. The chemical
potential of a substance in a mixture is also called the partial molar Gibbs Free Energy,
G i , where the line over the symbol indicates a partial molar value.
Pn
i
Gi
(
wG
)T , P ,n jzi
wni
(2.41)
If we add an infinitesimally small amount of substance, dni, to our mixture, the increase
in the Gibbs Free Energy, dG, normalized for the number of moles of material added
(dni) is the partial molar Gibbs Free Energy. We can think of adding a mol of substance
““i”” to a very large vat containing the mixture, and measuring the increase in the Gibbs
22
Free Energy. The change in the values of other extensive functions upon changing the
composition of a mixture can also depend on the composition. For example, partial molar
volume is defined as
Vi
(
wV
)T , P ,n jzi
wni
(2.42)
We will encounter this later when we discuss hydrodynamic properties of
macromolecules.
2.4 The chemical potential is dependent on log(concentration) of a substance.
The seemingly endless string of definitions and thermodynamic relationships are
only of value if we can express the parameters in terms of measurable quantities and use
the results for predictive purposes that can be compared to experimental results. Since we
are particularly interested in chemical and biochemical reactions, our focus is on the
expression of the chemical potential. We will now show that the chemical potential is
related to the logarithm of the concentration of a substance in a mixture. The relationship
can be expressed exactly for ideal gases, so we will start here and then generalize to
compounds and situations of biochemical interest. An ideal gas is one in which the
molecules behave as point masses (no molecular volume) and they do not interact with
each other. With these assumptions, the familiar equation of state ( PV
nRT ) can be
derived from the principle of maximizing entropy, as defined by Boltzmann (Equation
1.19). We will not, however, derive this, but accept the equation of state of an ideal gas as
an experimental relationship.
A Pure Ideal Gas: If our gas behaves as an ideal gas, the equation of state is
PV
nRT
(nN a )k BT
(2.43)
23
where n is the number of moles of the gas, R is the gas constant and is equal to
Avogodro’’s number (Na) x Boltzmann’’s constant (kB) = NakB. Note that nNa is equal to
the number of gas molecules in the sample.
Figure 2.8: Isothermal compression of a pure ideal gas increases the Gibbs Free
Energy in proportion to the logarithm of the pressure.
If we have a system of a pure gas, then the chemical potential is equal to the molar Gibbs
Free Energy (equation (2.40)). Let’’s calculate the change in the Gibbs Free Energy of the
gas if we compress or expand the gas in a closed container at constant temperature, going
from an initial state (Vi, Pi, T) to a final state (Vf, Pf, T) (Figure 2.8).
We start with the expression for the change in Gibbs Free Energy with pressure, derived
in equation (2.31)
§ wG ·
¨
¸
© wP ¹T ,n
V
(2.44)
24
Since we are interested in changes of Gibbs Free Energy and not the absolute value, we
can reference all values to an arbitrarily selected standard state, picked for convenience.
The following example will show how this works. For a gas, we can pick our standard
state to be the pure gas at a pressure of 1 bar ( P o
1bar ) at the temperature of our
experiment (T). At constant temperature, the change in free energy when we change the
pressure of our ideal gas from our standard state value (Po) to any final pressure (Pf) is
P
³ dG
P
Pf
Pf
o
o
G (T , P) G (T , P )
o
o
³ VdP ³
P
P
P
nRT
dP nRT ln fo
P
P
P pure (T , Pf ) Gm (T , Pf ) Gmo (T , P o ) RT ln
(2.45)
Pf
(2.46)
Po
This last equation tells us that the molar Gibbs Free Energy of a pure gas depends
logarithmically on the pressure. We can determine the value of the Gibbs Free Energy at
the initial state (Pi) also, and then calculate the difference between the initial and final
states by subtracting.
P pure (T , Pi ) Gm (T , Pi ) Gmo (T , P o ) RT ln
Pi
Po
(2.47)
Subtracting (2.47) from (2.46) yields the change in the molar Gibbs Free Energy ('Gm )
upon changing the pressure isothermally.
P pure (T , Pf ) P pure (T , Pi ) Gm (T , Pf ) Gm (T , Pi ) 'Gm
RT ln
Pf
Pi
(2.48)
Since we have a constant number of moles of gas present, n, the change in Gibbs Free
Energy (ǻG) is
'G
n'Gm
nRT ln
Pf
Pi
(2.49)
25
Since PV = nRT, and the gas concentration in moles/liter is c = n/V, we can also express
equation (2.49) as
'G
nRT ln
(n / V f )
(n / Vi )
nRT ln
cf
ci
(2.50)
The chemical potential of the gas increases if the pressure is increased, which means that
the escape tendency of the gas is increased at higher pressure. Compressing the gas into a
smaller volume also increases the concentration, which we can also think of as resulting
in the increase of the chemical potential. The reverse of all this is true. If we expand the
volume and decrease the pressure and concentration isothermally, the chemical potential
or escape tendency of the gas decreases.
2.5 The change in Gibbs Free Energy upon mixing two ideal gases.
Before we on to aqueous solutions, let’’s look at one more example with gases to
illustrate another aspect of the Gibbs Free Energy, which is that it will decrease as a
result of simply mixing two components. Now we will consider a mixture of ideal gases.
If we bring two containers together, each containing a different gas (e.g., nitrogen and
oxygen), we know that they will become completely mixed. This is a spontaneous
process, so we know that there must be a decrease in the Gibbs free energy of the system.
We also know that the entropy of the system must increase. The number of microscopic
states (W, in Boltzmann’’s equation 1.19)will increase because both the oxygen and
nitrogen can occupy regions of the volume that were not accessible prior to mixing.
Let’’s take a container with oxygen (gas ““a””) at 1 bar pressure and another
container with 9-times the amount of nitrogen (gas ““b””), also at a pressure of 1 bar. After
bringing these two containers together (Figure 2.9), we have a container with one part
26
oxygen and 9 parts nitrogen at a pressure of 1 bar. What is the change of the Gibbs Free
Energy?
Figure 2.9: Mixing the contents of two containers with different gases, such
as oxygen and nitrogen. Upon mixing there is a decrease in the Gibbs Free Energy
We start by modifying equation (2.46) by substituting the partial pressure of the
gas of interest in the mixture. For an ideal gas, if the total pressure is equal to P, the
partial pressure of component ““a”” (Pa) depends on the mole fraction of component ““a”” in
the gas mixture, Xa. If 10% of the gas consists of component ““a”” (i.e., Xa = 0.1) and 90%
is component ““b”” (Xb = 0.9) then the partial pressure due to component ““a”” is 10% of the
total pressure.
Pa
XaP
(2.51)
We will denote the chemical potential of component a in the mixture as
P amixture (T , P, na , nb ) Gm (T , P, na , nb ) Pmo (T ) RT ln
P
mixture
a
(T , P, na , nb )
Pa
Po
X P
P (T ) RT ln a o
P
(2.52)
o
m
In equation (2.52) we have written the molar Gibbs Free Energy of the pure gas in the
standard state Gmo (T , P o ) as P mo (T ) , a standard state chemical potential. Therefore,
27
ª
Pº
Pamixture (T , P, na , nb ) « Pao (T ) RT ln o » RT ln X a
P ¼
¬
(2.53)
Pamixture (T , P, na , nb ) Papure (T , P ) RT ln X a
The chemical potential of a component in a mixture of ideal gases is equal to the
chemical potential of the pure component at the specified temperature and total pressure
plus a term adjusting for the fact that the concentration is less than that of the pure
component. Since the mole fraction of component ““a”” is less than one (Xa < 1), the
RT ln X a term is negative, and the chemical potential of the component in the mixture is
less than that of the pure material under the same conditions.
We can calculate the Gibbs free energy change when we mix two gases (““a”” and
““b””), recalling that the chemical potential is equivalent to the molar Gibbs free energy,
equation (2.40). The two gases are at the same pressure and temperature prior to mixing,
and the mixing occurs at constant temperature and total pressure.
Before mixing, the total Gibbs Free Energy is the sum of the Gibbs Free Energy
of each of the pure gases. For each gas (““a”” or ““b””), the Gibbs Free Energy is equal to the
number of moles of the gas times the molar Gibbs Free Energy (or chemical potential).
Ga Gb
na P apure (T , P ) nb Pbpure (T , P )
(2.54)
After mixing the total Gibbs Free Energy is the sum of each component in the
mixture.
G mixture
na Pa (T , P, na , nb ) nb Pb (T , P, na , nb )
G mixture
na ª¬ Papure (T , P ) RT ln X a º¼ nb ª¬ Pbpure (T , P) RT ln X b º¼
G mixture
na Papure (T , P) nb Pbpure (T , P) na RT ln X a nb RT ln X b
(2.55)
28
The difference in Gibbs Free Energy between the two separate gases before and after
mixing is obtained by subtracting equation (2.54) from (2.55).
'Gmixing
na RT ln X a nb RT ln X b
nRT > X a ln X a X b ln X b @
(2.56)
where n=n a nb
This function for 'Gmixing (2.56) is plotted in Figure 2.10 as a function of Xa. Note that
since there are only two components, Xb = (1-Xa). The Gibbs Free Energy of mixing is
always negative so mixing is always spontaneous. The value of ǻGmixing is zero when
either Xa or Xb equals 1, and reaches a minimum when there are equal amounts of each
component.
We can also obtain an expression for the change in entropy upon mixing. Start with
equation (2.30).
ª wG º
« wT »
¬ ¼P
ª w'Gmixing º
S , therefore «
»
¬ wT ¼ P
'Smixing
(2.57)
'Smixing
nR > X a ln X a X b ln X b @
29
Figure 2.10: The Gibbs Free Energy of mixing two gases, assuming a total of 1 mole
(n = 1), T = 298 K. This is a plot of the function in equation (2.56).
Figure 2.11 shows that the entropy change is always positive and maximal when the
change in Gibbs free energy upon mixing is minimal. In this example,
'Stotal
'S sys 'S surr
'S sys . 'S surr
0 because no heat is exchanged with the
surroundings.
Finally, realize that if the gases in each container were the same, say oxygen in
each, then there would be no decrease in the Gibbs Free Energy and no increase in the
entropy of the system when the contents are mixed. This is because there is no distinction
between the kinds of molecules in either container. They are all identical and
indistinguishable.
30
Figure 2.11: Entropy of mixing two ideal gases, assuming a total of 1 mole (n = 1).
This is a plot of the function in equation (2.57).
2.6 The chemical potential of a substance in aqueous solution. In place of a mixture
of gases, we will now consider the chemical potential of a solute such as glucose,
dissolved in water. This is more typical of what we will be dealing with in biochemical
problems. By extension of the expression obtained with mixtures of gases, equation(2.55)
, we can write
Gtotal
nglu P glu nH 2O P H 2O
(2.58)
It is reasonable to express the chemical potential of water as the chemical potential of
pure water reduced in proportion to the mole fraction of water in our mixture.
PH O
2
nH 2O ª¬ P Hpure
(T , P ) RT ln X H 2O º¼
2O
(2.59)
31
However, it is not sensible to do the same for the glucose component, since that is a solid
in the pure state at room temperature and atmospheric pressure. How do we deal with
this? The strategy is to realize that we are only interested in changes of the Gibbs Free
Energy, so the problem can be avoided by the appropriate selection of standard states, as
we did with the gas in Section 2.4. This is described below.
2.61 Selection of the standard state and the use of molarity units of concentration
We have two interrelated issues to address at this point.
1. It is more convenient for biochemists to express our concentration in molar
units instead of mole fraction units.
2. We need to select convenient standard states to make calculations easy. There
is no need to be concerned about knowing the absolute value of the chemical potential
since we are always only interested in changes in the Gibbs Free Energy.
The first issue is handled by simply using molarity units, c = (mol solute)/(liter of
solution), in place of mole fraction. The use of molarity is convenient , but under some
circumstances this can result in small errors. This is because the volume of the solution
can itself be altered by the presence of the solute. Hence, two solutions with the same
mole fraction of different solutes might have different volumes and, therefore different
molar concentrations. Similarly, as the mole fraction of a solute increases, the change in
molar concentration may vary in strict proportion. More serious practitioners of
thermodynamics will use units of molality for this reason (moles of solute per 1000
grams of solvent). For most applications, molar units are suitable and that is what we
will use.
32
By convention, for biochemical systems, the standard state for any compound in
solution is defined as a 1 M solution of the solute at 1 bar pressure and 25oC (298.15K),
with the pH specified as pH 7 and, unless otherwise specified, an ionic strength of zero.
It is assumed that the standard state 1 M solution behaves as if it is a dilute solution, with
no complications due to molecular interactions often encountered in real solutions at high
concentration (but see Box 2.1). There are exceptions to this choice of standard state that
are used by biochemists. Most notably, the standard state of water is pure water (not 1 M
water) and for protons, typically, the standard state is 10-7 M (pH 7). There is no reason
why the standard states for all materials need to be the same, since it is purely arbitrary.
However, it does mean that one must be alert to what the standard states are that are
being used. We will explore this further in the next Chapter.
Once we select the standard state we can express the chemical potential (molar
Gibbs Free Energy) under any other condition if we know how the value of G changes
with whatever solution condition we are changing. We know by analogy to ideal
gases(e.g., equation , that the Gibbs Free Energy of a solute in solution will vary as the
ln(mole fraction) of the solute in solution
Xa
P (T , P, X a ) P o (T , P, X o ) ³ d P
Xo
(2.60)
P (T , P, X a ) P o (T , P, X o ) RT ln
Xa
Xo
In practice, we never need to know the actual value of the chemical potential at the
selected standard state concentration , P o (T , P, X o ) . Now, if we select the standard state
33
to be a concentration of 1 M, and if we assume that the molar concentration is related in a
simple way to mole fraction, such that ln
Xa
Xo
ln
ca
, we obtain
co
P (T , P, ca ) P o (T , P, c o ) RT ln
ca
co
(2.61)
The chemical potential of a component in solution is always given in relation to the
chemical potential in the standard state of that component. For a solution of glucose, we
now have
P glu
o
Defining cglu
o
P glu
RT ln
cglu
o
cglu
(2.62)
1M ,we write this as
P glu
o
P glu
RT ln cglu
(2.63)
It must be remembered that the concentration term (ln ci ) is actually normalized by the
standard state concentration (ln
ci
) and the term within the logarithm is, therefore,
1M
unitless. Generalizing this expression for the chemical potential of a reagent in a
biochemical system leads to the following equation, which is the starting point for all
biochemical thermodynamics.
Pi
Pio RT ln ci
(2.64)
We now have to tools to examine the thermodynamics of biochemical reactions,
which is the topic of the next Chapter.
34
Box 2.1: Deviations from ideality
An aqueous solution is far from an ideal gas. There are clearly interactions
between the molecules in any liquid. Hence, it is not evident that the Gibbs free energy
will be proportional to the logarithm of the concentration. Often, solutions do not behave
according to equation (2.64) because the molecules interact with each other. This is
handled empirically by correcting the concentration by an activity coefficient, Ȗ, which
must be experimentally determined and which corrects for non-ideal behavior. The
activity of a substance is defined as
ai
Ji(
ci
)
cio
(2.65)
and the chemical potential is now written as follows.
P glu
o
P glu
RT ln aglu
(2.66)
The activity coefficient, J i , is dependent on the concentration of the various
species present as well as the ionic strength. We can write the expression for the
chemical potential
Pi
Pio RT ln ai
[ Pio RT ln J i ] RT ln
ci
cio
(2.67)
Normally, the RT ln J i term is included in the standard state chemical potential.
For many in vitro biochemical applications performed in the laboratory, the
conditions of the experiments usually call for low concentrations of reagents, so the only
deviation of J i from unity is due to a dependence on ionic strength. Hence, both
Pi and Pio are functions of ionic strength.
35
Inside of a cell, the concentrations of some components can be very high, and
conditions are far from those leading to ideal behavior. We will need to take this into
account in dealing with thermodynamics of components in vivo. This is the major type of
problem in which one encounters any serious deviation from ideal behavior. Generally,
equation (2.64) is assumed without concern for the activity coefficient.
2.7 Gibbs Free Energy of formation
We saw in Chapter 1 (1.15) that one could obtain a standard molar heat of
formation of many biochemical species, starting with elements as the reference state (e.g.,
Table 1.2). The same can be done to define a standard Gibbs Free Energy of formation,
' f G o for biochemical species. Furthermore, since we know that 'G
'H T 'S for any
isothermal change of state (integrating equation (2.16)), we also can obtain the standard
state molar entropy of formation, ' f S if the values of ' f G o and ' f H o are known, since
' f Go
' f H o T' f So
(2.68)
We shall encounter these in the next Chapter in our discussion of the thermodynamics of
biochemical reactions.
2.8 Summary:
In Chapter 1, it was shown that in an isolated system, the criterion for any
spontaneous process is that the entropy must increase until it reaches a maximal value, at
which point equilibrium is reached. In this Chapter, we have extended this treatment to
systems that exchange energy, as heat and/or work, with the surroundings. For a system
held at constant pressure and constant temperature, a new state function, Gibbs Free
Energy is defined. Any spontaneous process at constant temperature and pressure must
proceed with a decrease in Gibbs free energy, and the equilibrium position is defined
36
when the Gibbs free energy is at a minimal value. The change in Gibbs free energy is
equal to the maximal amount of nonPV work that can be obtained from the process. The
change in the molar Gibbs free energy upon changing the concentration of a component
in solution is defined as the chemical potential, which depends on the logarithm of
concentration, or activity.
37