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.

revsys

sys

dqdST

(2.1)

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From this, it follows that for any irreversible pathway between the two states, sysq will

be less than sysTdS , i.e., sysq will have a smaller negative value than for heat

removed from the system or a larger positive value for heat added to the system (Figure

2.1).

revdq

sys sysTdS q (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 ( q, w, 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 sysq 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, sysq is negative and the maximum value (i.e., the

smallest negative value) is sysTdS .

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

surr sysq q (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 surrq , 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)

surrsurr

surr

qdST

(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

0sys surrdS dS (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 sys sys sysq dU w , we can substitute into equation (2.2) to yield

sys sys sysdU TdS w (2.6)

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For the special case of an isolated system (Figure 2.3 B), 0sys sys sysdU w q , so we

conclude that for any spontaneous process in an isolated system , as we saw in

Chapter 1. For any spontaneous process in an isolated system, the entropy of the system

must increase.

0sysTdS

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, sys sysU TS . We are now

considering the internal energy and entropy of the system itself, not the surroundings.

(2.7)

( )

( ) when

sys sys sys sys sys

sys sys sys sys

d U TS dU TdS S dT

d U TS dU TdS 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).

( ) ( ) (= w + w )at constant temperaturePV nonPVdA d U TS dU TdS w (2.9)

The change in the Helmholz Free Energy ( A) 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 ( A) 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

G is the limiting amount of nonPV work, it is always the case that

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( nonPVG w ) 0 . If the process occurs under conditions were no nonPV work is

actually accomplished, then . 0G The Helmholz Free Energy is also called the �“work function�”. For any process at

constant temperature,

dA w

A wor (2.10)

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 �–P V, 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 ( A), 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 ( G). We can rewrite equation (2.6) and explicitly divide the work into a

component done by a change in volume against the atmosphere ( )PVw PdV and all

work other than PV work ( )nonPVw . 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

sys PV nonPV nonPVdU TdS w w PdV w (2.11)

sys nonPVdU TdS PdV w (2.12)

For convenience, we define another state function, the Gibbs Free Energy,

sys sys sys sysG U TS PV (2.13)

from which it follows that the following is true for any small change in the Gibbs Free

Energy.

(2.14) sys sys sys sys sys sysdG dU TdS S dT PdV V dP

At constant temperature and pressure, 0 and 0,dT dP this simplifies to

sysdG dU TdS PdV (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

sysdG dH TdS (2.16)

By defining the Gibbs Free Energy function as we have, we re-write equation (2.12)

nonPVdG w (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 ( )nonPVw

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 ( )nonPVw .

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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)

( nonPVdG w ) 0 (2.19)

The equal sign applies only to the reversible process which is at equilibrium throughout

the entire process.

( ) 0 for reversible processes only.nonPVdG w (2.20)

For irreversible processes, meaning those that proceed spontaneously and irreversibly

towards equilibrium, the inequality must hold.

( ) 0nonPVdG w (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), 0nonPVw , the following must hold for any

irreversible (spontaneous) process.

0dG (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 0nonPVw , the Gibbs Free Energy will decrease until the

equilibrium condition is reached, defined by

0, at equilibrium (assuming =0).nonPVdG w (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).

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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 T we can write this in differential form S PV

(2.24) dG dU TdS SdT PdV VdP

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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.

,( ) ( )iP n T n

G GdG dT dPT P , i

(2.27)

Comparing equations (2.28) and(2.29), we have the following.

,( )iP n

G ST

(2.30)

,( )iT n

G VP

(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.

( , , )iG G T P n (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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, , , ,( ) ( ) ( )i i i jP n T n T P n

j j

G G GdG dT dP dnT P n 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.

, ,( )jn T

j

Gn i jP n (2.34)

We can now write i i

idG SdT VdP dn (2.35)

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.

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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 i

Q

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Under conditions of constant temperature and pressure, applicable to most biological

problems, expression (2.35) simplifies to

at constant T and P.i ii

dG dn (2.36)

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 , and ATP prot 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

potential in each side of the combined system, designated , , and A B A BATP ATP prot prot . 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.

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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?

( ) A BATP ATPdG dn dn 0

0

(2.37)

(2.38) ( )

at equilibrium.

B AATP ATP

B AATP ATP

dG dn

The same argument also holds for each component in the system, which tells us that at

equilibrium and B A B AATP ATP prot prot . Furthermore, the same must apply to the water,

which can also pass through the membrane separating the two chambers, so at

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equilibrium . If we had used a membrane separating to two chambers which

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.

B Awater water

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)

( B AATP ATPdG dn ) 0 (2.39)

Hence, if A BATP ATP , it follows that dn > 0, meaning that material will spontaneously

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

high to where the chemical potential is lower is given simply by . ( )B AATP 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 dVfdx

which

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 is equal to ( , 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 G, measured from the starting point to the equilibrium distribution.

)A BATP ATP

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The driving force is zero at the minimum in potential energy. We have a similar situation

with a charge (Q) in an electric potential, ( ). Restricting ourselves to only one

dimension (x), the force experienced by the charge is the result of differences in the

potential at different locations, eldf Q EQdx

, where the electric field is defined as

dEdx

. In three dimensions, E and fel are vector quantities with magnitude and

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 , ,( )i jT P n

j

Gn

which, in this example, is

. 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

(Figure 2.6) the difference in chemical potentials 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.

( A BATP ATP )

)( A BATP ATP

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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: ,i m iG n G . This is because the

Gibbs Free Energy is an extensive function and proportional to the size of the system.

Therefore,

,,

,

( ) i m ii T P

i i P T

n GG Gn n ,m 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,

iG , where the line over the symbol indicates a partial molar value.

, ,( )i

ini

GGn j iT P n (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

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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

, ,( )j i

i T P ni

VVn

(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 can be ( )PV nRT

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

( )a BPV nRT nN k T (2.43)

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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)

,T n

G VP

(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 ( ) 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 (P

1oP bar

o) to any final pressure (Pf) is

( , ) ( , ) lnf f

o o o

P PPfoo

P P P

PnRTdG G T P G T P VdP dP nRTP P

(2.45)

( , ) ( , ) ( , ) ln fpure o of m f m o

PT P G T P G T P RT

P (2.46)

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.

( , ) ( , ) ( , ) lnpure o o ii m i m o

PT P G T P G T P RTP

(2.47)

Subtracting (2.47) from (2.46) yields the change in the molar Gibbs Free Energy

upon changing the pressure isothermally.

( )mG

( , ) ( , ) ( , ) ( , ) ln fpure puref i m f m i m

i

PT P T P G T P G T P G RT

P (2.48)

Since we have a constant number of moles of gas present, n, the change in Gibbs Free

Energy ( G) is

ln fm

i

PG n G nRT

P (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

( / )

ln ln( / )

f f

i i

n V cG nRT nRT

n V c (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.

a aP X P (2.51)

We will denote the chemical potential of component a in the mixture as

( , , , ) ( , , , ) ( ) ln

( , , , ) ( ) ln

mixture o aa a b m a b m o

mixture o aa a b m o

PT P n n G T P n n T RTP

X PT P n n T RTP

(2.52)

In equation (2.52) we have written the molar Gibbs Free Energy of the pure gas in the

standard state as , a standard state chemical potential. Therefore, ( , )o omG T P ( )o

m T

27

( , , , ) ( ) ln ln

( , , , ) ( , ) ln

mixture oa a b a o

mixture purea a b a a

PT P n n T RT RT XP

T P n n T P RT X

a

(2.53)

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

ln aRT X 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).

(2.54) ( , ) ( , )pure purea b a a b bG G n T P n T P

After mixing the total Gibbs Free Energy is the sum of each component in the mixture.

(2.55)

( , , , ) ( , , , )

( , ) ln ( , ) ln

( , ) ( , ) ln ln

mixturea a a b b b a b

mixture pure purea a a b b b

mixture pure purea a b b a a b b

G n T P n n n T P n n

G n T P RT X n T P RT X

G n T P n T P n RT X n RT X

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).

ln ln ln ln

where n=n

mixing a a b b a a b b

a b

G n RT X n RT X nRT X X X X

n (2.56)

This function for mixingG (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 Gmixing 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).

, therefore

ln ln

mixingmixing

P P

mixing a a b b

GG S ST T

S nR X X X X (2.57)

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,

. because no heat is exchanged with the

surroundings.

total sys surr sysS S S S 0surrS

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

2 2total glu glu H O H OG n n

(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.

2 2 2 2

( , ) lnpureH O H O H O H On T P RT X (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

( , , ) ( , , )

( , , ) ( , , ) ln

a

o

Xo oa X

o o aa o

T P X T P X d

XT P X T P X RTX

(2.60)

In practice, we never need to know the actual value of the chemical potential at the

selected standard state concentration , . Now, if we select the standard state ( , , )o T P X o

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 lnao

ao

X cX c

, we obtain

( , , ) ( , , ) lno o aa o

cT P c T P c RTc

(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

ln gluoglu glu o

glu

cRT

c (2.62)

Defining ,we write this as 1ogluc M

lnoglu glu gluRT c (2.63)

It must be remembered that the concentration term is actually normalized by the

standard state concentration

(ln )ic

(ln )1

icM

and the term within the logarithm is, therefore,

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.

lnoi i iRT c (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, , which

must be experimentally determined and which corrects for non-ideal behavior. The

activity of a substance is defined as

( )ii i o

i

cac

(2.65)

and the chemical potential is now written as follows. lno

glu glu gluRT a (2.66)

The activity coefficient, 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

ln [ ln ] lno o ii i i i i o

i

cRT a RT RTc

(2.67)

Normally, the ln iRT 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 i from unity is due to a dependence on ionic strength. Hence, both

and oi i 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,

for biochemical species. Furthermore, since we know that of G 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 are known, since and of G

of H

oS (2.68) o of f fG H T

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