Chemistry 130

Chemistry 130

Gibbs Free Energy

Dr. John F. C. Turner

409 Buehler Hall

jturner@ion.chem.utk.edu

Chemistry 130

Equilibrium and energy

So far in chemistry 130, and in Chemistry 120, we have described chemical reactions thermodynamically by using

- the change in internal energy, U, which involves heat transferring in or out of the system only

or

- the change in enthalpy, H, which involves heat transfers in and out of the system as well as changes in work.

U applies at constant volume, where as H applies at constant pressure.

U

H

Chemistry 130

Equilibrium and energy

When chemical systems change,

either physically through melting, evaporation, freezing or some other physical process

variables (V, P, T)

or chemically by reactionvariables (ni)

they move to a point of equilibrium by either exothermic or endothermic processes.

Characterizing the change as exothermic or endothermic does not tell us whether the change is spontaneous or not.

Both endothermic and exothermic processes are seen to occur spontaneously.

Chemistry 130

Equilibrium and energy

Our descriptions of reactions and other chemical changes are on the basis of exothermicity or endothermicity – whether is negative or positive

is negative – exothermic is positive – endothermic

As a description of changes in heat content and work, these are adequate but they do not describe whether a process is spontaneous or not.

There are endothermic processes that are spontaneous – evaporation of water, the dissolution of ammonium chloride in water, the melting of ice and so on.

We need a thermodynamic description of spontaneous processes in order to fully describe a chemical system

H

H

H

Chemistry 130

Equilibrium and energy

A spontaneous process is one that takes place without any influence external to the system.

The opposite of a spontaneous change is a non-spontaneous change – one where there must be an external influence to force the change.

For any observed, spontaneous change, the reverse process is non-spontaneous.

Chemistry 130

Equilibrium and energy

If we use energy as the sole criterion of spontaneous change, we are using effectively a mechanical analogy – systems move to the local minimum in energy as the point of equilibrium.

In the example of a ball falling, we have one variable, position, in a field, the gravitational field of the earth, and there is no endothermic path.

Potential energy is converted into kinetic energy in flight and then into heat and sound (q and w) at impact.

Chemistry 130

Equilibrium and energy

In this case, minimization of the potential energy and conversion ultimately into heat defines the point of equilibrium and appears to be linked to the direction of spontaneous change. Early theories of chemical thermodynamics rested on the evolution of heat as the 'driving force' for a reaction, which fails.

Chemistry 130

Equilibrium and energy

It is certainly true that many reactions that are spontaneous are accompanied by the evolution of heat:

But some are not and are endothermic and yet are still spontaneous:

So, in the 'mechanical' view of chemical change, in the case of ammonium chloride, we have a system that spontaneously moves to a higher state of energy.

Physical changes such as melting and boiling are inherently endothermic and the same problem occurs.

NaOHs H2O

Naaq+ OHaq

- H = −44.51 kJ mol−1

NH4Cls H2O

NH4aq+ Claq

- H = 14.78.51 kJ mol−1

Chemistry 130

Equilibrium and energy

Both of these examples have an associated enthalpy change:

NaOHs H2O

Naaq+ OHaq

- H = −44.51 kJ mol−1

NH4Cls H2O

NH4aq+ Claq

- H = 14.78.51 kJ mol−1

Some processes do not result in a net change of energy in the system and are still spontaneous:

In an insulated system that cannot exchange heat with the surroundings, heat will spontaneously move to equalize a temperature gradient between two bodies.

q

T1 ≫ T2

T1 = T2

Chemistry 130

Equilibrium and energy

Similarly, there are no forces between a particles of a perfect gas and the internal energy of the perfect gas is independent of volume. Yet a perfect gas will always expand to fill the volume available, with no net change in energy.

Chemistry 130

Equilibrium and energy

So spontaneous changes can take place endothermically, exothermically or with no exchange of energy

q

T1 ≫ T2

T1 = T2

NaOHs H2O

Naaq+ OHaq

- H = −44.51 kJ mol−1

NH4Cls H2O

NH4aq+ Claq

- H = 14.78.51 kJ mol−1

Chemistry 130

Equilibrium and energy

We need a description of spontaneous change that includes the direction of the change, a description of the point of equilibrium quantitatively

Energy is not a good description – chemical changes are not mechanical.

Chemistry 130

Reversibility and irreversibility

Thermodynamically, we define a reversible change as one that takes place within an infinitesimal step from the point of equilibrium

If the point of equilibrium is defined by

then irreversible changes can occur via

Reversible changes occur infinitely slowly and maximize the amount of work that is possible from a system. They also do not occur in practice but are the theoretical limit for developing our description of spontaneous change and the point of equilibrium.

Gni ,P ,T

Gnidni ,P ,T Gni ,P ,T Gni ,PdP ,T

Gni ,P ,TdT

Chemistry 130

Reversibility and irreversibility

In practice, the maximum amount of work is never achievable.

Real changes are irreversible and the amount of work that can be extracted is always less than the maximum amount of work.

Gnidni , P ,T Gni ,P ,T Gni ,PdP ,T

Gni ,P ,TdT

Chemistry 130

Equilibrium, energy and entropy

The magnitude and sign of the change in enthalpy associated with a chemical or physical change does not reflect the spontaneity of the process. It is not a good measure.

However, any description of a molecular system such as a mole of a perfect gas is inherently statistical:

A mole contains particles and the number of ways that we can arrange a mole of particles is going to be of the order of

There are therefore many (!) ways of describing a chemical system while conserving the observed macroscopic properties – internal energy, pressure, temperature etc.

6.023 × 1023

Number of ways = W ~ 101023

Chemistry 130

Equilibrium, energy and entropy

If we have a large number of ways of describing the inside of the system so that the outside stays the same, then we have a large number of ways of distributing the energy of the system amongst these different configurations.

There are many equivalent ways of distributing the thermal energy of a system given a certain macroscopic energy.

A change is spontaneous when the number of ways of distributing the energy increases.

The point of equilibrium is when this number of ways is maximized.

Chemistry 130

Entropy

The measure of the number of ways of distributing energy that we use to describe this is the entropy of the system. We need entropy to be a state function and we cannot use heat, q, to do this because heat is not a state function.

Instead, we define the entropy, S, for a reversible change as

where qrev is the reversible heat transferred and T is the

thermodynamic temperature.

Entropy = S =qrev

T

Chemistry 130

Chemistry 130

Gibbs Free Energy

Dr. John F. C. Turner

409 Buehler Hall

jturner@ion.chem.utk.edu

Chemistry 130

Entropy Summary

1. A spontaneous change is one that takes place without any external action on the system and the reverse of a spontaneous change is non-spontaneous

3. Energy and the First Law of Thermodynamics does not predict the direction of spontaneous change

4. Spontaneous change occurs when the number of ways of distributing the energy associated with the change increases

5. We quantify this increase in the distribution of energy associated with a spontaneous change by the entropy of a system

6. Entropy is a state function and is defined by where qrev is the reversible heat change.

7. A reversible change is one that takes place infinitesimally close to the point of equilibrium

Entropy = S qrev

T

Chemistry 130

The Gibbs function

In order to predict the direction of spontaneous change, we need to consider the total entropy change in the universe.

We write this as

from our definition of entropy. We know that the heat change in the system is equivalent to the opposite of the heat change in the surroundings:

and we know, that for a system that can do work, qSystem=H

SUniverse = SSurroundings SSystem

SUniverse =qSurroundings

T

qSystem

T

qSurroundings

T= −

qSystem

T

Chemistry 130

The Gibbs function

Now we can write the change in the universe solely in terms of changes in the system.

This is important because the system is the part of the universe that we know enough about for an accurate description in principle.

The entropy then becomes

We define

where G is the Gibbs function.

SUniverse = −HT

SSystem

TSUniverse = −H TSSystem

−TSUniverse = H − TSSystem

qSurroundings

T= −

HT

G = H − TSSystem

Chemistry 130

The Gibbs function

The Gibbs function is a disguised form of entropy and has the units of energy; it is not an energy term in the First Law sense (H, U etc) but is a measure of the change in entropy of the universe.

For a spontaneous change,

i.e. the change in the Gibbs function must be zero or less than zero for a spontaneous change.

The Gibbs function allows us to define quantitatively the direction of spontaneous change in the universe – it allows us to determine what will take place amongst all the energetically possible changes allowed by the First Law

G 0

Chemistry 130

All conceivable chemical changes

The Gibbs function

The Gibbs function allows us to draw a 'map' of chemical change for the universe:

All possible chemical changes allowed by the First LawAll observed, spontaneous chemical changes allowed by the Second Law

You are here

Chemistry 130

The Gibbs function

We manipulate the Gibbs function and the entropy in the same way as we manipulate any other state function such as

The reference state that we use is the same as the standard state for

- the standard state of the pure material at 1 atm pressure. Under these conditions, we write

where the delta is the usual 'Products – Reactants'.

The units of the Gibbs function and the entropy are

H

G = G° = H°−T S°

Units G kJ mol−1

S J mol−1

Chemistry 130

The Gibbs function

There are three criteria for the Gibbs function in terms of sign:

The equilibrium position is defined by

Note that this is the standard change in the Gibbs function

Sign Direction of change G 0 positive non-spontaneous

G = 0 equilibrium

G 0 negative spontaneous

G° = −RT lnKeq

Chemistry 130

The Gibbs function

This relationship, allows us to determine

- the standard change in Gibbs function, given Keq

or

- the equilibrium constant, given the standard change in Gibbs function

It also explains why the equilibrium constant is sensitive to temperature, pressure and the number of moles of species present as

G° = −RT ln K eq

Gnidni , P ,T Gni ,P ,T Gni ,PdP ,T

Gni ,P ,TdT

Chemistry 130

The Gibbs function

We can also calculate the variation of the equilibrium constant with temperature. As

Then

and at two different temperatures,

Subtracting the two gives

G° = − RT ln K eq

ln Keq = − G°

RT

ln K1 = − G°

RT1

ln K2 = − G°

RT2

ln K2 − ln K1 = lnK2

K1 = −

G°

RT2

− − G°

RT1 = −

G°

RT2

G°

RT1

lnK2

K1 =

G°

RT1

− G°

RT2

= G°

R 1T1

−1T2

≈ H°

R 1T1

−1T2

van't Hoff equation

Chemistry 130

The Gibbs function

The van't Hoff equation shows the reason why the equilibrium constant is sensitive to temperature – it is based on the entropy of the system and is not simply a bookkeeping device.

Le Chatelier's principle therefore reveals the thermodynamic nature of chemical equilibrium.

ln K2

K1 =

H°

R 1T1

− 1T2

Chemistry 130

The Gibbs function

For an equilibrium between a gas and a pure liquid, we can calculate how the pressure will change with temperature.

As the equilibrium constant for

is given by

then

ln P2

P1 =

H°

R 1T1

− 1T2

Clausius-Clapyron equation

Al Ag

K =[ Ag]

[ Al]=

PA

1= PA

Chemistry 130

State functions reprise

From 120

State functions are extrinsic functions of variables such as temperature and pressure that define the thermodynamic state of a system.

The important state functions are:

Internal Energy U EntropyS

Enthalpy H Gibbs functionG

An extrinsic quantity is one that depends on the amount of material present, where as an intrinsic variable is one that does not depend on the quantity of material.

Intrinsic variables include: density and temperature

Chemistry 130

State functions reprise

ni

N~

exp{−Ei

kBT }∑

j

exp{−E j

kBT }

The energy level spacing and the temperature control the number of particles in the individual state i. The sum simply represents all possible states.

Any system above 0 K contains energy and can in principle do work with certain limitations. This energy is stored in modes inside the system; these modes can be translational, rotational, vibrational and electronic. The type of modes, the number of them and the energy separations between individual modes depends critically on the system concerned and the state of matter.

The availability of the modes depends on the temperature and is governed by the Boltzmann distribution. Ei is the energy of the

particular mode and ni is the population that is present in that

mode. N is the total number of particles in the system.

Chemistry 130

State functions reprise

ni

N~

exp{−Ei

kBT }∑

j

exp{−E j

kBT }The energy level spacing and the temperature control the number of particles in the individual state i. The sum simply represents all possible states.

Ei =

10Ei = 100

Ei = 1000

Ei = 10000

Chemistry 130

State functions reprise

Though we cannot hope to define the precise thermodynamic state of a system explicitly – the magnitude of Avogadro's number prevents this – we can measure straightforwardly changes in the state of the system by observing the heat and work involved in the chemical or physical change.

We define, in a system that can do no work and is at constant volume, the internal energy U.

U can change by the absorption or emission of heat with respect to the surroundings but in no other way.

If we allow the system to do work, i.e. it is at constant pressure, then we must account for the work done as well as heat changes, and in this case

Internal energy at constant pressure = q + w

We term this 'work-corrected' internal energy H, the enthalpy.

Chemistry 130

State functions reprise

As as state function, the manner in which the state is prepared is irrelevant. Only the initial and final states are important. The path between them is not important.

Thermodynamics has no bearing on the speed of chemical change – this is the province of chemical kinetics – but only on the magnitude (1st Law) and direction (2nd Law) of chemical change.

As the path is independent, we are free to choose any path that connects two state functions at will. Using this approach, we are able to calculate several quantities that are hard or difficult to measure, such as entropy or the Gibbs free energy, as well as predict and determine other thermodynamic quantities.

Chemistry 130

State functions reprise

Hess' Law of summation reflects this path independence. If we choose two different paths from one thermodynamic state to another, then the sum of any state function along those paths must be equivalent. This allows us to write a Hess cycle for a chemical change that will include the quantity that we are interested in, as well as other values that we know.

Chemistry 130

State functions reprise

Example: Determine the heat of reaction for the formation of carbon monoxide from graphite and oxygen:

We have two paths to the same material and so we can construct a Hess cycle.

Cgraphite, s +12

O2g COg Hr =

GivenCgraphite, s + O2g CO2g H1 = −393.5 kJ mol−1

COg +12

O2g CO2g H2 = −283.0 kJ mol−1

Chemistry 130

State functions reprise

Example: Determine the heat of reaction for the formation of carbon monoxide from graphite and oxygen:

Cgraphite,s O2g

COg 12

O2g CO2g

H1 H2

H rFollowing the direction of the arrows and remembering that the direction of the arrow tracks with the sign of we can construct the correct equations for the cycle:

H

H2 Hr = H1

Hr = H1− H2

Hr = −393.5 kJ mol−1−−283.0 kJ mol−1 Hr = −393.5 kJ mol−1283.0 kJ mol−1

Hr = −110.5 kJ mol−1

Chemistry 130

State functions reprise

This method is identical to the rearrangement of the equations algebraically.

The enthalpy of formation is the enthalpy for the formation of the material from the elements – we define the zero-point for enthalpy from the elements.

In this case, the heat of reaction is given by the difference between the total heats of formation of all the products less the total heats of formations of the reactants:

H1 = ∑ p HProducts−∑ R HReactants

Chemistry 130

State functions reprise

We use exactly the same approach with the Gibbs function. In the case of the Gibbs function of formation, we write

and treat the equations in the same manner. We do the same for any state function.

The Gibbs function can be dissected into the entropy change of the system and the enthalpic contribution. Therefore, we can calculate the change in entropy separately and the enthalpy separately, and then combine the two at the temperature of interest.

Gr = ∑ p GProducts−∑ R GReactants

Chemistry 130

State functions reprise

Once we have the standard Gibbs function for a reaction, we can calculate the equilibrium constant directly:

(note that this is true only for the standard change in Gibbs function).

Thus from the Gibbs function, we can find the equilibrium constant and vice versa.

The temperature dependence of the equilibrium constant

also allows us to calculate the Gibbs function at T2 given K1 at T1

G° = −RT ln K eq

ln K2

K1 =

H°

R 1T1

− 1T2