Chemistry 130
Gibbs Free Energy
Dr. John F. C. Turner
409 Buehler Hall
Chemistry 130 Equilibrium and energy
So far in chemistry 130, and in Chemistry 120, we have described chemical reactions thermodynamically by using
U the change in internal energy, U, which involves heat transferring in or out of the system only
or
H 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.
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 reaction variables (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 H is negative or positive
H is negative – exothermic H 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
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 nonspontaneous change – one where there must be an external influence to force the change.
For any observed, spontaneous change, the reverse process is nonspontaneous.
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:
+ −1 NaOHs Naaq OHaq H = −44.51 kJ mol H2 O But some are not and are endothermic and yet are still spontaneous:
+ −1 NH4 Cls NH4aq Claq H = 14.78.51 kJ mol H2 O 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.
Chemistry 130 Equilibrium and energy
Both of these examples have an associated enthalpy change: + −1 NaOH s Naaq OHaq H = −44.51 kJ mol H2O + −1 NH4 Cl s NH4aq Claq H = 14.78.51 kJ mol H2 O Some processes do not result in a net change of energy in the q system and are still spontaneous: T 1 ≫ T 2 In an insulated system that cannot exchange heat with the surroundings, heat will spontaneously move to equalize a temperature gradient between two bodies.
T 1 = T 2 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
+ −1 NaOHs Naaq OHaq H = −44.51 kJ mol H2 O + −1 NH4 Cls NH4aq Claq H = 14.78.51 kJ mol H2 O
q
T 1 ≫ T 2
T 1 = T 2
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 G ni , P ,T then irreversible changes can occur via
G nid ni , P ,T
G ni , P ,T G ni ,Pd P ,T G n , P , T d T i
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.
Chemistry 130 Reversibility and irreversibility
G n d n , P ,T i i
G ni , P ,T G ni ,Pd P ,T
G ni , P ,Td T
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.
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 6 . 0 2 3 × 1 0 2 3 particles and the number of ways that we can arrange a mole of particles is going to be of the order of
23 Number of ways = W ~ 1010 There are therefore many (!) ways of describing a chemical system while conserving the observed macroscopic properties – internal energy, pressure, temperature etc.
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
q Entropy = S = rev T
where qrev is the reversible heat transferred and T is the thermodynamic temperature.
Chemistry 130 Chemistry 130
Gibbs Free Energy
Dr. John F. C. Turner
409 Buehler Hall
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 nonspontaneous
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 q 6. Entropy is a state function and is defined by Entro p y = S rev T where qrev is the reversible heat change.
7. A reversible change is one that takes place infinitesimally close to the point of equilibrium 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
SUniverse = SSurroundings SSystem q q S = Surroundings System Universe T T
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: q q Surroundings = − System T T
and we know, that for a system that can do work, qSystem=H
Chemistry 130 The Gibbs function
Now we can write the change in the universe solely in terms of changes in the system. q H Surroundings = − T T This is important because the system is the part of the universe that we know enough about for an accurate description in principle. H The entropy then becomes S = − S Universe T System
TSUniverse = −H TSSystem
−TSUniverse = H − TSSystem
We define G = H − TSSystem where G is the Gibbs function.
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, G 0
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
Chemistry 130 The Gibbs function
The Gibbs function allows us to draw a 'map' of chemical change for the universe:
All conceivable chemical changes
All possible chemical changes allowed by the First Law
All 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 H
the standard state of the pure material at 1 atm pressure. Under these conditions, we write G = G° = H°−T S°
where the delta is the usual 'Products – Reactants'.
The units of the Gibbs function and the entropy are 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:
Sign Direction of change G 0 positive nonspontaneous
G = 0 equilibrium
G 0 negative spontaneous
The equilibrium position is defined by
° G = −RT ln K eq Note that this is the standard change in the Gibbs function
Chemistry 130 The Gibbs function
° This relationship, G = − R T l n K eq 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 n d n , P ,T i i
G ni , P ,T G ni , Pd P ,T
G ni , P ,Td T
Chemistry 130 The Gibbs function
We can also calculate the variation of the equilibrium constant with ° temperature. As G = −RT ln K eq G° Then ln K = − eq RT and at two different temperatures, G° G° ln K1 = − ln K 2 = − R T 1 R T 2 Subtracting the two gives
° ° ° ° K 2 G G G G ln K − ln K = ln = − − − = − 2 1 K RT RT R T RT 1 2 1 2 1 K G° G° G° 1 1 H° 1 1 ln 2 = − = − ≈ − K RT R T R T T R T T 1 1 2 1 2 1 2 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.
K H° 1 1 ln 2 = − K R T T 1 1 2 Le Chatelier's principle therefore reveals the thermodynamic nature of chemical equilibrium.
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
Al A g is given by
[ A g] P A K = = = P A [ Al] 1 then
P H° 1 1 ln 2 = − P R T T 1 1 2 ClausiusClapyron equation
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 Entropy S
Enthalpy H Gibbs function G
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
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. The energy level spacing and the temperature −E exp i control the number of particles in the individual k T ni B state i. The sum simply represents all possible ~ { } N −E states. exp j ∑ k T j { B }
Chemistry 130 State functions reprise
Ei = 10
Ei = 100
Ei = 1000
Ei = 10000
−E exp i n kB T i ~ { } The energy level spacing and the temperature N −E control the number of particles in the individual exp j ∑ k T state i. The sum simply represents all possible j { B } states. 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 'workcorrected' 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:
1 C + O CO H = graphite , s 2 2 g g r
Given −1 C graphite , s + O2 g CO2 g H1 = −393.5 kJ mol 1 CO + O CO H = −283.0 kJ mol−1 g 2 2 g 2 g 2
We have two paths to the same material and so we can construct a Hess cycle.
Chemistry 130 State functions reprise
Example: Determine the heat of reaction for the formation of carbon monoxide from graphite and oxygen:
Following the direction of the arrows and H 1 r remembering that the direction of the arrow CO O CO g 2 2 g 2 g tracks with the sign of H we can construct the correct equations for the cycle:
H2 Hr = H1 H2 H1 Hr = H1− H2
C graphite , s O2 g −1 −1 Hr = −393.5 kJ mol −−283.0 kJ mol −1 −1 Hr = −393.5 kJ mol 283.0 kJ mol −1 Hr = −110.5 kJ mol
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 zeropoint 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
Gr = ∑ p GProducts−∑ R GReactants 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.
Chemistry 130 State functions reprise
Once we have the standard Gibbs function for a reaction, we can calculate the equilibrium constant directly:
° G = −RT ln K eq
(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 K H° 1 1 ln 2 = − K R T T 1 1 2
also allows us to calculate the Gibbs function at T2 given K1 at T1
Chemistry 130