ChemicalChemical ThermodynamicsThermodynamics2013/20142013/2014

7th Lecture: Gibbs Energy and Fundamental EquationsValentim M B Nunes, UD de Engenharia

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What have we learned (from the 2What have we learned (from the 2ndnd Law!) Law!)

We have seen that, in isolated systems, the entropy change give us the direction of an spontaneous transformation:

ΔStotal > 0 – spontaneous transformationΔStotal = 0 – equilibrium (reversible process)

But, systems of interest generally are not isolated. It will be convenient to have a criteria for spontaneous change depending only on the system.

Gibbs (1839-1903) introduced a new state function, that plays a key role in chemistry and in the study of chemical equilibrium:

TSHG

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Josiah Willard Gibbs Josiah Willard Gibbs

Thermodynamicist!

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Criteria for Spontaneous Criteria for Spontaneous Change Change Consider a system in thermal equilibrium with its surroundings:

surrsyst TT

syst

systsystUniv

syst

surrsyst

surr

surrsystUniv

surrsystUniv

T

dQdSdS

T

dQdS

T

dQdSdS

dSdSdS

By the second Law,0

syst

systsyst T

dQdS

For any transformation in a closed system at constant p (p=pext)

dHTdST

dHdS or 0

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Criteria for Spontaneous Criteria for Spontaneous Change Change

Under constant T= Tsurr and constant p = pext, the criterion for spontaneity is:

0dGThis means that at constant p and constant T, equilibrium is achieved when the Gibbs energy is minimized.

Consider the process A B (keeping p and T constant)

ΔG < O – A B is spontaneousΔG = 0 – A and B are in equilibriumΔG > 0 – then B A is spontaneous

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What tell us G? (some What tell us G? (some mistakes!) mistakes!)

TSHG

minimummaximum

Wrong!

The second law is still valid. In some transformations the maximization of entropy can be obtained by “transferring” enthalpy to the surroundings

Snowing: gas liquid solid

ext

systext T

HS

If Text is low (cold air masses) then |ΔSext| > | ΔSsyst|

The Helmholtz The Helmholtz energyenergy

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If we define now the Helmholtz energy, A, as:

TSUA

The criterion for spontaneity under constant T = Tsurr and constant volume, V it will be:

0dA

Gibbs energy is, by far, more important that Helmholtz energy.

Physical meaning of Gibbs energy Physical meaning of Gibbs energy

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At constant temperature:

TdSdHdG

Since H = U + pV

TdSpdVVdpdWdQTdSVdppdVdUdG

pdVVdpdWdG

Now, considering that the total work is the sum of expansion work and other types of work

expansionexcept máx,dWpdVdW

As result, at constant p we finally obtain

máxdWdG

The Gibbs energy change represents the maximum available work, or the useful work that we can obtain from a given process.

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Standard Gibbs energy Standard Gibbs energy

As for enthalpy we can also obtain standard Gibbs energy of formation, ΔGºf (at pº = 1 bar). For any element in their reference state, ΔGºf=0.

From the definition of G = H – TS we obtain:

ººº STHG

and:0

,0

ifi

ir GG

These values are tabled, and are obtained from calorimetric or spectroscopic data!

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The Gibbs Energy and Fundamental The Gibbs Energy and Fundamental Equations Equations We can now combine the first an second law; as we have introduced all the state functions for closed systems

From the 1st Law: dWdQdU then:

pdVTdSdU

Since H = U + pV

VdpTdSdH

G = H-TS

SdTVdpdG

A=U-TS

SdTpdVdA

U= U(S,V)

H = H(S,p)

G = G(p,T)

A =A(T,V)

FUNDAMENTAL EQUATIONS: valid for reversible or irreversible changes for closed systems and only pV work.

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How G changes with p and THow G changes with p and T

Since G is a state function, and as we saw before G = G(T,p) then we can write:

dTT

Gdp

p

GdG

pT

And easily obtain:

ST

G

p

and Vp

G

T

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How G changes with p and THow G changes with p and T

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The Maxwell RelationsThe Maxwell Relations

Recalling that df = gdx + hdy is an exact differential if yx

x

h

y

g

and using the fundamental equations we obtain

pS

TV

VS

Tp

S

V

p

T

V

S

T

p

S

p

V

T

p

S

T

V

Maxwell RelationsMaxwell Relations: these equations allow us to establish useful thermodynamic relations between state properties!

U and H from equations of stateU and H from equations of state

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We can now relate U and H to p-V-T data. For instance for U, we can calculate the ΠT value:

STVTT V

U

V

S

S

U

V

U

pT

pT

VT

Generic equation of state!

For an ideal gas: T

p

V

nR

T

p

V

Then: 0

T

T V

U

This proves proves that for an ideal gas U = U(T), that is function of T only !

For a van der Waals gas:

02

mmT V

ap

bV

RT

V

U U=U(T,V)

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The variation of Gibbs Energy with temperatureThe variation of Gibbs Energy with temperature

As we saw previously:

ST

G

p

then

: T

HG

T

G

p

If we derivate now G/T we obtain:

2

111

T

G

T

G

TTTG

T

G

TT

G

T

or:2

1

T

G

T

HG

TT

G

T

And finally:

2T

H

TT

G

p

Gibbs-Helmholtz equation

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Gibbs Helmholtz equation and Chemical Gibbs Helmholtz equation and Chemical ReactionsReactions

Applying the Gibbs Helmholtz equation to a chemical reaction we obtain:

2T

H

TT

G

p

Where ΔH is the reaction enthalpy change.

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The variation of Gibbs Energy with pressureThe variation of Gibbs Energy with pressure

Recalling that Vp

G

T

And upon integration we obtain, at constant T:

VdpdG

f

i

p

p

if dppVpGpGG )(

For liquids and solids the molar volume, Vm, is small and approximately constant, so:

pVpGpG mif ~0

if pGpG

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Ideal GasesIdeal Gases

For a perfect gas (pV = nRT) we obtain: dpp

nRTpGpGf

i

p

p

if 1

or:

i

fif p

pnRTpGpG ln

Considering now that pi = pº (the standard pressure of 1 bar) and calculating the molar Gibbs energy, Gm (n=1), we obtain:

omm p

pRTGpG ln)( 0

The molar Gibbs energy is also called the chemical chemical potentialpotential, μ:

0lnp

pRTo

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Final remarkFinal remark

We have now all the “thermodynamic tools” that allow us to apply the thermodynamic principles to all kind of physical and chemical processes, like chemical equilibrium or phase equilibrium. That will be the syllabus for the next lectures…