ME 083ME 083

Thermodynamic Aside:Thermodynamic Aside:Gibbs Free EnergyGibbs Free Energy

Professor David M. SteppProfessor David M. Stepp

Mechanical Engineering and Materials ScienceMechanical Engineering and Materials Science189 Hudson Annex189 Hudson Annex

david.stepp@duke.edudavid.stepp@duke.edu549-4329 or 660-5325549-4329 or 660-5325

24 February 200324 February 2003

From Last Time….From Last Time….

Gibbs Free Energy: Gibbs Free Energy: G = H – T*SG = H – T*SRecall our Arrhenius relationship for the equilibrium number Recall our Arrhenius relationship for the equilibrium number of vacancies (defects):of vacancies (defects):

Frenkel defects (vacancy plus interstitial defect)Frenkel defects (vacancy plus interstitial defect)

Tk

Q

totvB

v

eNN

Energy of a System:Energy of a System:

Three Categories of Energy:Three Categories of Energy:

State Function Internal Energy:State Function Internal Energy:

A State FunctionA State Function

(Depends only on the current condition of the system)(Depends only on the current condition of the system)

Kinetic (motion), Potential (position), Kinetic (motion), Potential (position),

and INTERNAL (molecular motions)and INTERNAL (molecular motions)

– How can we change a material’s (or system’s) How can we change a material’s (or system’s) internal energy?internal energy?

∆∆U = Q + W + W’U = Q + W + W’

THERMODYNAMIC ASIDE:THERMODYNAMIC ASIDE: GIBBS FREE ENERGY GIBBS FREE ENERGY

State Function Internal Energy (U):State Function Internal Energy (U):

∆∆U = Q + W + W’U = Q + W + W’

(The First Law of Thermodynamics)(The First Law of Thermodynamics)

Q: Heat flow (into system)Q: Heat flow (into system)

W: P-V work (on system)W: P-V work (on system)

W’: Other work (on system)W’: Other work (on system)

Energy ConservationEnergy Conservation

Processes in nature have a natural direction of changeProcesses in nature have a natural direction of change

DOES NOTDOES NOT

SPONTANEOUSSPONTANEOUS OCCUROCCUR

State Function Entropy (S) State Function Entropy (S) ∆S∆SP P ≥ 0≥ 0

The Entropy of a system may increase or decrease The Entropy of a system may increase or decrease during a process.during a process.

The Entropy of the universe, taken as a system plus The Entropy of the universe, taken as a system plus surroundings, can only increase.surroundings, can only increase.

(The Second Law of Thermodynamics)(The Second Law of Thermodynamics)

““Entropy is Time’s Arrow”Entropy is Time’s Arrow”

Note: The laws of thermodynamics are empirical, Note: The laws of thermodynamics are empirical, based on considerable experimental evidence.based on considerable experimental evidence.

One Can Show That:One Can Show That:

For Reversible Processes QFor Reversible Processes QREVREV = TdS = TdS

– QQREVREV denotes the maximum heat absorbed denotes the maximum heat absorbed

– Note the cyclic path integral (reversibility)Note the cyclic path integral (reversibility)

With this, the combined Notation for the First and With this, the combined Notation for the First and Second Law can be expressed:Second Law can be expressed:

ddU =TU =TddS + S + ddWWREVREV + + ddW’W’

BUT BUT dWdWREVREV = F*dx = F*(A/A)*dx = F/A*(-dV) = -PdV = F*dx = F*(A/A)*dx = F/A*(-dV) = -PdV

dU = TdS – PdV + dW’dU = TdS – PdV + dW’ Combined statement ofCombined statement of

First and Second LawsFirst and Second Laws

The Third Law of Thermodynamics:The Third Law of Thermodynamics:

The Entropy of all substances is the same at 0 KThe Entropy of all substances is the same at 0 K

Both Entropy and Temperature Have Absolute Values Both Entropy and Temperature Have Absolute Values (Both have an empirically observed zero point)(Both have an empirically observed zero point)

State Function Enthalpy (H) State Function Enthalpy (H) H = U + P*VH = U + P*V

So: So: dH = dU + PdV + VdPdH = dU + PdV + VdP

Consider the special case where Consider the special case where dP = 0 and dW’ = 0dP = 0 and dW’ = 0::

dHdHPP = TdS = TdSPP = dQ = dQREV,PREV,P

State Function Gibbs Free EnergyState Function Gibbs Free Energy

G = H – TS = U + PV - TS G = H – TS = U + PV - TS

dG = dU + PdV + VdP – TdS – SdTdG = dU + PdV + VdP – TdS – SdT

= (TdS – PdV + dW’) + PdV + VdP – TdS – SdT= (TdS – PdV + dW’) + PdV + VdP – TdS – SdT

= VdP – SdT + dW’= VdP – SdT + dW’

Special Case of dT = 0 and dP = 0: Special Case of dT = 0 and dP = 0: dGdGT,PT,P = dW’ = dW’T,PT,P

Or, if W’ = 0: Or, if W’ = 0: dGP = -SdTdGP = -SdT

At constant Temperature and Pressure, the (change At constant Temperature and Pressure, the (change in) Gibbs Free Energy reflects all non-mechanical in) Gibbs Free Energy reflects all non-mechanical work done on the system.work done on the system.

Back to Crystals…Back to Crystals…

Consider Frenkel Defects (vacancy + interstitial)Consider Frenkel Defects (vacancy + interstitial)

The Free Energy of the Crystal can be written asThe Free Energy of the Crystal can be written as– the Free Energy of the perfect crystal (Gthe Free Energy of the perfect crystal (G00) )

– plus the free energy change necessary to create n plus the free energy change necessary to create n interstitials and vacancies (n*interstitials and vacancies (n*∆g)∆g)

– minus the entropy increase that derives from the different minus the entropy increase that derives from the different possible ways in which defects can be arranged (∆Spossible ways in which defects can be arranged (∆SCC))

∆∆G = G-GG = G-G00 = n*∆g - T∆S = n*∆g - T∆SCC

n = the number of defectsn = the number of defects

∆∆g = g = ∆∆h - T∆S (the energy to create defects)h - T∆S (the energy to create defects)

The Configurational Entropy, The Configurational Entropy, ∆S∆SCC, is proportional to , is proportional to

the number of ways in which the defects can be the number of ways in which the defects can be arranged arranged (W).(W).

∆∆SSCC = k = kBB* ln(W)* ln(W)

(The Boltzmann Equation)(The Boltzmann Equation)

Note: this constitutes a connection between atomistic Note: this constitutes a connection between atomistic and phenomenological descriptions (thru statistics).and phenomenological descriptions (thru statistics).

Perfect Crystal:Perfect Crystal:

N atoms which are indistinguishable can only be N atoms which are indistinguishable can only be placed in one way on the N lattice sites:placed in one way on the N lattice sites:

∆∆SSCC = k*ln(N!/N!) = k*ln(1) = 0 = k*ln(N!/N!) = k*ln(1) = 0

Crystal with Vacancies and InterstitialsCrystal with Vacancies and Interstitials

With N normal lattice sites (and equal interstitial sites),With N normal lattice sites (and equal interstitial sites),

nnii (number of interstitial atoms) can be arranged in: (number of interstitial atoms) can be arranged in:

distinct waysdistinct ways

Similarly, the vacant sites can be arranged in:Similarly, the vacant sites can be arranged in:

distinct waysdistinct ways

!)!(

!

ii nnN

N

!n)!nN(

!N

vv

Recalling that the equilibrium number of vacancies Recalling that the equilibrium number of vacancies obeys an Arrhenius relationship:obeys an Arrhenius relationship:

Remember: Remember: ∆g = ∆h - T∆s∆g = ∆h - T∆s

Therefore:Therefore: Assuming configurational Assuming configurational entropy entropy in in creatingcreating defects is defects is negligiblenegligible

Now compare this with our Now compare this with our equation for Frenkel Defects:equation for Frenkel Defects:

kT

hks

kTg

eeeN

n 222

kT

h

eN

n 2

kTQ

T

I

T

VIsV

eN

N

N

N 2/

The Configurational Entropy of these non-The Configurational Entropy of these non-interacting defects is:interacting defects is:

Probability theory for statistically independent events: Probability theory for statistically independent events:

PPABAB = P = PAA * P * PBB

Stirling’s Approximation for Large Numbers:Stirling’s Approximation for Large Numbers:

ln(x!) ln(x!) ≈≈ x * ln(x) - x x * ln(x) - x

Recalling that nRecalling that nii = n = nvv = n (for Frenkel defects): = n (for Frenkel defects):

!!

!

!!

!ln

vviiC nnN

N

nnN

NkS

n

nNn

nN

NNkSC lnln2

At equilibrium, the free energy is a minimum with At equilibrium, the free energy is a minimum with respect to the number of defectsrespect to the number of defects

Note: Entropy (per defect) is a maximum at this pointNote: Entropy (per defect) is a maximum at this point

Recall: Recall: ∆G = G – G∆G = G – G00 = n * ∆g – T∆S = n * ∆g – T∆SCC

G = GG = G00 + n * (∆h – T∆s) - T∆S + n * (∆h – T∆s) - T∆SCC

Next Time…Next Time…

The Statistical Interpretation of EntropyThe Statistical Interpretation of Entropy

∆∆SSCC = k = k * ln(W)* ln(W)

Configurational Entropy is proportional to the number Configurational Entropy is proportional to the number of ways in which defects can be arrangedof ways in which defects can be arranged