❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦

■❋❙❈✴❯❙P

❆✉❧❛ ✵✼

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶ ✴ ✷✻

❖✉t❧✐♥❡

✶ ❈♦♥❞✐çã♦ P❛r❛ ♦ ❊q✉✐❧í❜r✐♦

✷ ❋✉♥çã♦ ❞❡ P❛rt✐çã♦

✸ ❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ ❚❡♠♣❡r❛t✉r❛

✹ ❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ Pr❡ssã♦

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷ ✴ ✷✻

❈♦♥❞✐çã♦ P❛r❛ ♦ ❊q✉✐❧í❜r✐♦

■♥tr♦❞✉çã♦

❈♦♥s✐❞❡r❡ ♦ ❡q✉✐❧í❜r✐♦ ❡♥tr❡ ❛s ❡s♣é❝✐❡s ❆ ❡ ❇ ❞❛❞♦ ♣♦r✿

Ak−→ B ✭✶✮

❆ ❝♦♥st❛♥t❡ ❞❡ ❡q✉✐❧í❜r✐♦ é ❞❛❞❛ ♣❡❧❛ r❛③ã♦ ❡♥tr❡ ❛s ❝♦♥❝❡♥tr❛çõ❡s ♥♦❡q✉✐❧í❜r✐♦✿

K =[B]

[A]✭✷✮

❆ q✉❛♥t✐❞❛❞❡ q✉❡ ♣r❡❞✐③ ♦ ❡q✉✐❧í❜r✐♦ é ♦ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦✳ ❊♠t❡♠♣❡r❛t✉r❛ ❡ ♣r❡ssã♦ ✜①❛✱ ❛ ❢✉♥çã♦ ❡①tr❡♠❛ é ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❞❡ ●✐❜❜s✿

dG = −SdT + Vdp + µAdNA + µBdNB ✭✸✮

♦♥❞❡ NA ❡ NB sã♦ ♦s ♥ú♠❡r♦s ❞❡ ♣❛rtí❝✉❧❛s ♥♦s ❡st❛❞♦s A ❡ B ❡ µA ❡ µBsã♦ s❡✉s ♣♦t❡♥❝✐❛✐s q✉í♠✐❝♦s✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✸ ✴ ✷✻

❈♦♥❞✐çã♦ P❛r❛ ♦ ❊q✉✐❧í❜r✐♦

■♥tr♦❞✉çã♦

dG = −SdT + Vdp + µAdNA + µBdNB ✭✹✮

❚❡♥❞♦ ♣ ❡ ❚ ✜①♦s✱ ❛ ❝♦♥❞✐çã♦ ♣❛r❛ ♦ ❡q✉✐❧í❜r✐♦ s❡rá✿

dG = µAdNA + µBdNB = ✵ ✭✺✮

❙❡ ❛s ♠♦❧é❝✉❧❛s ❡stã♦ ♥♦s ❡st❛❞♦s ❆ ♦✉ ❇✱ Ntot é ❝♦♥st❛♥t❡✿

NA + NB = Ntot = cte ⇒ dNA + dNB = ✵ ⇒ dNA = −dNB ✭✻✮

❊ ♣♦❞❡♠♦s r❡s❝r❡✈❡r ❛ ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ ❝♦♠♦✿

(µA − µB)dNA = ✵ ✭✼✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✹ ✴ ✷✻

❈♦♥❞✐çã♦ P❛r❛ ♦ ❊q✉✐❧í❜r✐♦

■♥tr♦❞✉çã♦

(µA − µB)dNA = ✵ ✭✽✮

❯♠❛ ✈❡③ q✉❡ dNA ❞❡✈❡ s❡r ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱ (µA − µB) ❞❡✈❡ s❡r ✐❣✉❛❧ ❛③❡r♦✱ ♦✉✿

µA = µB ✭✾✮

✐✳❡✳✱ ❛ ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ é q✉❡ ♦s ♣♦t❡♥❝✐❛✐s q✉í♠✐❝♦s ❞❛s ❡s♣é❝✐❡s A ❡B s❡❥❛♠ ✐❣✉❛✐s✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✺ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❖ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ s❡ r❡❧❛❝✐♦♥❛ ❝♦♠ ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦ ❛tr❛✈és ❞❛r❡❧❛çã♦✿

µ =

(

∂F

∂N

)

V ,T

✭✶✵✮

❙❡F = −kT lnQ ✭✶✶✮

✱❧♦❣♦✿∂F

∂N= −kT

(

∂lnQ

∂N

)

✭✶✷✮

P❛r❛ NA ♣❛rtí❝✉❧❛s ❞♦ t✐♣♦ A✱ t❡♠♦s q✉❡ QA = qNAA /NA!✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✻ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❯s❛♥❞♦ ❛ ❛♣r♦①✐♠❛çã♦ ❞❡ ❙t✐r❧✐♥❣✱ N! ≈ (N/e)N ✿

QA =qNAA

NA!=

(

eqA

NA

)NA

✭✶✸✮

❡

lnQA = NA ln

(

eqA

NA

)

✭✶✹✮

❊ ✜♥❛❧♠❡♥t❡∂ lnQA∂NA

= ln

(

qA

NA

)

✭✶✺✮

❱♦❧t❛♥❞♦ ❛♦ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦✿

µA = −kT ln

(

qA

NA

)

µB = −kT ln

(

qB

NB

)

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✼ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

P❡❧❛ ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦ µA = µB ✿

qA

NA=

qB

NB✭✶✻✮

❙❡ K = NB/NA✿

K =NB

NA=

qB

qA=

(

q′Bq′A

)

e−(ǫ✵B−ǫ✵A)/kT ✭✶✼✮

❝♦♠

q′ = eǫ✵/kTq = ✶+ e−(ǫ✶−ǫ✵)/kT + e−(ǫ✷−ǫ✵)/kT + ...+ e−(ǫt−ǫ✵)/kT ✭✶✽✮

❆ss✉♠✐♥❞♦ q✉❡ ♥ã♦ ❤á ✐♥t❡r❛çã♦ ❡♥tr❡ ❛s ♣❛rtí❝✉❧❛s✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r ❛❝♦♥st❛♥t❡ ❞❡ ❡q✉✐❧í❜r✐♦ ❑ ❛ ♣❛rt✐r ❞❛s ♣r♦♣r✐❡❞❛❞❡s ❛tô♠✐❝❛s ❞❡ ❆ ❡ ❇✉s❛♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ♣❛rt✐çã♦✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✽ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❱❛r✐❛♥❞♦ ❛ ❡st❡q✉✐♦♠❡tr✐❛

❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❛ r❡❛çã♦

aA+ bBk−→ cC ✭✶✾✮

❝♦♠ a✱ b ❡ c ✐♥❞✐❝❛♥❞♦ ❛s ❡st❡q✉✐♦♠❡tr✐❛s ❞❛s ❡s♣é❝✐❡s✳ ❈♦♠ T ❡ p❝♦♥st❛♥t❡✱ t❡♠♦s ❛ ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✿

dG = µAdNA = µBdNB + µCdNC = ✵ ✭✷✵✮

❖ ❡q✉✐❧í❜r✐♦ ❡stá s✉❥❡✐t♦ à ❝♦♥❞✐çã♦ ❞❡ r❡str✐çã♦✿

dNC = (c)dξ dNA = −(a)dξ dNB = −(b)dξ

♦♥❞❡ ξ é ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ♣r♦❣r❡ss♦ ♦✉ ♠❡❞✐❞❛ ❞❛ r❡❛çã♦✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✾ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❱❛r✐❛♥❞♦ ❛ ❡st❡q✉✐♦♠❡tr✐❛

❱♦❧t❛♥❞♦ à ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✿

(cµC − aµA − bµB)dξ = ✵ ✭✷✶✮

◆♦ ❡q✉✐❧í❜r✐♦✿

cµC = aµA + bµB ✭✷✷✮

❙✉❜st✐t✉✐♥❞♦ (µ) ♣♦r [−kT ln(q/N)]✱ t❡♠♦s✿

c

[

−kT ln

(

qC

NC

)]

= a

[

−kT ln

(

qA

NA

)]

+ b

[

−kT ln

(

qB

NB

)]

✭✷✸✮

⇒

(

qC

NC

)c

=

(

qA

NA

)a (qB

NB

)b

✭✷✹✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✵ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❱❛r✐❛♥❞♦ ❛ ❡st❡q✉✐♦♠❡tr✐❛

❘❡❛rr❛♥❥❛♥❞♦✿

(

(qC )c

(qA)a(qB)b

)

=NcC

NaANbB

= K ✭✷✺✮

❖✉✱ ❡s❝r❡✈❡♥❞♦ ❞❡ ♦✉tr❛ ❢♦r♠❛✿

K =

(

q′cCq′aA q

′bB

)

e−(cǫ✵C−aǫ✵A−bǫ✵B)/kT ✭✷✻✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✶ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊①❡♠♣❧♦

❯♠ r❡❝❡♣t♦r ❘ s❡ ❛ss♦❝✐❛ ❛ ✉♠ ❧✐❣❛♥t❡ ▲ ♣❛r❛ ❢♦r♠❛r ✉♠ ❝♦♠♣❧❡①♦ ❈✿

R + Lk−→ C ✭✷✼✮

❍❛✈❡♥❞♦ NR ✱ NL ❡ NC ♠♦❧é❝✉❧❛s ❡♠ ✉♠ ✈♦❧✉♠❡ V ✭❜❛✐①❛ ❝♦♥❝❡♥tr❛çã♦✮✱✈❛♠♦s ❞❡t❡r♠✐♥❛r ❛ ❝♦♥st❛♥t❡ ❞❡ ❛✜♥✐❞❛❞❡ ❡♠ ❢✉♥çã♦ ❞❛s ❢✉♥çõ❡s ❞❡♣❛rt✐çã♦ s❛❜❡♥❞♦ q✉❡ ❞❛ ✐♥t❡r❛çã♦ r❡s✉❧t❛ ✉♠❛ ❡♥❡r❣✐❛ ǫ✵ < ✵✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✷ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

P❡rt✉r❜❛çã♦ ❞♦ ❊q✉✐❧í❜r✐♦

❙❡ ♦ ❡st❛❞♦ ❡q✉✐❧í❜r✐♦ é ❞❡✜♥✐❞♦ ♣❡❧♦ ♠í♥✐♠♦ ❞❡ ❡♥❡r❣✐❛ ❧✐✈r❡✱ ♣❡rt✉r❜❛çõ❡s❞♦ ❡q✉✐❧í❜r✐♦ r❡s✉❧t❛♠ ❡♠ ❛✉♠❡♥t♦ ♥❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❞♦ s✐st❡♠❛✳ ❆ r❡s♣♦st❛❞♦ s✐st❡♠❛ à ♣❡rt✉r❜❛çã♦ é ♦ r❡t♦r♥♦ ❛♦ ❡st❛❞♦ ❞❡ ❡q✉✐❧í❜r✐♦✳

AK−→ B ✭✷✽✮

dG = (µB − µA)dξ ✭✷✾✮

❝♦♠ ξ s❡♥❞♦ ❛ ❝♦♦r❞❡♥❛❞❛ ❞❡ r❡❛çã♦✳ P❛r❛ r❡t♦r♥❛r ❛♦ ❡q✉✐❧í❜r✐♦✱ dG ≤ ✵✱✐♠♣❧✐❝❛♥❞♦ q✉❡ (µB − µA) ❡ dξ tê♠ q✉❡ t❡r s✐♥❛✐s ♦♣♦st♦s✳ ❙❡ ❛♣❡rt✉r❜❛çã♦ ❛✉♠❡♥t❛ ♦ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ❞❡ ❇ ✭µB > µA✮✱ ❛ ❞✐r❡çã♦ ♣❛r❛♦ ❡q✉✐❧í❜r✐♦ é dξ < ✵✱ ♦✉ s❡❥❛✱ r❡❞✉③✐♥❞♦ NB ✳

P♦t❡♥❝✐❛❧ q✉í♠✐❝♦ → t❡♥❞ê♥❝✐❛ ❞❡ ❡s❝❛♣❡✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✸ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

Pr✐♥❝í♣✐♦ ❞❡ ▲❡ ❈❤❛t❡❧✐❡r

P♦t❡♥❝✐❛❧ q✉í♠✐❝♦ → t❡♥❞ê♥❝✐❛ ❞❡ ❡s❝❛♣❡✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✹ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

❆ ❞❡♣❡♥❞ê♥❝✐❛ ❞❛ ❝♦♥st❛♥t❡ ❞❡ ❡q✉✐❧í❜r✐♦ K (T ) ❝♦♠ ❛ t❡♠♣❡r❛t✉r❛ ♣♦❞❡❡♥s✐♥❛r s♦❜r❡ ❛ ❡♥t❛❧♣✐❛✴❡♥tr♦♣✐❛ ❞❛ r❡❛çã♦✳ ❈♦♠♦ ❡①❡♠♣❧♦✱ ♣❛r❛ ❡q✉✐❧í❜r✐♦❞❡ ❞♦✐s ❡st❛❞♦s t❡♠♦s✿

AK−→ B ✭✸✵✮

◆♦ ❡q✉✐❧í❜r✐♦✱ µA = µB ❡ K = NB/NA✳ ❯s❛♥❞♦ N = pV /kT ✿

K =NB

NA=

pBVkT

kTpAV=

pB

pA✭✸✶✮

❏á ♦ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ é ❞❛❞♦ ♣♦r✿

µ = −kT ln( q

N

)

✭✸✷✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✺ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

❙❡ ❡s❝r❡✈❡r♠♦s q ❡♠ ❢✉♥çã♦ ❡①♣❧✐❝✐t❛♠❡♥t❡ ❞♦ ✈♦❧✉♠❡ q = q✵V ✱ ❡❛s✉♠✐♥❞♦ V = NkT/p ✭❣❛s ✐❞❡❛❧✮✿

q

N=

q✵NkT

Np=

q✵kT

p✭✸✸✮

µ = −kT ln

(

qokT

p

)

= −kT ln(q✵kT ) + kT ln p ✭✸✹✮

❙❡ ❝❤❛♠❛r♠♦s −kT ln(q✵kT ) ❞❡ µ✵✿

µ = µ✵ + kT ln p ✭✸✺✮

♦♥❞❡ µ✵ é ♦ ♣♦t❡♥❝✐❛❧ q✉í♠✐❝♦ ♥♦ ❡st❛❞♦ ♣❛❞rã♦✱ ❞❡❝♦♠♣♦st♦ ♥❛ ♣❛rt❡ q✉❡❞❡♣❡♥❞❡ ❞❛ ♣r❡ssã♦ ❡ ♥❛ ♣❛rt❡ q✉❡ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ♣r❡ssã♦ ✭µ✵✮✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✻ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

❱♦❧t❛♥❞♦ à ❝♦♥❞✐çã♦ ❞❡ ❡q✉✐❧í❜r✐♦✿

µA = µB ✭✸✻✮

µ✵A + kT ln pA = µ✵B + kT ln pB ✭✸✼✮

❘❡❛rr❛♥❥❛♥❞♦✿

lnKp = ln

(

pB

pA

)

=−(µ✵B − µ

✵A)

kT= −

∆µ✵

kT✭✸✽✮

❈♦♠♦

µj =

(

∂G

∂Nj

)

T ,p,Ni 6=j

G = H − TS

❧♦❣♦✿

µj =

(

∂H

∂Nj

)

T ,p,Ni 6=j

− T

(

∂S

∂Nj

)

T ,p,Ni 6=j

= hj − Tsj ✭✸✾✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✼ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

µj =

(

∂H

∂Nj

)

T ,p,Ni 6=j

− T

(

∂S

∂Nj

)

T ,p,Ni 6=j

= hj − Tsj ✭✹✵✮

❆q✉✐✱ h ❡ s sã♦ ❡♥t❛❧♣✐❛ ❡ ❡♥tr♦♣✐❛ ♠♦❧❛r❡s ♣❛r❝✐❛✐s✳P♦❞❡♠♦s r❡s❝r❡✈❡r ∆µ ❡♠ ❢✉♥çã♦ ❞❡st❛s ❣r❛♥❞❡③❛s✿

∆µ✵ = ∆h✵ − T∆s✵ ✭✹✶✮

◆♦✈❛♠❡♥t❡✱ h ❡ s ✐♥❞✐❝❛♠ ❛ ❡♥t❛❧♣✐❛ ❡ ❡♥tr♦♣✐❛ ♣♦r ♠♦❧é❝✉❧❛ ♦✉ ♣♦r ♠♦❧✱❞❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r ❛ µ✱ q✉❡ ❡①♣r❡ss❛ ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ♣♦r ♠♦❧é❝✉❧❛ ♦✉ ♠♦❧✳ ❖sí♠❜♦❧♦ ✵ ✐♥❞✐❝❛ ❡q✉✐❧í❜r✐♦ ❛ p = ✶ ❛t♠✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✽ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

❱♦❧t❛♥❞♦ ❛♦ ❡q✉✐❧í❜r✐♦✿

lnKp = −∆µ✵

kT= −

∆h✵ − T∆s✵

kT✭✹✷✮

❉❡r✐✈❛♥❞♦ ♦s ❞♦✐s ❧❛❞♦s ❞❛ ❡q✉❛çã♦ ❝♦♠ r❡❧❛çã♦ ❛ ❚✱ t❡♠♦s✿

(

∂ lnKp∂T

)

= −∂

∂T

(

∆h✵ − T∆s✵

kT

)

✭✹✸✮

❙❡ ∆h✵ ❡ ∆s✵ sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s ❞❛ t❡♠♣❡r❛t✉r❛✱ ❛ ❡q✉❛çã♦ ♣♦❞❡ s❡rs✐♠♣❧✐✜❝❛❞❛ ♣❛r❛✿

(

∂ lnKp∂T

)

=∆h✵

kT ✷✭✹✹✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✶✾ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

∆h✵ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❛ ♣❛rt✐r ❞❛ ❘❡❧❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛✳ ❈♦♠♦d(✶/T ) = −(✶/T ✷)dT ✿

(

∂ lnKp∂(✶/T )

)

= −∆h✵

k✭✹✺✮

❙❡ r❡♣r❡s❡♥t❛r♠♦s ❣r❛✜❝❛♠❡♥t❡ lnK ✈❡rs✉s (✶/T )✱ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛s❡rá −∆h✵/k ✳

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✵ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛

❆ r❡❧❛çã♦ ❧✐♥❡❛r ❝♦♥✜r♠❛ q✉❡ ∆h✵ ❞❡✈❡ s❡r ✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ t❡♠♣❡r❛t✉r❛✳◆❡st❡ ❝❛s♦✱ ❛ ✐♥t❡❣r❛çã♦ ❞❛ ❡q✉❛çã♦ ❛♥t❡r✐♦r r❡s✉❧t❛ ❡♠✿

ln

[

Kp(T✷)

Kp(T✶)

]

=−∆h✵

k

(

✶

T✷−

✶

T✶

)

✭✹✻✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✶ ✴ ✷✻

❋✉♥çã♦ ❞❡ P❛rt✐çã♦

❊q✉❛çã♦ ❞❡ ✈❛♥✬t ❍♦✛ ✲ ❊①❡♠♣❧♦

ln

[

Kp(T✷)

Kp(T✶)

]

=−∆h✵

k

(

✶

T✷−

✶

T✶

)

✭✹✼✮

T✶ = ✶✺✵✵K ⇒ ✶/T✶ = ✻.✻✻× ✶✵−✹

⇒ lnK = −✶✸.✶✹✼

T✷ = ✷✷✺✼K ⇒ ✶/T✷ = ✹.✹✸× ✶✵−✹

⇒ lnK = −✻.✹

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✷ ✴ ✷✻

❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ ❚❡♠♣❡r❛t✉r❛

❊q✉❛çã♦ ❞❡ ●✐❜❜s✲❍❡❧♠❤♦t③

❈♦♠♦ ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ●✭❚✮ ❞❡♣❡♥❞❡ ❞❛ t❡♠♣❡r❛t✉r❛❄

G = H − TS ⇒ H = G + TS ✭✹✽✮

❯s❛♥❞♦ S = −(∂G/∂T )p✱ t❡♠♦s✿

H = G − T

(

∂G

∂T

)

p

✭✹✾✮

❱❛♠♦s ❛♥❛❧✐s❛r ❛ r❡❧❛çã♦✿

(

∂(G/T )

∂T

)

p

=✶

T

(

∂G

∂T

)

p

−G

T ✷= −

✶

T ✷

[

G − T

(

∂G

∂T

)

p

]

✭✺✵✮

❙✉❜s✐t✉✐♥❞♦ ❛ r❡❧❛çã♦ ❞❡ ❍ ♥❛ ❡q✉❛çã♦ ❛❝✐♠❛ ✭t❡r♠♦ ❡♠ ❝♦❧❝❤❡t❡s✮✿(

∂(G/T )

∂T

)

p

= −H(T )

T ✷✭✺✶✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✸ ✴ ✷✻

❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ ❚❡♠♣❡r❛t✉r❛

❊q✉❛çã♦ ❞❡ ●✐❜❜s✲❍❡❧♠❤♦t③

❊q✉❛çã♦ ❞❡ ●✐❜❜s✲❍❡❧♠❤♦t③

(

∂(G/T )

∂T

)

p

= −H(T )

T ✷✭✺✷✮

❉❡ ♠❛♥❡✐r❛ s✐♠✐❧❛r✱ ♣❛r❛ ❛ ❡♥❡r❣✐❛ ❧✐✈r❡ ❞❡ ❍❡❧♠❤♦t③✱ t❡♠♦s✿

(

∂(F/T )

∂T

)

V

= −U(T )

T ✷✭✺✸✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✹ ✴ ✷✻

❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ Pr❡ssã♦

❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ Pr❡ssã♦

❆ ❛♣❧✐❝❛çã♦ ❞❡ ♣r❡ssã♦ ♣♦❞❡ ❞❡s❧♦❝❛r ♦ ❡q✉✐❧í❜r✐♦✳ P❛r❛ ♦ ❡q✉✐❧í❜r✐♦ AK−→ B ✿

∂ lnK (p)

∂p=

∂

∂p

[

−(µ✵B − µ

✵A)

kT

]

= −✶

kT

(

∂∆µ✵

∂p

)

✭✺✹✮

P❛r❛ ❣❛s❡s ✐❞❡❛✐s✱ dK/dp é ③❡r♦✳ P❛r❛ ♦✉tr♦s s✐st❡♠❛s µ ❞❡♣❡♥❞❡ ❞❡ p✳(

∂µ

∂p

)

T

=

(

∂V

∂N

)

T ,p

= υ ✭✺✺✮

❖♥❞❡ υ é ♦ ✈♦❧✉♠❡ ♣♦r ♠♦❧é❝✉❧❛ ♦✉ ♠♦❧✳(

∂(µ✵B − µ✵A)

∂p

)

T

= υ✵B − υ✵A = ∆υ

✵ ✭✺✻✮

❙✉❜st✐t✉✐♥❞♦✿(

∂ lnK (p)

∂p

)

T

= −∆υ✵

kT✭✺✼✮

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✺ ✴ ✷✻

❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ Pr❡ssã♦

❊q✉✐❧í❜r✐♦ ❉❡♣❡♥❞❡♥t❡ ❞❛ Pr❡ssã♦ ✲ ❊①❡♠♣❧♦

(

∂ lnK (p)

∂p

)

T

= −∆υ✵

kT✭✺✽✮

❙❡ B ❡stá ♥✉♠ ❡st❛❞♦ ❞❡ ✈♦❧✉♠❡ ♠❡♥♦r✱ υB − υA < ✵✱ ❧♦❣♦ ♦ ❛✉♠❡♥t♦ ❞❛♣r❡ssã♦ ✈❛✐ ❞❡s❧♦❝❛r ♦ ❡q✉✐❧í❜r✐♦ ❞❡ ❆ ♣❛r❛ ❇✳

❍❛❧♦t❛♥♦✿ ▼❛✐s s♦❧ú✈❡❧ ❡♠ ♠❡✐♦s ❤✐❞r♦❢ó❜✐❝♦s q✉❡ ❡♠ á❣✉❛✳

♣✭❛t♠✮ lnK

✵ ✼✳✽✹

✷✽✵ ✼✳✻

❆❧❡ss❛♥❞r♦ ❙✳ ◆❛s❝✐♠❡♥t♦ ✭■❋❙❈✴❯❙P✮ ❊q✉✐❧í❜r✐♦ ◗✉í♠✐❝♦ ❆✉❧❛ ✵✼ ✷✻ ✴ ✷✻

Condição Para o EquilíbrioFunção de PartiçãoEquilíbrio Dependente da TemperaturaEquilíbrio Dependente da Pressão