Fundamental Equations of Thermodynamics

Fundamental Equations of Thermodynamics

한양대학교

4.1 Fundamental Equation for the Internal Energy

The entropy is a criterion of weather a change in an isolated system is spontaneous.

What would be the criterion at constant T and V or constant T and P?

The first law of thermodynamics dU =dq+dw

The second law of thermodynamics d

d q

S ≥ T

For a closed system with only reversible PV work dw = −P Vd and dq =T Sd dU =T Sd −P Vd

In 1876, Gibbs added the chemical potential µ_iterms.

1 1 2 2

1

d d d d d d d d

Ns

i i i

U T S P V µ n µ n T S P V µ n

=

= − + + +^= − +

∑

The chemical potential µ_i is a measure of the potential a species has to move from one phase to another or undergo a chemical reaction.

The total differential of U

한양대학교

1

d d d d

Ns

i i i

U T S P V µ n

=

= − +

∑

U(S, V, {n_i})

,{ } ,{ } 1 , ,

d d d d

s

i i j i

N

i

V n S n i i S V n n

U U U

U S V n

S V ₌ n _≠

 

∂ ∂ ∂

   

= ∂  + ∂  +

∑

The equation of state

,{ } ,{ } , ,

i i j i

i

V n S n i S V n n

U U U

T P

S V µ n

≠

 

∂ ∂ ∂

   

= ∂  = −∂  = ∂ 

The Maxwell relations

,{ } ,{ } , , ,{ }

, , ,{ } , , , ,

i i j i i

j i i i j j i

i

S n V n i S V n n V n

i i j

i S V n n S n j S V n n i S V n n

T P T

V S n S

P

n V n n

µ

µ µ µ

≠

≠ ≠ ≠

  ∂

∂ ∂ ∂  

  = −  =

∂   ∂  ∂   ∂ 

       

  ∂

 ∂  ∂  ∂  

−_{ } =_{ } _ _ =_{ }

∂  ∂  ∂ ∂

     

Spontaneity Criterion with the Internal Energy

ext

1

d d and d d d

Ns

i i i

q T S w P V µ n

=

≤ = − +

∑

ext

1

d d d d d d

Ns

i i i

U q w T S P V µ n

=

= + ≤ − +

∑

At constant S, V, and {n_i}, U must be at a minimum. (d ) , ,{ } 0

S V ni

U ≤

1

d d d d

Ns

i i i

U T S P V µ n

=

= − +

∑

At constant T, P, and µ_i

1

Ns

i i i

U TS PV µ n

=

= − +

∑

Euler’s theorem

k=1 ( ,^{1 2}, , ) ¹

j i

N

N i

i i x x

k f x x x x f

= x ≠

 ∂ 

=

∑



4.2 Legendre Transformation

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A linear change in variables to define a new function from a mathematical function by subtracting one or more products of conjugate variables.

1

1

Define

d d d d

d d d d d d d

s

s

N

i i i

N

i i i

U T S P V n

H U P V V P T S V P n H U PV

µ

µ

=

=

= − +

= + +

=

+ +

+

=

∑

∑

The equation of state

,{ } ,{ } , ,

i i j i

i

P n T n i P T n n

H H H

T V

S P µ n

≠

 

∂ ∂ ∂

   

= ∂  = ∂  =∂ 

At constant S, P, and {n_i}, H must be at a minimum. (d ) _{, ,{ }} 0

S P ni

H ≤

At constant T, P, and µ_i

1

Ns

i i i

H TS µ n

=

= +

∑

Legendre Transformation U → A

1

1

Define

d d d d

d d d d d d d

s

s

N

i i i

N

i i i

U T S P V n

A U T S S T S T P V n A U TS

µ

µ

=

=

= − +

= − −

=

− +

−

= −

∑

∑

The equation of state

,{ } ,{ } , ,

i i j i

i

V n T n i T V n n

A A A

S P

T V µ n

≠

 

∂ ∂ ∂

   

= −∂  = −∂  =∂ 

At constant T, V, and {n_i}, A must be at a minimum. (d ) _{, ,{ }} 0

T V ni

A ≤

At constant T, P, and µ_i

1

Ns

i i i

A PV µ n

=

= − +

∑

Legendre Transformation U → G

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1

1

Define

d d d d

d d d d d d d d d

s

s

N

i i i

N

i i i

U T S P V n

G U P V V P T S S T G U PV TS H T

S T V P S

n µ

µ

=

=

= − +

= + + − − = −

=

+ +

+ − = −

∑

∑

The equation of state

,{ } ,{ } , ,

i i j i

i

P n T n i T P n n

G G G

S V

T P µ n

≠

 

∂ ∂ ∂

   

= −∂  = ∂  =∂ 

At constant T, P, and {n_i}, A must be at a minimum. (d ) _{, ,{ }} 0

T P ni

G ≤

At constant T, P, and µ_i

1

Ns

i i i

G µ n

=

=

∑

i Gi

µ =

Thermodynamic relations based on the chemical potential

G(T, P, {n_i}) If a thermodynamic potential is known as a function of its natural variables, all of the thermodynamic functions can be calculated.

At constant T and P, µ_i =Gⁱ

and

P T

T P

P

T

G G

S V

T P

G G

U G PV TS G P T

P T

H G TS G T G T A G PV G P G

P

∂ ∂

   

= − ∂  = ∂ 

∂ ∂

   

= − + = −  ∂  −  ∂ 

∂ 

= + = − ∂ 

∂ 

= − = −  ∂ 

Gibbs Energy

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A criterion for spontaneous change at constant T and P without considering the surroundings

G = H – TS

The Gibbs Free Energy ∆_{G < 0}

∆_{G = 0}

∆_{G > 0}

∆_{S > 0}

∆_{S = 0}

∆_{S < 0}

Spontaneous Equilibrium Impossible

universe system surroundings

system surroundings

system universe system

universe system system system

S S S

S H

T

S S H

T

T S T S H G

∆ = ∆ + ∆

∆ = −∆

∆ = ∆ − ∆

∆ = ∆ − ∆ ≡ −∆

Works other than the PV work

 Surface work

 Elongation work

dw=γdAS

dw= f Ld

s

S 1

S 1

, ,{ },

s , ,{ },

d d d d d d

d d d d d d

s

s

i

i

N

i i i

N

i i i

T P n A

T P n L

U T S P V n f L A

G S T V P n f L A

f G

L G A

µ γ

µ γ

γ

=

=

= − + + +

= − + + + +

∂ 

=  ∂ 

 ∂ 

= ∂ 

∑

∑

Maximum Work

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d d d

d d d

d( ) d

(d ) d (d ) d

T

T

U q w

U T S w

U TS w

A w

A w

= +

− + ≥ −

− − ≥ −

− ≥ −

≤

The decrease in A is an upper bound on the total work done in the surroundings

ext nonpv

ext nonpv

nonpv

, nonpv

, nonpv

d d d d d d

d d d d

d( ) d

(d ) d

(d ) d

T P

T P

U q w q P V w

U P V T S w

U PV TS w

G w

G w

= + = − +

− − + ≥ −

− + − ≥ −

− ≥ −

≤

The decrease in G is an upper bound on the non-PV work done on the surroundings

4.3 Temperature Effect on the Gibbs Energy

,{ }

= < 0

P ni

G S

T

∂  −

 ∂ 

 

,{ }_i P n

G H TS H T G T

∂ 

= − = +  ∂ 

2

,{ } ,{ }

/ 1

i i

P n P n

G T G G

T T T T

∂ ∂

  = − +  

 ∂  ∂

   

2

,{ }

/

P ni

H T G T

T

∂ 

= −  ∂ 

2

,{ }

( / )

P ni

H T G T

T

∂ ∆ 

∆ = −  ∂ 

the Gibbs–Helmholtz equation

4.4 Pressure Effect on the Gibbs Energy

한양대학교

,{ }

0

T ni

G V

P

∂  =

 ∂ 

 >



2 2

1 1

2 1

2 1

d d

d

G P

G P

P P

G V P

G G V P

=

= +

∫ ∫

∫

If the volume is nearly independent of pressure, as it is for a liquid or solid, G₂ =G₁ +V P( ₂ −P₁) For an ideal gas, nRT

V = P

2

2 1

1

d d ln

ln

ln

G P

G P

G nRT P

G G nRT P P G G G nRT P

P

=

= +

∆ = − =

∫

∫

 

Dependence of the Gibbs energy of formation of an ideal gas on the pressure of the gas

Pressure Effect on the Gibbs Energy

Example 4.1 Derive the molar thermodynamic properties for an ideal gas.

ln P G G RT

= ^ + P_

P

T

T P

P

S G

T V G

P

G G

U G PV TS G P T

P T

H G TS G T G T A G PV G P G

P

∂ 

= −∂ 

∂ 

=  ∂ 

∂ ∂

   

= − + = −  ∂  − ∂ 

∂ 

= + = −  ∂ 

∂ 

= − = −  ∂ 

ln

ln

P

P G

S S nR S

P T V RT

P

U U G T S RT H RT H H G T S

A A RT P P

∂ 

= − = − ∂ 

=

= = + − = −

= = +

= +

  



   

  





Pressure Effect on the Gibbs Energy

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Example 4.2 An ideal gas at 27℃ expands isothermally and reversibly from 10 to 1 bar.

Calculate q per mole and w per mole and ∆U, , , , .∆H ∆G ∆A and ∆S

1 1 1

2 1

1 2

(8.314 J K mol )(300.15 K) ln10 5746 J mol

lnV ln P 1

w RT RT

V P

− − −

− = −

= − = − =

5746 J mol 1

A w − ⁻

∆ = = 0

∆ =U

0 5746 5746 J mol 1

q = ∆ − =U w + = ⁻

( ) 0 0 0

H U PV

∆ = ∆ + ∆ = + =

1 1 1

1 10

(8.314 J K mol )(300.15 K) ln 1 5746 J mol 10

d ln 1

G V P RT 10 ^{− −} = − ⁻

∆ =

∫

1

1 1

rev 5746 J mol

19.14 J K mol 300.15 K

S q T

− = − −

∆ = =

1

1 1

0 (5746 J mol )

19.14 J K mol 300.15 K

H G

S T

− − −

+ =

∆ − ∆

∆ = =

Pressure Effect on the Gibbs Energy

Example 4.3 An ideal gas at 27℃ expands isothermally into an evacuated vessel from 10 to 1 bar. Calculate q per mole and w per mole and ∆U, , , , .∆H ∆G ∆A and ∆S

0 w =

1 1 1

2 1

24.63 L

(8.314 J K mol )(300.15 K) ln 5746 J mol 2.463 L

lnV A RT

V

− − = − −

∆ = − = −

0

∆ =U

0 0 0

q = ∆ − =U w + =

( ) 0 0 0

H U PV

∆ = ∆ + ∆ = + =

1 1 1

1 10

(8.314 J K mol )(300.15 K) ln 1 5746 J mol 10

d ln 1

G V P RT 10 ^{− −} = − ⁻

∆ =

∫

1

1 1

0 (5746 J mol )

H G + ⁻ _{− −}

∆ − ∆ =

∆ = =

한양대학교

Pressure Effect on the Gibbs Energy

Example 4.4∆_fG^◦(CH₃OH, g) = –161.96 kJ mol^–1, ∆_fG^◦(CH₃OH, l) = –166.27 kJ mol^–1and ρ(CH₃OH, l) = 0.7914 g/cm³at 298.15 K (a) Calculate ∆_fG(CH₃OH, g) at 10 bar at 298.15 K assuming the methanol vapor as an ideal gas (b) Calculate ∆_fG(CH₃OH, l) at 10 bar at 298.15 K.

1 1 1 1

f f

161.96 kJ mol + (8.314 10 kJ K mol )(298.15 K) ln 10 = 156.25 kJ mol

(a) ln

G G RT P P

− − − −

− × −

∆ = ∆ +

=

 

1 3 3

6 3 1

32.04 g mol 1 m 100 cm 0.7914 g cm

40.49 10 m mol

(b)

V

−

−

− −

×

 

= ×  

=

f f

1

6 3 1 5

3 1

1

166.27 kJ mol

(40.49 10 m mol )(9 10 Pa) +

10 J kJ 166.23 kJ mol

( )

G G V P P

−

− −

−

−

−

× ×

−

∆ = ∆ + −

=

=

 

The Gibbs energy of formation of gaseous and liquid methanol at 298.15K

4.5 Fugacity and Activity

ln P G G RT

= ^ + P_ for an ideal gas, G. N. Lewis devised fugacityf(T,P) for real gases.

0

ln lim 1

P

f f

G G RT

P ^→ P

= ^ + _ =

id id

At constant temperature, dG =V Pd for a real gas and dG =V d P for an ideal gas.

The difference in Gibbs energy between a real gas and an ideal gas can be integrated from some low pressure, P* to the pressure of interest, P.

id id

*

id

*

id * *id id

* *

* *id id id

*

0 id

As

d( ) ( )d

( ) ( ) ( )d

0, , ( ) ( )d

ln 1 ( ) 1

exp ( )d

d or

P P

P P

P P

P

P P

P P

P

G G V V P

G G G G V V P

P G G G G

f

V V P

f V V P

P V V P

P RT

RT φ

− = −

− − − = −

→ → − = −

  = −

  

 

= =  − 

∫

∫

∫

∫

∫

∫

The Fugacity Coefficient

한양대학교

Using the compressibility factor, PV Z = RT

id

0 0 0

1 1 1

exp ( )d exp ( )d exp d

P P P

f ZRT RT Z

V V P P P

P RT RT P P P

φ = = ^

∫

∫

∫

With the virial equation, PV 1 ' ' ²

Z B P C P

= RT = + + +

( )

0 0

1 '

ln d ' ' d '

2

P P

f Z C P

P B C P P B P

P P

=

∫

∫

For a van der Waals gas, 1

1 a

Z b P

RT RT

 

= +  −  +

0 0

1 1

ln d d

P P

f Z a a P

P b P b

P P RT RT RT RT

−    

=

∫

∫

exp a P

f P b

RT RT

  

=  −  

Fugacity

Example 4.7 The van der Waals coefficients of nitrogen are a = 1.408 L²bar mol^–2and b = 0.03913 L mol^–1. Estimate the fugacity of nitrogen at 50 bar and 298K.

1.408 50

(50 bar) exp 0.03913

(0.083145)(298) (0.083145)(298) 48.2 bar

exp a P

f P b

RT RT

−

  

=  −  

  

=   

=

Activity

한양대학교

ln _i

i i RT a

µ = µ^ + to express the chemical potential of a species in a mixture in the reference state for which

for an ideal gas, for a real gas

for solutions where is for a pure solid or liquid if

the activity coefficie the pres

nt

1

1

i i i

i i

i i

i i

i i i

i

i

a

P f

a a

P P

a m

a

µ µ

γ γ

= =

= =

=

=



 

sure is close enough to the standard state pressure

If the effect of pressure is not negligible, assuming the molar volume to be constant at all reasonable pressures, for a pure solid or liquid

( , ) ( ) ( )

ln ( )

( )

exp

T P T V P P RT a V P P

V P P

a RT

µ = µ + −

= −

 − 

=  

 

 





Activity

Example 4.8 What is the activity of liquid water at 1, 10, and 100 bar at 298K, assuming that the molar volume is constant?

1

1 1

( )

exp

At 1 bar, 1

(0.018 kg mol )(10 bar 1 bar)

At 10 bar, exp 1.007

(0.083145 L bar K mol )(298 K) At 100 bar, 1.075

V P P

a RT

P a

P a

P a

−

− −

 − 

=  

 

= =

 − 

= =   =

 

= =



4.6 The Chemical Potential

한양대학교

, ,{ _{j i}} , ,{ _{j i}} , ,{ _{j i}} , ,{ _{j i}}

i

i S V n i S P n i T V n i T P n

U H A G

n n n n

µ

≠ ≠ ≠ ≠

∂  ∂   ∂  ∂ 

= ∂  =∂  =∂  =∂ 

the partial molar Gibbs e At constant and T P, µi =Gi nergy

a species diffuses spontaneously from the phase with higher chemical potential to the phase with lower chemical potential

µ_i(β) µ_i(α)

( )

[

]

d < 0, ( ) > ( ) d = 0, ( ) = ( )

i i i i i i i

T P

i i

i i

G n n n

G G

µ α µ β µ β µ α

µ α µ β µ α µ β

= − + = −

at equilibrium

The Chemical Potential

1

d d d d

Ns

i i

i

G S T V P µ n

=

= − + +

∑

For a mixture of ideal gases

,{ } ,{ }

i i

i i

i i

P n T n

S V

T P

µ µ

∂ ∂

   

− = ∂  = ∂ 

,{ }i ,{ }i ,{ }i

i i i i

i i

i i

T n T n T n

P

V V RT x

P P P P P

µ ^ µ ^{  } µ ^

∂ ∂ ∂ ∂

   

= = = ∂  =  ∂  ∂  =  ∂ 

,{ }_i i

i T n i

RT

P P

µ

∂  =

 ∂ 

 

d d

i i

i

P i

i P i

RT P

P

µ

µ µ =

∫

∫

ln ⁱ

i i

S S R P

= ^ − P_

4.7 Partial Molar Properties

한양대학교

At constant T, P, and µ_i

1 1

=

s s

N N

i i i i

i i

G n µ n G

= =

=

∑ ∑

All extensive properties of a one-phase system are additive.

,{ }

1 1

,{ }

1 1

the partial molar entropy

the partial molar volume

s s

i

s s

i

N N

i i i i

P n

i i

N N

i i i i

T n

i i

i

i

S n S n

T

V n V n

P

S

V µ

µ

= =

= =

∂ 

= −  ∂  =

∂ 

=  ∂  =

∑ ∑

∑ ∑

G(T, P, {n_i})

1

d d d d

Ns

i i i

G S T V P µ n

=

= − + +

∑

2

,{ }

/

P ni

H T G T

T

∂ 

= −  ∂ 

the Gibbs–Helmholtz equation

2

,{ }

1 1

the partial molar enthalpy

( / )

s s

i

N N

i i i i

P n

i i

i

H T T n H n

T H

µ

= =

∂ 

= −

∑

∑

Partial Molar Properties

, ,{ } , ,{ } , ,{ }

,{ }

,{ }

, ,{ } , ,{ } , ,{ }

j i j i j i

j i

j i j i j i

j i

i T P n i T P n i T P n

i i i

i

P n

P n

i T P n i T P n i T P n

G H TS

G H S

n n T n

G H T S

S G

T

S G G

n n T T n

µ

≠ ≠ ≠

≠

≠ ≠ ≠ ≠

= −

∂  =∂  −  ∂ 

∂  ∂  ∂ 

     

≡ = −

∂ 

− =  ∂ 

 

 

 ∂  ∂ ∂   ∂ ∂  

−∂  = ∂ ∂   = ∂ ∂   _{,{ }}

,{ }

P nj i

i i

i

P n

S G

T T

µ

≠

≠



∂  ∂ 

− = ∂  =  ∂ 

Partial Molar Properties

한양대학교

Ideal mixture at low pressures, mixtures of real gases are assumed to behave as ideal gases.

The chemical potential of a species in a ideal gas mixture

ln ⁱ

i i i i

RT y P P y P µ = µ^ + P_ ≡

and

ln ln ln ln

i i t i

t

i i i i i t i i i i t

n n X X

n

P P

G n RT n y n RT n y RT y y R n G

P

G n

T P

µ µ

µ = =

 

= + + =  + +  =

=



∑

∑ ∑ ∑ ∑

∑

∑

 

 

( )

,{ }

,{ }

ln ln ln ln

i

i

i i

i i i i t i i i t

i i

i i i i t

P

t n

n

i

i i i

T

P P

n S R n y n R n y S R y y R n

S G

T H G

P P S

n T S n H n y H n H n RT n V

P TS

V G

P

µ

 

= − − =  − −  =

∂ 

= −∂ 

= +

∂

 

= + = = =

 =

=  ∂  =

 

∑ ∑ ∑ ∑ ∑

∑ ∑ ∑

∑ ∑

 

 

   



Every gas is as vacuum to every other gas.

Ideal Gas Mixture

At constant pressure P₁= P₂= P

( ) ( )

0 1 1 1 2 2 2

mix 1 1 1 1 1 2 2 2 2 2

mix mix 0 1 1 2 2 1 1 2 2

ln ln

ln ln ln ln

ln ln _t ln ln

P P

G n n RT n n RT

P P

P P

G n n RT y n RT n n RT y n RT

P P

G G G RT n y n y n RT y y y y

µ µ

µ µ

= + + +

= + + + + +

∆ = − = + = +

 

 

 

 

( ) ( )

1 2

0 1 1 2 2

1 2

mix 1 1 1 1 2 2 2 2

mix mix 0 1 1 2 2 1ln 1 2 2

ln ln

ln ln ln ln

ln ln _t ln

P P

S n S n R n S n R

P P

P P

S n S n R y n R n S n R y n R

P P

S S S R n y n y n R y y y y

= − + −

= − − + − −

∆ = − = − + =− +

 

 

 

 

mixH 0

∆ =

4.8 Gibbs-Duhem Equation

한양대학교

1

1

1

1 1

d d d d

' 0

d ' d d d d d d d 0

s

s

s

s s

N i i i

N

i i i

N i i i

N N

i i i i

i i

U PV TS n

U P V T S n

U U PV TS n

U U P V V P T S S T n n

µ

µ

µ

µ µ

=

=

=

= =

= − + +

= − + +

= + − − =

= + + − − − − =

∑

∑

∑

∑ ∑

the complete Legendre transformation

1

d d d 0

Ns

i i

i

V P S T n µ

=

− −

∑

4.9 Maxwell Relations

( , ) d d d

( , ) d d d

( , ) d d d

( , ) d

S V

S P

T S

T P

U S V U T S P V

V S

T V

H S P H T S V P

P S

S P

A T V A S T P V

V T

G T P

∂ ∂

   

= − ∂  = −∂ 

 

∂ ∂

 

= + ∂  =∂ 

 ∂  ∂ 

= − − ∂  = ∂ 

d d

T P

S V

G S T V P

P T

∂  ∂ 

= − + −∂  =∂ 

For a mole of a substance

Internal Pressure of a Real Gas

한양대학교

2

2 2

a van der Waals gas,

T T V

T

U S P

T P T P

V V T

RT a P V b V

U R R RT a a

V V b P V b V b V V

∂  = ∂  − = ∂  −

 ∂  ∂  ∂ 

   

= −

−

 

∂  = − = − −  =

 ∂  − − −

   

Example 4.11 What is the molar internal energy change of propane isothermally expanding from 10 to 30 L?

2 2

1 1 2

1 2

2 2 5 1 3 3 1 2 6 2

6 2 1

3 3 1 3 3 1

(8.779 L bar mol )(10 Pa bar )(10 m L ) = 0.8779 Pa m mol

1 1

0.8779 Pa m mol 58.5 J mol

10 10 m mol 30 10 m mol

1 1

d d

( )

U V

U V

a

U U a V a

V V V

U

− − − − −

− −

− − − −

=

− =

× ×

 

∆ = = =  − 

 

∆ =  

∫ ∫

Isothermal Expansion of a van der Waals Gas

2 2

1 1

2

2 1

d 1 d ln

T V

S V

S V

RT a P V b V

S P R

V T V b

V b

S S R V R

V b V b

= −

−

∂  = ∂  =

∂  ∂  −

 

∆ = = = −

− −

∫ ∫

The molar entropy of isothermal expansion of a van der Waals gas

Cubic Expansion Coefficient and Isothermal Compressibility

한양대학교 Using the cyclic rule,

1 1 1

P V

T

T T V

P

P P T T

V

V P T

T V

P

U S P T P

T P T P

V V V

V V T

C C P U

V

V T V T P V P

V

α κ

α κ

α κ

κ

∂ 

∂ 

 ∂  = −  =

∂  ∂ 

 



   

∂ ∂ ∂

∂ 

 

∂  ₌ ∂  _{− = }∂ _ −

    ∂

=   = = − = −

− =

 

   

∂ ∂ ∂ ∂

       

 

∂  ∂  ∂ 

   

− = +  ∂ ∂

2

T P

V TV T

α κ

   ∂  =

   ∂ 

   

 