Capítulo 09

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UFABC - Princípios de Termodinâmica Curso 2017.1

Prof. Germán Lugones

CAPÍTULO 9

Criticalidade

T hre e w orl ds, M. C . Esch er , 1 95 5.

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Ponto crítico:

Se nos deslocamos ao longo da linha

de coexistência líquido-vapor de uma

substância pura na direção de altas

pressões e temperaturas elevadas, a

densidade do vapor aumenta e a

densidade do líquido diminui.

No fim, há um ponto que determina o

fim da linha de coexistência, onde as

densidades das duas fases se tornam

iguais.

Este ponto, chamado de ponto crítico,

corresponde ao estado em que as

duas fases se tornam o idênticas.

9.4 Lugares geométricos de las fases: ecuación de 155

Otra representación de los estados correspondientes a cada una de las fases

posibles de un sistema se deduce de la figura 9.5, en la que se representa en función

de

T y P. La superficie puede ser entrecruzada, como se muestra en la figura,

y

la curva a lo largo de la cual se cortan las dos ramas de la superficie (como el

punto

D o el O de la figura 9.4) determina el lugar geométrico de los estados para

los cuales ocurre la transición de fase. Supongamos que proyectamos ahora la curva

de intersección de las dos ramas de la superficie sobre el plano

P-T.

O bien que,

en cada sección p-P, como se indica en la figura 9.4, proyectamos el punto D o

el O sobre el eje

P ; los puntos asi proyectados determinan una curva tal como la

representada en la figura 9.10, y resulta evidente la distinción entre las dos regiones,

correspondientes a fases diferentes.

Figura 9.10

La terminación de la curva de fases corresponde al vértice de la región de mezcla

en la figura 9.9 (por encima del cual las isotermas son monótonas) y a la desaparición

del característico entrecruzamiento de la superficie a temperatura alta en la fi-

gura 9.5. El estado singular así determinado en cada una de las figuras es el

punto

crítico.

La

temperatura crítica

y

la

presión crítica

son los valores de la temperatura

y la presión en el punto crítico.

T

Figura 9.11

Si un sistema tiene mas de dos fases, el diagrama de fases puede tener un aspecto

semejante al del agua, que se ilustra en la figura 9.1

1. La posible complejidad de

tales diagramas está rigurosamente limitada por la regla de las fases de Gibbs,

explicada en la sección 9.6.

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Perto do ponto crítico, a linha de coexistência é representada por uma linha reta:

onde A é uma constante positiva identificada como (−∂P/∂T)_!calculada no ponto crítico.

Usando a equação de Clausius-Clapeyron, vemos que as diferenças de entropia molar e volume molar em ambas as fases em coexistência estão relacionados por:

Portanto, no ponto crítico, não apenas os volumes molares v_G e v_L se tornam idênticos, mas também as entropias molares s_G e s_L ficam iguais. Consequentemente, o calor latente fica nulo no ponto crítico ⟼ esse ponto corresponde a uma transição de fase de segunda ordem (aparece uma descontinuidade apenas na derivada segunda da energia livre de Gibbs G).

120 8 Criticality

In the phase diagram, the critical point .T

_c

; p

c

/

lies therefore in the terminal point

of the liquid-vapor coexistence line. Near the critical point the coexistence line is

represented by the semi-straight line

p

! p

c

D A.T ! T

c

/;

T < T

c

;

(8.1)

where A is a constant strictly positive which is identified as .@p=@T /

_v

calculated at

the critical point. Using the Clausius-Clapeyron equation, we see that the differences

in molar entropy and molar volume in both phases in coexistence are related by

.s

G

! s

L

/

D A.v

G

! v

L

/:

(8.2)

Therefore, at the critical point, not only the molar volumes v

_G

and v

_L

become

iden-tical, but the molar entropies s

_G

and s

_L

become identical as well and, consequently,

the boiling latent heat `

_e

vanishes.

8.1.2 Liquefaction

Most pure substances that are gases under normal conditions of temperature and

pressure can be liquefied by compression alone, that is, can pass into the liquid state

when subjected isothermally to sufficiently high pressures. This occurs with carbon

dioxide, ammonia, ethane, propane, butane and other gases. These substances have

a critical temperature higher than the room temperature, so that the coexistence

line can be reached by isothermal compression. Carbon dioxide, for example,

can be liquefied at a temperature of 20

ı

C under the pressure of 5.73 MPa. Other

gases, on the other hand, such as helium, neon, argon, krypton, xenon, hydrogen,

oxygen, nitrogen, carbon monoxide and methane cannot be liquefied at room

temperature, no matter how large the applied pressure. These substances have

critical temperatures below the room temperature. A compression alone at room

temperature is insufficient to meet the liquid-vapor coexisting line.

Figure

8.1 shows the carbon dioxide isotherms at various temperatures above

and below the critical temperature, which occurs at 31.04

ı

C. It is seen that, below

the critical temperature, the vapor-liquid coexistence may occur and therefore

liquefaction may happen by compression alone. Along a subcritical isotherm, the

volume decreases by compression and presents a jump at the transition. Along a

supercritical isotherm, the volume increases continuously with pressure, without any

discontinuity. The critical point is reached by compression alone along the critical

isotherm when pressures reaches 72.85 atm. The density of the carbon dioxide at the

critical point, where the liquid and vapor become identical, is 0.468 g/cm

3

_{. The first}

experimental measurements around a critical point were made in carbon dioxide by

Andrews, who proved to be possible to continuously convert steam into liquid and

vice-versa, bypassing the critical point through an appropriate path.

120 8 Criticality

In the phase diagram, the critical point .T

c

; p

c

/

lies therefore in the terminal point

of the liquid-vapor coexistence line. Near the critical point the coexistence line is

represented by the semi-straight line

p

! p

_c

D A.T ! T

_c

/;

T < T

_c

;

(8.1)

where A is a constant strictly positive which is identified as .@p=@T /

v

calculated at

the critical point. Using the Clausius-Clapeyron equation, we see that the differences

in molar entropy and molar volume in both phases in coexistence are related by

.s

_G

! s

_L

/

D A.v

_G

! v

_L

/:

(8.2)

Therefore, at the critical point, not only the molar volumes v

G

and v

L

become

iden-tical, but the molar entropies s

G

and s

L

become identical as well and, consequently,

the boiling latent heat `

e

vanishes.

8.1.2 Liquefaction

Most pure substances that are gases under normal conditions of temperature and

pressure can be liquefied by compression alone, that is, can pass into the liquid state

when subjected isothermally to sufficiently high pressures. This occurs with carbon

dioxide, ammonia, ethane, propane, butane and other gases. These substances have

a critical temperature higher than the room temperature, so that the coexistence

line can be reached by isothermal compression. Carbon dioxide, for example,

can be liquefied at a temperature of 20

ı

C under the pressure of 5.73 MPa. Other

gases, on the other hand, such as helium, neon, argon, krypton, xenon, hydrogen,

oxygen, nitrogen, carbon monoxide and methane cannot be liquefied at room

temperature, no matter how large the applied pressure. These substances have

critical temperatures below the room temperature. A compression alone at room

temperature is insufficient to meet the liquid-vapor coexisting line.

Figure

8.1 shows the carbon dioxide isotherms at various temperatures above

and below the critical temperature, which occurs at 31.04

ı

C. It is seen that, below

the critical temperature, the vapor-liquid coexistence may occur and therefore

liquefaction may happen by compression alone. Along a subcritical isotherm, the

volume decreases by compression and presents a jump at the transition. Along a

supercritical isotherm, the volume increases continuously with pressure, without any

discontinuity. The critical point is reached by compression alone along the critical

isotherm when pressures reaches 72.85 atm. The density of the carbon dioxide at the

critical point, where the liquid and vapor become identical, is 0.468 g/cm

3

_{. The first}

experimental measurements around a critical point were made in carbon dioxide by

Andrews, who proved to be possible to continuously convert steam into liquid and

vice-versa, bypassing the critical point through an appropriate path.

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O ponto crítico pode ser determinado pelas condições:

i.e., no diagrama Pv, o ponto crítico é um ponto estacionário e um ponto de inflexão (ponto de sela).

A partir da equação de Van der Waals,

Obtemos:

Ponto crítico

8.2 van der Waals Theory

125 it is required that K

₁

.T /

and K

₂

.T /

are analytical in T within the same region. As a

consequence, the analytical continuation of K

₁

.T /

and K

₂

.T /, if they exist, should

be the same. But in this case the procedure just presented becomes equivalent to

Maxwell construction.

In summary, the placement of the horizontal segment of the isotherm in a

position different from that given by the Maxwell construction would make f .T; v/

a nonalytic function inside a single-phase region.

8.2.3 Critical Point

Increasing the temperature along the coexistence line, the difference !v D v

G

! v

L

between the molar volumes decreases and vanishes at a certain temperature that we

call critical temperature T

_c

_{. At this temperature, !v D 0 and the two phases become}

identical. The corresponding point .T

_c

; p

_c

/

in the phase diagram marks the end of

the coexistence line. Above the critical temperature the van der Waals fluid exhibits

a single phase.

The critical point can be determined by

@p

@v

D 0

and

@

2

p

@v

2

D 0;

(8.14)

because the critical point is both a stationary point and an inflexion point, as seen in

Figs.

8.1 and

8.2 . From the van der Waals equation, we get

@p

@v

D !

RT

.v

! b/

2

C

2a

v

3

;

(8.15)

and

@

2

p

@v

2

D

2RT

.v

! b/

3

!

6a

v

4

;

(8.16)

from which we obtain the critical molar volume

v

_c

D 3b;

(8.17)

and the critical temperature

T

_c

D

8a

27bR

;

(8.18)

which substituted in van der Waals equation gives the critical pressure

p

c

D

a

27b

2

:

(8.19)

8.2 van der Waals Theory 125

it is required that K

1

.T /

and K

2

.T /

are analytical in T within the same region. As a

consequence, the analytical continuation of K

1

.T /

and K

2

.T /, if they exist, should

be the same. But in this case the procedure just presented becomes equivalent to

Maxwell construction.

In summary, the placement of the horizontal segment of the isotherm in a

position different from that given by the Maxwell construction would make f .T; v/

a nonalytic function inside a single-phase region.

8.2.3 Critical Point

Increasing the temperature along the coexistence line, the difference !v D v

G

! v

L

between the molar volumes decreases and vanishes at a certain temperature that we

call critical temperature T

c

. At this temperature, !v D 0 and the two phases become

identical. The corresponding point .T

c

; p

c

/

in the phase diagram marks the end of

the coexistence line. Above the critical temperature the van der Waals fluid exhibits

a single phase.

The critical point can be determined by

@p

@v

D 0

and

@

2

p

@v

2

D 0;

(8.14)

because the critical point is both a stationary point and an inflexion point, as seen in

Figs.

8.1 and

8.2 . From the van der Waals equation, we get

@p

@v

D !

RT

.v

! b/

2

C

2a

v

3

;

(8.15)

and

@

2

p

@v

2

D

2RT

.v

! b/

3

!

6a

v

4

;

(8.16)

from which we obtain the critical molar volume

v

c

D 3b;

(8.17)

and the critical temperature

T

c

D

8a

27bR

;

(8.18)

which substituted in van der Waals equation gives the critical pressure

p

c

D

a

27b

2

:

(8.19)

~ v (cm3_/g) 0 2 4 6 8 10 30 40 50 60 70 80 90 100 p (atm) 40,09 34,72 32,05 31,01 29,93 28,05 25,07 19,87 10,82 2,85 1 2 3 ₄ ~ v (cm3_/g) 70 71 72 73 74 75 p (atm) 32,05 31,52 31,32 31,19 31,01 30,41 29,93

Fig. 8.1 Isotherms of carbon dioxide (CO₂) in the diagram pressure p versus specific volume _Qv, for various values of temperature (in ıC), obtained experimentally by Michels et al. (1937)

Table

8.1 shows experimental data corresponding to the critical point of several

pure substances. In addition to critical temperature, pressure and density, the table

presents also the compressibility factor Z

c

D p

c

v

c

=RT

c

determined at the critical

point.

8.2 van der Waals Theory

8.2.1 van der Waals Equation

van der Waals theory provides a quantitative description of the liquid-vapor phase

transition and of the corresponding critical point. Although it does not describe

accurately the behavior of the thermodynamic properties near the critical point it

provides an appropriate description of the phase coexistence and its relation with

the critical point.

The van der Waals equation is given by

p

D

RT

v

! b

!

a

v

2

;

(8.3)

where a and b are constants. It describes approximately the behavior of gases and of

the liquid-vapor transition. For convenience we call a system obeying this equation

a van der Waals fluid or van der Waals gas, although it also describe the liquid

phase. From the microscopic point of view, a fluid that undergoes a liquid-vapor

transition must consist of molecules that attract each other over long distances and

repel at short distances. A van der Waals fluid is to be understood as composed

of hard spherical attractive molecules. The repulsion is the result of the rigidity

of the molecules and is related to the parameter b. The attraction is related to the

parameter a, which must be understood as a measure the force of attraction between

molecules.

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Igualando a zero essas derivadas obtemos ▪_{o volume crítico molar}

▪_{a temperatura crítica}

Substituindo na equação de Van der Waals, obtemos a pressão crítica:

Eliminando a e b das equações anteriores, obtemos:

Encontramos, assim, que, apesar de que P_c, v_c, e T_c variam de sistema para sistema (através dos parâmetros a e b), a quantidade K=pcvc/(RTc) tem um valor comum,

universal para todos elas.

Os resultados experimentais mostram que de fato K é quase o mesmo para um grande grupo de substâncias; por exemplo, o seu valor para o tetracloreto de carbono, éter etílico, e formato de etilo é 3.677, 3.814, 3.895 respectivamente (i.e., os valores são próximos, embora não exatamente os mesmos). ⟼ universalidade.

it is required that K

1

.T /

and K

2

.T /

are analytical in T within the same region. As a

consequence, the analytical continuation of K

₁

.T /

and K

₂

.T /, if they exist, should

be the same. But in this case the procedure just presented becomes equivalent to

Maxwell construction.

In summary, the placement of the horizontal segment of the isotherm in a

position different from that given by the Maxwell construction would make f .T; v/

a nonalytic function inside a single-phase region.

8.2.3 Critical Point

Increasing the temperature along the coexistence line, the difference !v D v

G

! v

L

between the molar volumes decreases and vanishes at a certain temperature that we

call critical temperature T

_c

_{. At this temperature, !v D 0 and the two phases become}

identical. The corresponding point .T

c

; p

c

/

in the phase diagram marks the end of

the coexistence line. Above the critical temperature the van der Waals fluid exhibits

a single phase.

The critical point can be determined by

@p

@v

D 0

and

@

2

p

@v

2

D 0;

(8.14)

because the critical point is both a stationary point and an inflexion point, as seen in

Figs.

8.1 and

8.2 . From the van der Waals equation, we get

@p

@v

D !

RT

.v

! b/

2

C

2a

v

3

;

(8.15)

and

@

2

p

@v

2

D

2RT

.v

! b/

3

!

6a

v

4

;

(8.16)

from which we obtain the critical molar volume

v

_c

D 3b;

(8.17)

and the critical temperature

T

c

D

8a

27bR

;

(8.18)

which substituted in van der Waals equation gives the critical pressure

p

_c

D

a

27b

2

:

(8.19)

it is required that K1.T / and K2.T / are analytical in T within the same region. As a

consequence, the analytical continuation of K1.T / and K2.T /, if they exist, should

be the same. But in this case the procedure just presented becomes equivalent to Maxwell construction.

In summary, the placement of the horizontal segment of the isotherm in a position different from that given by the Maxwell construction would make f .T; v/ a nonalytic function inside a single-phase region.

8.2.3 Critical Point

Increasing the temperature along the coexistence line, the difference !v D vG ! vL

between the molar volumes decreases and vanishes at a certain temperature that we call critical temperature Tc. At this temperature, !v D 0 and the two phases become

identical. The corresponding point .Tc; pc/ in the phase diagram marks the end of

the coexistence line. Above the critical temperature the van der Waals fluid exhibits a single phase.

The critical point can be determined by @p

@v D 0 and

@2p

@v2 D 0; (8.14)

because the critical point is both a stationary point and an inflexion point, as seen in Figs.8.1 and 8.2. From the van der Waals equation, we get

@p @v D ! RT .v ! b/2 C 2a v3 ; (8.15) and @2p @v2 D 2RT .v ! b/3 ! 6a v4 ; (8.16)

from which we obtain the critical molar volume

vc D 3b; (8.17)

and the critical temperature

Tc D

8a

27bR; (8.18)

which substituted in van der Waals equation gives the critical pressure pc D

a

27b2: (8.19)

it is required that K1.T / and K2.T / are analytical in T within the same region. As a

consequence, the analytical continuation of K1.T / and K2.T /, if they exist, should

be the same. But in this case the procedure just presented becomes equivalent to Maxwell construction.

In summary, the placement of the horizontal segment of the isotherm in a position different from that given by the Maxwell construction would make f .T; v/ a nonalytic function inside a single-phase region.

8.2.3 Critical Point

Increasing the temperature along the coexistence line, the difference !v D vG ! vL

between the molar volumes decreases and vanishes at a certain temperature that we call critical temperature Tc. At this temperature, !v D 0 and the two phases become

identical. The corresponding point .Tc; pc/ in the phase diagram marks the end of

the coexistence line. Above the critical temperature the van der Waals fluid exhibits a single phase.

The critical point can be determined by @p

@v D 0 and

@2p

@v2 D 0; (8.14)

because the critical point is both a stationary point and an inflexion point, as seen in Figs.8.1 and 8.2. From the van der Waals equation, we get

@p @v D ! RT .v ! b/2 C 2a v3 ; (8.15) and @2p @v2 D 2RT .v ! b/3 ! 6a v4 ; (8.16)

from which we obtain the critical molar volume

vc D 3b; (8.17)

and the critical temperature

Tc D

8a

27bR; (8.18)

which substituted in van der Waals equation gives the critical pressure pc D

a

27b2: (8.19)

126 8 Criticality

Eliminating a and b from these three equations we reach the following relation

p

_c

v

_c

RT

_c

D

3

8 :

(8.20)

The compressibility factor Z of a fluid is defined by Z D pv=RT . Therefore,

from this equation we conclude that at the critical point Z

c

D 3=8 for a fluid

that satisfies the van der Waals equation. In Table

8.1 we present the values of

Z

_c

obtained from the critical properties of several pure substances. Although the

experimental values for Z

c

D p

c

v

c

=RT

c

are not equal to 3=8 D 0:375, they are

close to each other, mainly those related to the noble gases.

8.2.4 Expansion Around the Inflexion Point

For temperatures near the critical temperature, the molar volumes of the liquid and

vapor phases, v

L

and v

G

, are close to each other and close to the inflexion point v

0

of the van der Waals isotherm. Therefore, if we want to determine v

_L

and v

_G

near

the critical temperature, it is reasonable to approximate the van der Waals isotherm

and the free energy by an expansion around v D v

0

.

The expansion of p.v/ up to cubic terms gives

p

D p

₀

C A.v ! v

₀

/

C B.v ! v

₀

/

3

;

(8.21)

where v

0

is given by p

00

.v

0

/

D 0, p

0

D p.v

0

/

, A D p

0

.v

0

/

and B D p

000

.v

0

/=6.

The corresponding free energy f .v/ is given by

f

D f

₀

! p

₀

.v

! v

₀

/

!

A

2 .v

! v

0

/

2

!

B

4 .v

! v

0

/

4

_:

_(8.22)

Notice that f

0

, v

0

, p

0

, A and B depend only on temperature. The expansions above

are valid for small values of jv ! v

0

j. For temperatures near T

c

, the constants can

be obtained as explicit functions of temperature. The expansion of these constants

around T

c

gives

v

₀

D v

_c

.1

C 2

T

! T

c

T

c

/;

(8.23)

p

₀

D p

_c

.1

C 4

T

! T

c

T

_c

/;

(8.24)

A

D !6

p

c

v

_c

T

! T

_c

T

_c

(8.25)

and

B

D !

3p

c

2v

3 c

:

(8.26)

(6)

A universalidade pode ser observada também de outra maneira.

Escrevemos a equação de Van der Waals em termos das seguintes variáveis reduzidas:

Obtemos:

Esta também é uma equação universal, i.e., a mesma para todas a s s u b s t â n c i a s , j á q u e a s constantes a e b desapareceram ( t e o r e m a d o s e s t a d o s correspondentes).

12.2 Condensation of a van der Waals gas 409

pertaining to T = Tc, which, of course, passes through the critical point is referred to as the

critical isotherm of the system; it is straightforward to see that the critical point is a point

of inflection of this isotherm, so that both (@P/@v)_T and (@2P/@v2)_T vanish at this point. Using (1), we obtain for the coordinates of the critical point

P_c ₌ a

27b2, vc = 3b, Tc =

8a

27bR, (2)

so that the number

K ⌘ RTc/Pcvc = 8/3 = 2.666.... (3)

We thus find that, while Pc,vc, and Tc vary from system to system (through the interaction

parameters a and b), the quantity K has a common, universal value for all of them — so long as they all obey the same (i.e., van der Waals) equation of state. The experimental results for K indeed show that it is nearly the same over a large group of substances; for instance, its value for carbon tetrachloride, ethyl ether, and ethyl formate is 3.677, 3.814, and 3.895, respectively — close, though not exactly the same, and also a long way from the van der Waals value. The concept of universality is, nonetheless, there (even though the van der Waals equation of state may not truly apply).

It is now tempting to see if the equation of state itself can be written in a universal form. We find that this indeed can be done by introducing reduced variables

P_r ₌ P P_c , vr = v v_c, Tr = T T_c. (4)

Using (1) and (2), we readily obtain the reduced equation of state

✓

P_r ₊ 3 v_r2

◆

(3v_r _{1) = 8T}_r, (5)

which is clearly universal for all systems obeying van der Waals’ original equation of state (1); all we have done here is to rescale the observable quantities P, v, and T in terms of their critical values and thereby “push the interaction parameters a and b into the back-ground.” Now, if two different systems happen to be in states characterized by the same values of v_r and T_r, then their P_r would also be the same; the systems are then said to be in “corresponding states” and, for that reason, the statement just made is referred to as the “law of corresponding states.” Clearly, the passage from equation (1) to equation (5) takes us from an expression of diversity to a statement of unity!

We shall now examine the behavior of the van der Waals system in the close neighbor-hood of the critical point. For this, we write

P_r _{= 1 + ⇡, v}_r _{= 1 + , T}_r _{= 1 + t.} (6)

of inflection of this isotherm, so that both (@P/@v)T and (@2P/@v2)T vanish at this point.

Using (1), we obtain for the coordinates of the critical point

P_c ₌ a

27b2, vc = 3b, Tc =

8a

27bR, (2)

so that the number

K ⌘ RTc/Pcvc = 8/3 = 2.666.... (3)

We thus find that, while P_c,v_c, and T_c vary from system to system (through the interaction parameters a and b), the quantity K has a common, universal value for all of them — so long as they all obey the same (i.e., van der Waals) equation of state. The experimental results for K indeed show that it is nearly the same over a large group of substances; for instance, its value for carbon tetrachloride, ethyl ether, and ethyl formate is 3.677, 3.814, and 3.895, respectively — close, though not exactly the same, and also a long way from the van der Waals value. The concept of universality is, nonetheless, there (even though the van der Waals equation of state may not truly apply).

P_r ₌ P P_c, vr = v v_c, Tr = T T_c . (4)

Using (1) and (2), we readily obtain the reduced equation of state

✓

Pr + 3

v2_r

◆

(3vr 1) = 8Tr, (5)

which is clearly universal for all systems obeying van der Waals’ original equation of state (1); all we have done here is to rescale the observable quantities P, v, and T in terms of their critical values and thereby “push the interaction parameters a and b into the back-ground.” Now, if two different systems happen to be in states characterized by the same values of vr and Tr, then their Pr would also be the same; the systems are then said to be in

“corresponding states” and, for that reason, the statement just made is referred to as the “law of corresponding states.” Clearly, the passage from equation (1) to equation (5) takes us from an expression of diversity to a statement of unity!

(7)

Perto do ponto crítico, podemos expandir as grandezas como segue:

Substituindo na equação de Van der Waals temos:

Eq. (*)

Consideremos as seguintes aproximações:

Expansão 1: " em função de #

•Estamos ao longo da isoterma crítica t=0. •Estamos muito perto do ponto crítico

Obtemos:

que pode ser escrita na forma:

Expansão na vizinhança do ponto crítico

128 8 Criticality 0 1 2 3 T / T_c 0 0.5 1.0 1.5 2.0 κ T / κ 0 0 0.5 1.0 1.5 2.0 T / T_c 0 2 4 6 8 c v / R

a

b

Fig. 8.3 van der Waals fluid. (a) Isothermal compressibility !_T of the fluid along the critical isochoric (T > T_c) and of the liquid along the coexistence line (T < T_c), where !₀ _{D 1=p}_c. (b) Isochoric molar heat capacity c_v along the critical isochoric line. The jump of c_v at T _{D T}_c is equal to 9R=2

8.2.5 Compressibility

At temperatures above the critical temperature, the isotherms are strictly monotonic decreasing so that the isothermal compressibility !_T _{D !.1=v/.@v=@p/ is positive} (Fig. 8.3). The largest value of !_T along an isotherm occurs at the inflexion point. At the critical temperature it diverges because the derivative @p=@v vanishes at this point. According to (8.21), the pressure varies with the molar volume, along the critical isotherm, and around the critical point, according to the equation

p D p_c ! 3pc 2v3 c

.v ! v_c/3: (8.37) Let us determine next the behavior of !T around the critical point, along the

coexistence line and its extension, defined by (8.29), which coincides with the inflexions points of the isotherms. From (8.21), we get

1

!T D !v

@p

@v D !vŒA C 3B.v ! v0/

2_": _(8.38)

For temperatures above T_c, the molar volume along the extension of the coexistence line is given by v D v0, where v0 depends on T according to (8.23), so that

1

!_T D !vcA D 6pc.

T ! T_c

T_c /; (8.39)

of inflection of this isotherm, so that both (@P/@v)_T and (@2P/@v2)_T vanish at this point. Using(1), we obtain for the coordinates of the critical point

Pc = a

27b2, vc = 3b, Tc =

8a

27bR, (2)

so that the number

K ⌘ RTc/Pcvc = 8/3 = 2.666.... (3)

We thus find that, while P_c,v_c, and T_c vary from system to system (through the interaction parameters a and b), the quantity K has a common, universal value for all of them — so long as they all obey the same (i.e., van der Waals) equation of state. The experimental results for K indeed show that it is nearly the same over a large group of substances; for instance, its value for carbon tetrachloride, ethyl ether, and ethyl formate is 3.677, 3.814, and 3.895, respectively — close, though not exactly the same, and also a long way from the van der Waals value. The concept of universality is, nonetheless, there (even though the van der Waals equation of state may not truly apply).

P_r ₌ P Pc, vr = v vc, Tr = T Tc. (4)

Using(1) and (2), we readily obtain the reduced equation of state

✓

Pr + 3

v2_r

◆

(3vr 1) = 8Tr, (5)

which is clearly universal for all systems obeying van der Waals’ original equation of state (1); all we have done here is to rescale the observable quantities P, v, and T in terms of their critical values and thereby “push the interaction parameters a and b into the back-ground.” Now, if two different systems happen to be in states characterized by the same values of v_r and T_r, then their P_r would also be the same; the systems are then said to be in “corresponding states” and, for that reason, the statement just made is referred to as the “law of corresponding states.” Clearly, the passage from equation (1) to equation (5) takes us from an expression of diversity to a statement of unity!

Pr = 1 + ⇡, vr = 1 + , Tr = 1 + t. (6)

410 Chapter 12

.

Phase Transitions: Criticality, Universality, and Scaling

Equation (5)then takes the form

⇡⇣2 + 7 + 8 2_{+ 3}3⌘_{+ 3}3 _{= 8t}⇣1 + 2 + 2⌘. (7)

First of all, along the critical isotherm (t = 0) and in the close vicinity of the critical point (|⇡|,| | ⌧ 1), we obtain the simple, asymptotic result

⇡ ⇡ 3

2 3, (8)

which is indicative of the “degree of flatness” of the critical isotherm at the critical point. Next, we examine the dependence of on t as we approach the critical point from below. For this, we write(7) in the form

3 3_{+ 8(⇡} t) 2_{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

Now, a close look at the (symmetric) shape of the coexistence curve near its top (where |t| ⌧ 1) shows that the three roots 1, 2, and 3 of equation (9), which arise from the

limiting behavior of the roots v1,v2, and v3 of the original equation of state (1)as T ! Tc ,

are such that | 2| ⌧ | 1,3| and | 1| ' | 3|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

so that one of the roots, ₂, of equation (9) essentially vanishes while the other two are given by

2

+ 8t + 4t ' 0. (9a)

We expect the middle term here to be negligible (as will be confirmed by the end result), yielding

1,3 ⇡ ±2|t|1/2; (11)

note that the upper sign here pertains to the gaseous phase and the lower sign to the liquid phase.

Finally, we consider the isothermal compressibility of the system which, in terms of reduced variables, is determined essentially by the quantity (@ /@⇡)t. Retaining only the

dominant terms, we obtain from (7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

For t > 0, we approach the critical point along the critical isochore ( = 0); equation (12), with the help ofequation (10), then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

Phase Transitions: Criticality, Universality, and Scaling

Equation (5)

then takes the form

⇡ ⇣2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

First of all, along the critical isotherm (t = 0) and in the close vicinity of the critical point

(

|⇡|,| | ⌧ 1), we obtain the simple, asymptotic result

⇡ ⇡ 3 2

3_, ₍₈₎

which is indicative of the “degree of flatness” of the critical isotherm at the critical point.

Next, we examine the dependence of on t as we approach the critical point from below.

For this, we write

(7)

in the form

3 3 _{+ 8(⇡} t) 2 _{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

Now, a close look at the (symmetric) shape of the coexistence curve near its top (where

|t| ⌧ 1) shows that the three roots

1

,

2

, and

3

of

equation (9)

, which arise from the

limiting behavior of the roots v

₁

,v

₂

, and v

₃

of the original equation of state

(1)

_{as T ! T}

_c

,

are such that |

2

| ⌧ |

1,3

| and |

1

| ' |

3

|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

so that one of the roots,

₂

, of

equation (9)

essentially vanishes while the other two are

given by

2

+ 8t + 4t ' 0. (9a)

We expect the middle term here to be negligible (as will be confirmed by the end result),

yielding

1,3 ⇡ ±2|t|1/2; (11)

note that the upper sign here pertains to the gaseous phase and the lower sign to the liquid

phase.

Finally, we consider the isothermal compressibility of the system which, in terms of

reduced variables, is determined essentially by the quantity (@ /@⇡)

_t

. Retaining only the

dominant terms, we obtain from

(7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

For t > 0, we approach the critical point along the critical isochore ( = 0);

equation (12)

,

with the help of

equation (10)

, then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

Phase Transitions: Criticality, Universality, and Scaling

Equation (5)

then takes the form

⇡⇣2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

First of all, along the critical isotherm (t = 0) and in the close vicinity of the critical point

(

|⇡|,| | ⌧ 1), we obtain the simple, asymptotic result

⇡ ⇡ 3 2

3_, ₍₈₎

which is indicative of the “degree of flatness” of the critical isotherm at the critical point.

Next, we examine the dependence of on t as we approach the critical point from below.

For this, we write

(7)

in the form

3 3 _{+ 8(⇡} t) 2 _{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

Now, a close look at the (symmetric) shape of the coexistence curve near its top (where

|t| ⌧ 1) shows that the three roots

1

,

2

, and

3

of

equation (9)

, which arise from the

limiting behavior of the roots v

1

,v

2

, and v

3

of the original equation of state

(1)

as T ! T

c

,

are such that |

2

| ⌧ |

1,3

| and |

1

| ' |

3

|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

so that one of the roots,

₂

, of

equation (9)

essentially vanishes while the other two are

given by

2

+ 8t + 4t ' 0. (9a)

We expect the middle term here to be negligible (as will be confirmed by the end result),

yielding

1,3 ⇡ ±2|t|1/2; (11)

note that the upper sign here pertains to the gaseous phase and the lower sign to the liquid

phase.

Finally, we consider the isothermal compressibility of the system which, in terms of

reduced variables, is determined essentially by the quantity (@ /@⇡)

t

. Retaining only the

dominant terms, we obtain from

(7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t . (12)

For t > 0, we approach the critical point along the critical isochore ( = 0);

equation (12)

,

with the help of

equation (10)

, then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

(8)

Expansão 2: # em função de t A Eq. (*)

pode ser escrita na forma (desprezando o termo "#3_):

Eq (**)

Esta Eq. tem tres raizes, #1, #2, #3; porém, uma delas é praticamente zero.

Para mostrar isso usamos a Eq. (*) para obter " ≈ 4 t (detalhes em sala de aula).

Portanto, o termo 2("-4t) é desprezível. Logo, uma das raizes é #2 ≈0, e as outras duas

raizes são obtidas da Eq. (**), que fica na forma: Substituindo " ≈ 4 t, obtemos:

(detalhes em sala de aula)

410 Chapter 12

.

⇡ ⇣2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

⇡ ⇡ 3 2

3_, ₍₈₎

which is indicative of the “degree of flatness” of the critical isotherm at the critical point. Next, we examine the dependence of on t as we approach the critical point from below. For this, we write (7) in the form

3 3 _{+ 8(⇡} t) 2 _{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

limiting behavior of the roots v1,v2, and v3 of the original equation of state(1) as T ! Tc ,

⇡ ⇡ 4t, (10)

2 _{+ 8t + 4t ' 0.} _(9a)

1,3 ⇡ ±2|t|1/2; (11)

dominant terms, we obtain from (7) ✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

Phase Transitions: Criticality, Universality, and Scaling

Equation (5)

then takes the form

⇡ ⇣

2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

First of all, along the critical isotherm (t = 0) and in the close vicinity of the critical point

(

_{|⇡|,| | ⌧ 1), we obtain the simple, asymptotic result}

⇡ ⇡ 3

2

3_, ₍₈₎

which is indicative of the “degree of flatness” of the critical isotherm at the critical point.

Next, we examine the dependence of on t as we approach the critical point from below.

For this, we write

(7)

in the form

3 3 _{+ 8(⇡} t) 2 _{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

Now, a close look at the (symmetric) shape of the coexistence curve near its top (where

|t| ⌧ 1) shows that the three roots

1

,

2

, and

3

of

equation (9)

, which arise from the

limiting behavior of the roots v

₁

,v

₂

, and v

₃

of the original equation of state

(1)

_{as T ! T}

_c

,

are such that |

2

| ⌧ |

1,3

| and |

1

| ' |

3

|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

so that one of the roots,

₂

, of

equation (9)

essentially vanishes while the other two are

given by

2 _{+ 8t + 4t ' 0.} _(9a)

We expect the middle term here to be negligible (as will be confirmed by the end result),

yielding

1,3 ⇡ ±2|t|1/2; (11)

note that the upper sign here pertains to the gaseous phase and the lower sign to the liquid

phase.

Finally, we consider the isothermal compressibility of the system which, in terms of

reduced variables, is determined essentially by the quantity (@ /@⇡)

_t

. Retaining only the

dominant terms, we obtain from

(7)

✓_@ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

For t > 0, we approach the critical point along the critical isochore ( = 0);

equation (12)

,

with the help of

equation (10)

, then gives

✓_@ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

⇡ ⇣

2 + 7 + 8 2+ 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

⇡ ⇡ 3 2

3_, ₍₈₎

3 3_{+ 8(⇡} t) 2_{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

limiting behavior of the roots v₁,v₂, and v₃ of the original equation of state (1) _{as T ! T}_c , are such that | 2| ⌧ | 1,3| and | 1| ' | 3|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

2_{+ 8t + 4t ' 0.} _(9a)

1,3 ⇡ ±2|t|1/2; (11)

dominant terms, we obtain from(7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 16t. (12)

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

⇡⇣2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

⇡ ⇡ 3

2 3, (8)

3 3_{+ 8(⇡} t) 2_{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

limiting behavior of the roots v₁,v₂, and v₃ of the original equation of state (1) _{as T ! T}_c , are such that | 2| ⌧ | 1,3| and | 1| ' | 3|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

2_{+ 8t + 4t ' 0.} _(9a)

1,3 ⇡ ±2|t|1/2; (11)

Finally, we consider the isothermal compressibility of the system which, in terms of reduced variables, is determined essentially by the quantity (@ /@⇡)_t. Retaining only the dominant terms, we obtain from(7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

For t > 0, we approach the critical point along the critical isochore ( = 0);equation (12), with the help ofequation (10), then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

(9)

A equação anterior pode ser escrita na forma:

Expansão 3: $T em função de t

A compressibilidade isotérmica é determinada pela grandeza

Considerando apenas os termos dominantes, obtemos (demonstração em sala):

Para t>0, nos aproximamos do ponto crítico ao longo da isócora crítica (#=0). Usando "_{≈ 4t a equação anterior fica:}

✓ v vc 1 ◆ ⇡ ±2 _TT c 1 1/2

410 Chapter 12

.

Phase Transitions: Criticality, Universality, and Scaling

Equation (5)

then takes the form

⇡ ⇣

2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

First of all, along the critical isotherm (t = 0) and in the close vicinity of the critical point

(

|⇡|,| | ⌧ 1), we obtain the simple, asymptotic result

⇡ ⇡ 3

2

3_, ₍₈₎

which is indicative of the “degree of flatness” of the critical isotherm at the critical point.

Next, we examine the dependence of on t as we approach the critical point from below.

For this, we write

(7)

in the form

3 3 _{+ 8(⇡} t) 2 _{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

Now, a close look at the (symmetric) shape of the coexistence curve near its top (where

|t| ⌧ 1) shows that the three roots

1

,

2

, and

3

of

equation (9)

, which arise from the

limiting behavior of the roots v

₁

,v

₂

, and v

₃

of the original equation of state

(1)

_{as T ! T}

_c

,

are such that |

2

| ⌧ |

1,3

| and |

1

| ' |

3

|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

so that one of the roots,

₂

, of

equation (9)

essentially vanishes while the other two are

given by

2 _{+ 8t + 4t ' 0.} _(9a)

We expect the middle term here to be negligible (as will be confirmed by the end result),

yielding

1,3 ⇡ ±2|t|1/2; (11)

note that the upper sign here pertains to the gaseous phase and the lower sign to the liquid

phase.

Finally, we consider the isothermal compressibility of the system which, in terms of

reduced variables, is determined essentially by the quantity (@ /@⇡)

_t

. Retaining only the

dominant terms, we obtain from

(7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 16t . (12)

For t > 0, we approach the critical point along the critical isochore ( = 0);

equation (12)

,

with the help of

equation (10)

, then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

⇡⇣2 + 7 + 8 2 + 3 3⌘+ 3 3 = 8t⇣1 + 2 + 2⌘. (7)

⇡ ⇡ 3 2

3_, ₍₈₎

3 3 _{+ 8(⇡} t) 2 _{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

limiting behavior of the roots v1,v2, and v3 of the original equation of state (1) as T ! Tc ,

⇡ ⇡ 4t, (10)

so that one of the roots, 2, of equation (9) essentially vanishes while the other two are

given by

2

+ 8t + 4t ' 0. (9a)

1,3 ⇡ ±2|t|1/2; (11)

dominant terms, we obtain from(7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

For t > 0, we approach the critical point along the critical isochore ( = 0); equation (12), with the help of equation (10), then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

410 Chapter 12

.

Equation (5) then takes the form

⇡⇣2 + 7 + 8 2_{+ 3}3⌘_{+ 3}3 _{= 8t}⇣1 + 2 + 2⌘. (7)

⇡ ⇡ 3

2 3, (8)

which is indicative of the “degree of flatness” of the critical isotherm at the critical point. Next, we examine the dependence of on t as we approach the critical point from below. For this, we write (7)in the form

3 3_{+ 8(⇡} t) 2_{+ (7⇡} _{16t) + 2(⇡ 4t) ' 0.} (9)

Now, a close look at the (symmetric) shape of the coexistence curve near its top (where |t| ⌧ 1) shows that the three roots 1, 2, and 3 of equation (9), which arise from the limiting behavior of the roots v₁,v₂, and v₃ of the original equation of state(1)_{as T ! T}_c , are such that | 2| ⌧ | 1,3| and | 1| ' | 3|. This means that, in the region of interest,

⇡ ⇡ 4t, (10)

2 _{+ 8t + 4t ' 0.} _(9a)

1,3 ⇡ ±2|t|1/2; (11)

Finally, we consider the isothermal compressibility of the system which, in terms of reduced variables, is determined essentially by the quantity (@ /@⇡)_t. Retaining only the dominant terms, we obtain from (7)

✓ @ @⇡ ◆ t ⇡ 2 7⇡ + 9 2 _16t. (12)

For t > 0, we approach the critical point along the critical isochore ( = 0); equation (12), with the help of equation (10), then gives

✓ @ @⇡ ◆ t!0+ ⇡ 1 6t. (13)

(10)

Para t<0, nos aproximamos do ponto crítico ao longo da curva de coexistência (ao longo da qual temos #2_{≈−4t). Neste caso obtemos:}

Resumindo, podemos escrever a compressibilidade como:

onde os coeficientes A+ e A−, relacionados com o comportamento da

compressibilidade acima e abaixo da temperatura crítica, verificam A+ / A−=2. Veja que

$_T_{é divergente em T=T}_c_.

12.3 A dynamical model of phase transitions 411

For t < 0, we approach the critical point along the coexistence curve (on which 2 _{⇡ 4t);} we now obtain ✓ @ @⇡ ◆ t!0 ⇡ 1 12|t|. (14)

For the record, we quote here results for the specific heat, C_V, of the van der Waals gas (Uhlenbeck, 1966; Thompson, 1988) C_V _⇡ 8 > < > : (C_V)_ideal+ 9 2Nk ✓ 1 + 28₂₅t ◆ (t  0) (15a) (C_V)_ideal (t > 0), (15b)

which imply a finite jump at the critical point.

Equations (8), (11), (13), (14), and (15) illustrate the nature of the critical behavior dis-played by a van der Waals system undergoing the gas–liquid transition. While it differs in several important respects from the critical behavior of real physical systems, it shows up again and again in studies pertaining to other critical phenomena that have apparently nothing to do with the gas–liquid phase transition. In fact, this particular brand of behav-ior turns out to be a benchmark against which the results of more sophisticated theories are automatically compared.

12.3 A dynamical model of phase transitions

A number of physico-chemical systems that undergo phase transitions can be represented, to varying degrees of accuracy, by an “array of lattice sites, with only nearest-neighbor interaction that depends on the manner of occupation of the neighboring sites.” This simple-minded model turns out to be good enough to provide a unified, theoretical basis for understanding a variety of phenomena such as ferromagnetism and antiferromag-netism, gas–liquid and liquid–solid transitions, order–disorder transitions in alloys, phase separation in binary solutions, and so on. There is no doubt that this model considerably oversimplifies the actual physical systems it is supposed to represent; nevertheless, it does retain the essential physical features of the problem — features that account for the prop-agation of long-range order in the system. Accordingly, it does lead to the onset of a phase transition in the given system, which arises in the nature of a cooperative phenomenon.

We find it convenient to formulate our problem in the language of ferromagnetism; later on, we shall establish correspondence between this language and the languages appropriate to other physical phenomena. We thus regard each of the N lattice sites to be occupied by an atom possessing a magnetic moment µ, of magnitude gµ_Bp_{[J(J + 1)],} which is capable of (2J + 1) discrete orientations in space. These orientations define “dif-ferent possible manners of occupation” of a given lattice site; accordingly, the whole lattice is capable of (2J + 1)N different configurations. Associated with each configuration is an energy E that arises from mutual interactions among the neighboring atoms of the

that is,

!

T

D

1 6p

c

.

T

c

T

! T

c

/:

(8.40)

If we wish to determine !

T

along the critical isochoric we substitute v D v

c

into

(

8.38 ). However, the second term inside the square brackets is of higher order

compared to the first and can be neglected. Therefore, the result (

8.40 ) is also valid

for the critical isochoric near the critical point.

For temperatures below T

c

, along the coexistence line, we should specify whether

the compressibility is being determined for the liquid (v D v

L

) or for the vapor

(v D v

G

). In both cases, however, according to (

8.31 ), we have

.v

! v

0

/

2

D

!A

B

D 4.

T

c

! T

T

c

/v

_c2

;

(8.41)

so that

1 !

T

D 2v

c

A

D 12p

c

.

T

c

! T

T

c

/;

(8.42)

that is,

!

T

D

1 12p

c

.

T

c

T

c

! T

/;

(8.43)

for both phases. The results (

8.40 ) and (

8.43 ) show that the isothermal

compress-ibility diverges at the critical point according to

!

T

D A

˙

jT

c

! T j

!1

;

(8.44)

where the coefficients A

_C

and A

_!

, related to the behaviors of the compressibility

above and below the critical temperature, differ by a factor of 2, that is, A

_C

=A

_!

D 2.

8.2.6 Molar Heat Capacity

Let us determine the isochoric molar heat capacity along the critical isochoric near

the critical point (Fig.

8.3 ). Above the critical temperature, there is just a single

phase whose internal energy is given by (

8.8 ). Since c

v

D .@u=@T /

v

, then

c

v

D c

for

T > T

c

:

(8.45)

Below the critical temperature, the two phases coexist and along the critical

isochoric the internal energy is given by

u D x.cT !

a

v

L

/

C .1 ! x/.cT !

a

v

G

/;

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128 8 Criticality 0 1 2 3 T / T_c 0 0.5 1.0 1.5 2.0 κT / κ0 0 0.5 1.0 1.5 2.0 T / T_c 0 2 4 6 8 c v / R

a

b

Fig. 8.3 van der Waals fluid. (a) Isothermal compressibility !T of the fluid along the critical

isochoric (T > Tc) and of the liquid along the coexistence line (T < Tc), where !0 D 1=pc.

(b) Isochoric molar heat capacity cv along the critical isochoric line. The jump of cv at T D Tc is

equal to 9R=2

8.2.5 Compressibility

At temperatures above the critical temperature, the isotherms are strictly monotonic decreasing so that the isothermal compressibility !_T _{D !.1=v/.@v=@p/ is positive} (Fig. 8.3). The largest value of !_T along an isotherm occurs at the inflexion point. At the critical temperature it diverges because the derivative @p=@v vanishes at this point. According to (8.21), the pressure varies with the molar volume, along the critical isotherm, and around the critical point, according to the equation

p D p_c ! 3pc 2v3 c

.v ! v_c/3: (8.37) Let us determine next the behavior of !_T around the critical point, along the coexistence line and its extension, defined by (8.29), which coincides with the inflexions points of the isotherms. From (8.21), we get

1

!_T D !v @p

@v D !vŒA C 3B.v ! v0/

2_": _(8.38)

For temperatures above T_c, the molar volume along the extension of the coexistence line is given by v D v0, where v0 depends on T according to (8.23), so that

1 !T D !v cA D 6pc. T ! T_c Tc /; (8.39)

(a) Compressibilidade isotérmica de um fluido de van der Waals. Para T > Tc o

resultado corresponde ao gás longo da isócora crítica. Para T < Tc o resultado

corresponde ao líquido ao longo da linha de coexistência. A constante $0 é

dada por $0 = 1/pc.

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Em torno do ponto crítico, várias grandezas termodinâmicas têm a forma de uma lei de potência da forma:

onde y podem ser a compressibilidade ou a capacidade calorífica molar, por exemplo. O expoente x é denominado expoente crítico.

No caso da transição de fase líquido-vapor são necessários 6 expoentes críticos, normalmente denominados: α, α’ %, &, &’, ' que descrevem as seguintes grandezas:

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Os valores dos expoentes críticos para diversas substâncias são apresentados na Tabela abaixo. Os resultados mostram que existe uma boa concordância entre os valores de um expoente dado para diversas substâncias, indicando um carácter universal do comportamento crítico.

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Os expoentes α, α’, &, &’ são positivos, indicando que a capacidade calorífica e a compressibilidade divergem.

A figura mostra a divergência da capacidade calorífica molar do argônio e do dióxido de carbono ao longo da linha isocórica crítica como função da temperatura._{8.3 Critical Behavior} ₁₃₁

T (K) 146 148 150 152 154 0 50 100 150 200 250 c v (J/mol K) Ar 10 20 30 40 50 θ (o_C) 0 50 100 150 200 250 300 cv (J/mol K) CO₂ a b

Fig. 8.4 Molar heat capacity cv along the critical isochoric line as a function of temperature,

(a) for the argon, with experimental data obtained by Voronel et al. (1973), and (b) for carbon dioxide, with experimental data obtained by Beck et al. (2002)

jp ! pcj " j! ! !cjı or jp ! pcj " jv ! vcjı: (8.51)

We consider finally the isochoric molar heat capacity cv. We assume that this

quantity behaves along the critical isochoric according to

cv " jT ! Tcj!˛; (8.52)

both above and below the critical temperature. In general we expect a weak divergence for the isochoric molar heat capacity. It is possible that this divergence is of the logarithmic type

c_v " ln jT ! T_cj: (8.53)

The notation a.x/ " b.x/ used to characterize the critical behavior, where both a and b diverge or both vanish when x ! 0, means that the ratio a=b approaches a finite constant or, equivalently, that ln a= ln b ! 1 when x ! 0.

If the critical behavior corresponds in fact to a power law, the critical exponent is defined by the power law. It is convenient, however, to define the critical exponent more broadly to include other behaviors that are not strict power laws. Thus, we define the critical exponent ", associated with a quantity a.x/, by

" D lim

x!0

ln a.x/

ln x ; (8.54)

or equivalently by a " x"_{. With this definition we see that a logarithm divergence}

is associated to an exponent zero.

The exponents ˇ, #, ı and ˛ contained in the power laws (8.49)–(8.52), called critical exponents, concisely characterize the critical behavior. Table 8.2 presents experimental values of the exponents for the critical point related to the liquid-vapor

Molar heat capacity cv along the critical isochoric line as a function of temperature, (a) for the argon, with experimental data obtained by Voronel et al. (1973), and (b) for carbon dioxide, with experimental data obtained by Beck et al. (2002)

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De maneira análoga, os expoentes críticos podem ser definidos para outras transições de fase, como, por exemplo a transição de fase ferromagnética, transições ordem-desordem em ligas metálicas, etc.

É possível mostrar, que muitas transições de fase de segunda ordem podem ser descritas pelos mesmos expoentes críticos.