Entropía

  Clausius comprendio� en 1865 que él había

descubierto una nueva propiedad termodinámica y

decidio� nombrarla entropía, la cual esta� designada

por S y definida como

dS a dQTb

int rev

!!1kJ>K 2

Entropía

  Características

  Se define el cambio de entropía mas que la medida de la

entropía misma.

  Se puede asumir la entropía 0 en un estado inicial y se

calcula el incremento de la entropía en el estado final

  Datos tabulados Proceso

irreversible

Proceso

reversible

1

2

0.3 0.7 S, kJ/K

∆S = S2

– S1 = 0.4 kJ/K

T

Entropía

  Procesos Isotérmicos

  Recuerde que los procesos isotérmicos de transferencia

de calor son interna- mente reversibles,

¢S

2

1

a dQTb

int rev

2

1

a dQT0

bint rev

1

T0

2

1

1dQ 2 int rev

2

Entropía

  Ejemplo.- Un dispositivo compuesto por cilindro

émbolo contiene una mezcla de líquido y vapor de

agua a 300 K. Durante un proceso a presión constante

se transfieren al agua 750 kJ de calor. Como resultado,

la parte líquida en el cilindro se vaporiza. Determine el

cambio de entropía del agua durante este proceso.

Entropía

  Conclusiones

  Es un sistema cerrado, únicamente se transfiere calor

  La temperatura es constante y se cumple la relación Q/

T

  No depende de la naturaleza del fluido, únicamente de

las condiciones de Q y T

Entropía

  Principio del Incremento de la Entropía

  Transferencia de entropía por medio de calor

■

Proceso 1-2 (reversible o irreversible)

1

2

Proceso 2-1 (internamente reversible)

FIGURA 7-5

■

en la figura 7-5. De la desigualdad de Clausius,

!2

1

dQ

T! !

1

2

a dQTb

int rev

# 0

" dQT # 0

dSdQ

T

Entropía

  Principio del Incremento de la Entropía

  El signo de la desigualdad en las relaciones precedentes

es un constante recordatorio de que el cambio de

entropía de un sistema cerrado durante un proceso

irreversible siempre es mayor que la transferencia de

entropía

dSdQ

T

¢Ssis # S2 ! S1 # "2

1

dQ

T$ Sgen

Entropía

  Conclusiones

  La cantidad de entropía Sgen no puede ser negativa

  Sgen depende del proceso no es una propiedad del

sistema

  Si no se presenta transferencia de entropía el cambio de

entropía dará cuenta de la Sgen

Entropía

  Para un sistema aislado se tendría que el cambio de entropía

  La entropía de un sistema aislado durante un proceso siempre se incrementa o, en el caso límite de un proceso reversible, permanece constante. En otros términos, nunca disminuye. Esto es conocido como el principio de incremento de entropía

¢Saislado % 0

Entropía

  Es posible considerar a un sis- tema y sus alrededores

como dos subsistemas de un sistema aislado, y el cam-

bio de entropía de éste durante un proceso resulta de la

suma de los cambios de entropía del sistema y sus

alrededores, la cual es igual a la generación de entropía

porque un sistema aislado no involucra transferencia

de entropía. Es decir,

Entropía

Subsistema

1

Subsistema

3

Subsistema

2

Subsistema

N

(Aislado)

∆Stotal = ∆Si > 0 i=1

N

Σ

Sgenerada ! ¢Stotal ! ¢Ssistema " ¢Salrededores # 0

Entropía

  Como ningún proceso real es verdaderamente

reversible, es posible con- cluir que alguna entropía se

genera durante un proceso y por consiguiente la

entropía del universo, la cual puede considerarse como

un sistema aislado, esta� incrementándose

continuamente.

Entropía

  El principio de incremento de

entropía no implica que la de

un sistema no pueda disminuir.

El cambio de entropía de un

sistema puede ser negativo

durante un proceso pero la

generación de entropía no.

S gen • 7 0 proceso irreversible

! 0 proceso reversible

6 0 proceso imposible

Alrededores

Sistema

∆Ssistema = –2 kJ/K

∆Salrededores = 3 kJ/K

Sgenerada = ∆Stotal = ∆Ssistema + ∆Salrededores = 1 kJ/K

Q

Segunda Ley Análisis

  Una vez entendidos los conceptos de motores o máquinas

reversibles, el siguiente paso es desarrollar el

entendimiento analítico que fundamenta lo estudiado.

Segunda Ley

Preposición 1

Preposición 2

Preposición 3

Preposición 4

Preposición 1

  Para un ciclo reversible que absorbe calor a una

temperatura t1 y entrega calor a una temperatura t2,

demuestre que el radio de Q1/Q2 depende únicamente

de las temperaturas

The Second Law of Thermodynamics 75

3. 7 Second Law: The Analytical Statement

Once we understand the concept of a reversible heat engine, we are ready

to develop the analytical statement of the second law. This is carried out

with the use of the ensuing four 'Propositions' discussed briefly here (for

an in depth presentation see Denbigh p.26):

I.If a heat engine performs a reversible cycle by:

*absorbing an amount of heat Q1 at a temperature t 1, and

* rejecting an amount of heat Q2 at a temperature t2,

then the ratio (Q11Qv is a function of t1 and t2 only, i.e.:

IQll I Qzl = f(tl ,tz)

where the absolute values are used to eliminate concern with signs.

II. This ratio is given by:

I Qll /U1)

I Qzl f(t2)

where f (t) is the thermodynamic temperature, which can be set equal to

the 'ideal gas' temperature, i.e.:

IQII Tl

IQzl T2

III. The entropy S, defined by:

dS = dQr T

where the subscript r stands for reversible, is a state function.

IV.For any natural (spontaneous) process, the total entropy change

(.dS10r) equal to:

* the entropy change of the system (L1Ssys), plus

*that of the surroundings (L1Ssw.),

is larger than zero. It is equal to zero, only when the process is rever-

sible, and it can never be negative:

L1Sror = L1Ssys + L1Ssur 0

Preposición 2

  El radio Q1/Q2 está dado por la expresión f(t2)/f(t1)

donde f(t) es la temperatura termodinámica de un gas

ideal

The Second Law of Thermodynamics 75

3. 7 Second Law: The Analytical Statement

Once we understand the concept of a reversible heat engine, we are ready

to develop the analytical statement of the second law. This is carried out

with the use of the ensuing four 'Propositions' discussed briefly here (for

an in depth presentation see Denbigh p.26):

I.If a heat engine performs a reversible cycle by:

*absorbing an amount of heat Q1 at a temperature t 1, and

* rejecting an amount of heat Q2 at a temperature t2,

then the ratio (Q11Qv is a function of t1 and t2 only, i.e.:

IQll I Qzl = f(tl ,tz)

where the absolute values are used to eliminate concern with signs.

II. This ratio is given by:

I Qll /U1)

I Qzl f(t2)

where f (t) is the thermodynamic temperature, which can be set equal to

the 'ideal gas' temperature, i.e.:

IQII Tl

IQzl T2

III. The entropy S, defined by:

dS = dQr T

where the subscript r stands for reversible, is a state function.

IV.For any natural (spontaneous) process, the total entropy change

(.dS10r) equal to:

* the entropy change of the system (L1Ssys), plus

*that of the surroundings (L1Ssw.),

is larger than zero. It is equal to zero, only when the process is rever-

sible, and it can never be negative:

L1Sror = L1Ssys + L1Ssur 0

The Second Law of Thermodynamics 75

3. 7 Second Law: The Analytical Statement

Once we understand the concept of a reversible heat engine, we are ready

to develop the analytical statement of the second law. This is carried out

with the use of the ensuing four 'Propositions' discussed briefly here (for

an in depth presentation see Denbigh p.26):

I.If a heat engine performs a reversible cycle by:

*absorbing an amount of heat Q1 at a temperature t 1, and

* rejecting an amount of heat Q2 at a temperature t2,

then the ratio (Q11Qv is a function of t1 and t2 only, i.e.:

IQll I Qzl = f(tl ,tz)

where the absolute values are used to eliminate concern with signs.

II. This ratio is given by:

I Qll /U1)

I Qzl f(t2)

where f (t) is the thermodynamic temperature, which can be set equal to

the 'ideal gas' temperature, i.e.:

IQII Tl

IQzl T2

III. The entropy S, defined by:

dS = dQr T

where the subscript r stands for reversible, is a state function.

IV.For any natural (spontaneous) process, the total entropy change

(.dS10r) equal to:

* the entropy change of the system (L1Ssys), plus

*that of the surroundings (L1Ssw.),

is larger than zero. It is equal to zero, only when the process is rever-

sible, and it can never be negative:

L1Sror = L1Ssys + L1Ssur 0

Preposición 3

  Se dedujo una nueva propiedad que se definió como

entropía la cual está definida por la siguiente expresión

y que se trata de una variable de estado

The Second Law of Thermodynamics 75

3. 7 Second Law: The Analytical Statement

Once we understand the concept of a reversible heat engine, we are ready

to develop the analytical statement of the second law. This is carried out

with the use of the ensuing four 'Propositions' discussed briefly here (for

an in depth presentation see Denbigh p.26):

I.If a heat engine performs a reversible cycle by:

*absorbing an amount of heat Q1 at a temperature t 1, and

* rejecting an amount of heat Q2 at a temperature t2,

then the ratio (Q11Qv is a function of t1 and t2 only, i.e.:

IQll I Qzl = f(tl ,tz)

where the absolute values are used to eliminate concern with signs.

II. This ratio is given by:

I Qll /U1)

I Qzl f(t2)

where f (t) is the thermodynamic temperature, which can be set equal to

the 'ideal gas' temperature, i.e.:

IQII Tl

IQzl T2

III. The entropy S, defined by:

dS = dQr T

where the subscript r stands for reversible, is a state function.

IV.For any natural (spontaneous) process, the total entropy change

(.dS10r) equal to:

* the entropy change of the system (L1Ssys), plus

*that of the surroundings (L1Ssw.),

is larger than zero. It is equal to zero, only when the process is rever-

sible, and it can never be negative:

L1Sror = L1Ssys + L1Ssur 0

80 Applied Chemical Engineering Thermodynamics

where the subscript r depicts the reversible character of the operation, and

call it Entropy. (As we will see in Chapter 4 this term was introduced by

Clausius in 1865.)

We will demonstrate next that entropy is a state junction, i.e. that there

is indeed more to the ratio (Q!T) than meets the eye.

To this purpose consider a gas that undergoes a Carnot cycle by absor-

bing an amount of heat Q1 at T1 and rejecting an amount of heat Q2 at T2•

To calculate the entropy changes of the gas going through this cycle, we

notice that they occur only in the heat absorption and heat rejection steps

since - according to Eq.3.10.1 - there is no entropy change in the two

adiabatic ones:

and:

QJ Q2 ..1S = ..1S1 + ..1S2 = - +- (3.10.2)

T1 T2

Thus - according to Eq.3.5.4- the total entropy change of the gas going

through the whole cycle is zero:

fas = ..1s = o (3.10.3)

Eq.3.10.3 applies to any reversible cycle, i.e. other than those involving

two isothermal and two adiabatic steps (for proof see, among others,

Denbigh), and demonstrates that entropy is a state function.

Consider, for example, a body going from state A to state B following

path 'a', or path 'b', reversibly in both cases. Application of Eq.3.10.3

to the cycle: A to B through path 'a' and then back to A through path 'b'

gives:

J: dS(a) + J dS(b) = 0; or,

J: dS(a) = - dS(b) = J: dS(b)

The entropy change of the body is thus independent of the path followed

- provided it is a reversible one- and, consequently, entropy is a state

function.

Proposición 4

  Para un proceso natural o espontáneo, el cambio total

de entropía es igual a:

  El cambio de entropía el sistema

  El cambio de entropía de los alrededores

  Es mayor a cero, excepto para procesos reversibles en

donde es cero, y jamás es menor que cero

The Second Law of Thermodynamics 75

3. 7 Second Law: The Analytical Statement

Once we understand the concept of a reversible heat engine, we are ready

to develop the analytical statement of the second law. This is carried out

with the use of the ensuing four 'Propositions' discussed briefly here (for

an in depth presentation see Denbigh p.26):

I.If a heat engine performs a reversible cycle by:

*absorbing an amount of heat Q1 at a temperature t 1, and

* rejecting an amount of heat Q2 at a temperature t2,

then the ratio (Q11Qv is a function of t1 and t2 only, i.e.:

IQll I Qzl = f(tl ,tz)

where the absolute values are used to eliminate concern with signs.

II. This ratio is given by:

I Qll /U1)

I Qzl f(t2)

where f (t) is the thermodynamic temperature, which can be set equal to

the 'ideal gas' temperature, i.e.:

IQII Tl

IQzl T2

III. The entropy S, defined by:

dS = dQr T

where the subscript r stands for reversible, is a state function.

IV.For any natural (spontaneous) process, the total entropy change

(.dS10r) equal to:

* the entropy change of the system (L1Ssys), plus

*that of the surroundings (L1Ssw.),

is larger than zero. It is equal to zero, only when the process is rever-

sible, and it can never be negative:

L1Sror = L1Ssys + L1Ssur 0