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