Download 1 8. Entropy (Hiroshi Matsuoka) Why do we need entropy? There

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8. Entropy
(Hiroshi Matsuoka)
Why do we need entropy? There are two main reasons. First, entropy is a “state variable”
directly related to “heat” flowing into a system so that we can estimate the heat by keeping track
of how much the entropy of the system changes. In this chapter, we will focus mostly on this
aspect of entropy.
Secondly, entropy allows us to distinguish irreversible processes from
reversible ones through the second law of thermodynamics. In Ch.11, we will discuss this side
of entropy.
8.1 Entropy as a state variable
What is entropy?
According to the first law of thermodynamics applied to an infinitesimal quasi-static process,
the internal energy U of a system changes infinitesimally by
dU = ! Q + ! W = ! Q " PdV ,
qs
qs
qs
where ! Qq s is an infinitesimal amount of heat flowing quasi-statically into the system while
! W q s is an infinitesimal amount of work done quasi-statically on the system. Note that the
qs
quasi-static work ! W is related to an infinitesimal change dV of the system’s volume, a state
variable, by
! W q s = " PdV .
qs
Using this equation, we can calculate a finite amount of work W done quasi-statically on the
qs
system through a finite process by integrating ! W :
Vi
W
qs
Vi
= " ! W = # " P(V $)dV $ .
qs
Vi
Vi
Now, wouldn’t it be nice if we can find a state variable that is related to ! Qq s in a similar
way so that we can calculate a finite amount of heat Qq s flowing quasi-statically into the system
through a finite process by integrating ! Qq s? The state variable we wish to find is in fact the
entropy of the system whose change dS is related to ! Qq s by
! Qq s = TdS ,
2
where T is the absolute temperature of the system, which plays a role similar to that of ! P in
! Wqs.
Using this relation, we can also find the entropy change dS from the infinitesimal quasi-static
heat ! Qq s:
!Q q s
,
dS =
T
which implies that when dividing ! Qq s, which is not a change of a state variable, by T, which is a
state variable, we get a change of the entropy, which we claim to be a state variable. Clearly,
this is a claim that needs to be justified. It turns out that to justify this relation we need the
second law of thermodynamics. We will justify this relation later when we discuss the second
law. Until then, we simply accept this claim and regard the entropy as a state variable.
Entropy defined on the macroscopic level by the relation dS = !Qq s T is not a concept we
can intuitively understand, although this relation tells us, at least, that the entropy of a system
increases when some heat flows into the system and that with the same amount of heat flowing
into the system, the resulting increase of the entropy becomes less and less at higher
temperatures.
One important message of the relation dS = !Qq s T is that if we block any quasi-static heat
exchange between a system and its outside so that ! Qq s = 0 , the entropy of the system remains
constant. In other words, the entropy stays constant in a quasi-static adiabatic (with no heat
transfer) process. When we discuss a quasi-static adiabatic process as a part of a heat engine, we
can then use the entropy of the engine to characterize the adiabatic process.
Entropy on the microscopic level
For an isolated system, the internal energy of the system is simply the total energy carried by
the constituent molecules or atoms of the system so that the internal energy is equal to an energy
eigenvalue, E(V ,N ) , that we find by solving the Schrödinger equation for the system. What is
the entropy of the system on the microscopic level? Roughly speaking, the entropy of the system
counts the number of ways of distributing the total energy of the system among the constituent
molecules or atoms. More precisely, we will find in Ch.15 that the entropy is given by
S = k B lnW (U, V, N ) ,
where W (U, V, N ) is the number of energy eigenstates whose energy eigenvalues, E(V ,N ) , are
all near U or in an interval [U, U + !U ] , where ! U << U . Each such energy eigenstate
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corresponds to a way of distributing the total energy among the constituent molecules or atoms.
As W increases with U and V, the entropy also increases with U and V.
Entropy is extensive
We can show that entropy is an “extensive” variable by noting that if we combine two
identical systems each of which receives an infinitesimal amount of heat ! Qq s, then the
combined system receives the total amount 2! Qq s of heat so that the entropy of the combined
system changes by
2! Qq s
!Q q s
dScombined =
=2
= 2dS = d( 2S ) ,
T
T
which implies that the entropy of the combined system is twice as much as the entropy of each
subsystem:
Scombined = 2S .
The entropy therefore scales with the system size and is an extensive variable so that it can be
expressed as
S = ns ,
where n is the mole number of the system and s is the corresponding molar entropy, which is a
function of two intensive variables.
The fundamental equation of thermodynamics: dU = TdS ! PdV
With the relation ! Qq s = TdS , we can now rewrite the first law for an infinitesimal quasistatic process, where the mole number n of a system is kept constant, as
dU = ! Qq s + ! W q s = TdS " PdV ,
which we call “the fundamental equation” of thermodynamics simply because almost all the
thermodynamic equations or formulas follow from this equation. As the relation ! Qq s = TdS is a
consequence of the second law of thermodynamics, this fundamental equation is a result of both
the first and the second laws of thermodynamics.
Using the fundamental equation, we can also see that the entropy S must be extensive since
dS ! n :
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dS =
!1
1
P
1
P
P $
dU + dV = d (nu ) + d (nv ) = n# du + dv & .
"T
T
T
T
T
T %
Comparing this equation for dS in terms of dU and dV with the following calculus equation,
" !S %
" !S %
dS = $
' dU + $
' dV ,
# !U & V ,n
# ! V & U ,n
we also find
" !S %
1
$
' = >0
# !U & V ,n T
and
" !S %
P
$
' = >0,
# !V & U, n T
which implies that the entropy increases with the internal energy U and the volume V. However,
as we will see below, the entropy may not always increase when the volume is increased while
the temperature, instead of the internal energy, is kept constant.
Entropy and heat capacities
As an infinitesimal change in the entropy of a system is related to an infinitesimal amount of
quasi-static heat ! Qq s flowing into the system by
! Qq s = TdS ,
the heat capacity at constant volume CV and the heat capacity at constant pressure CP of the
system can be expressed in terms of partial derivatives of the entropy as follows.
If the volume of the system is kept constant during an infinitesimal quasi-static process, we
qs
know that the heat ! QV ,n that flows into the system is related to CV and the temperature change
dT inside the system by
! QVq ,ns = CV dT .
According to the following calculus equation,
" !S %
" !S %
dS = $ ' dT + $ ' dV ,
# !T & V , n
# !V & T, n
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if the volume is kept constant so that dV = 0 , then the entropy change dSV ,n during this
infinitesimal quasi-static process must satisfy
" !S %
dSV ,n = $ ' dT
# ! T & V ,n
so that
which implies
# "S &
qs
CV dT = ! QV ,n = TdSV , n = T % ( dT ,
$ " T ' V ,n
" !S %
CV = T $ ' .
# !T & V ,n
Similarly, if the pressure P of the system is kept constant during an infinitesimal quasi-static
qs
process, we know that the heat ! QP, n that flows into the system is related to CP and the
temperature change dT inside the system by
! QP,q sn = CP dT .
According to the following calculus equation,
" !S %
" !S%
dS = $ ' dT + $ ' dP ,
# !T & P ,n
# !P & T, n
if the pressure is kept constant so that dP = 0 , then the entropy change dSP ,n during this
infinitesimal quasi-static process must satisfy
" !S %
dSP ,n = $ ' dT
# ! T & P, n
so that
which implies
# "S &
CP dT = !QqP,sn = TdSP ,n = T % ( dT ,
$ "T ' P,n
" !S %
CP = T $ ' .
# !T & P ,n
As the entropy is extensive so that S = ns , where n is the mole number, if we divide the both
sides of these equations by n, we find
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" !s %
cv = T $ '
# !T & v
and
" !s %
c P = T$ ' .
# !T& P
As CV > 0 and cv > 0 for any macroscopic system, we then find
" !S %
$ ' >0
# !T & V , n
and
" !s %
$ ' >0
# !T & v
so that S = S (T, V,n ) and s = s(T.v ) are increasing functions of temperature. Similarly, as
CP > 0 and c P > 0 for any system, we also find
" !S %
$ ' >0
# !T & P,n
and
" !s %
$ ' >0
# !T & P
so that S = S (T, P, n ) and s = s(T. P) are increasing functions of temperature.
State variables and molar heat capacities
We have thus found that the heat capacity at constant volume of a system is related to a
partial derivative of both the internal energy and the entropy of the system:
" !U %
" !S %
CV = $
' = T$ ' .
# !T & V ,n
# ! T & V ,n
The corresponding equations hold for the molar variables:
" !u %
" !s %
cv = $ ' = T $ ' .
# !T&v
# !T & v
Similarly, the heat capacity at constant pressure of the system is related to a partial derivative
of both the enthalpy and the entropy of the system:
" !H %
" !S %
CP = $
' = T$ ' .
# !T & P ,n
# !T & P ,n
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The corresponding equations hold for the molar variables:
" !h %
" !s %
cP = $
' = T$ ' .
# !T &P
# !T & P
The volume and pressure dependence of entropy
In Sec.8.2, we will show that we can express dS in terms of dT and dV as
dS =
CV
!
dT +
dV
T
"T
dS =
CP
dT ! V"dP .
T
or in terms of dT and dP as
Comparing “the first dS equation” with the following calculus equation,
" !S %
" !S %
dS = $ ' dT + $ ' dV ,
# !T & V , n
# !V & T, n
we find
" !S %
C
$ ' = V and
# !T & V , n T
" !S %
(
$
' =
.
# !V & T ,n ) T
Since ! can be negative for some systems, the entropy may not always increase with the volume
while the temperature is kept constant.
Comparing “the second dS equation” with the following calculus equation,
" !S %
" !S%
dS = $ ' dT + $ ' dP ,
# !T & P ,n
# !P & T, n
we find
" !S %
C
$ ' = P and
# !T & P,n T
" !S %
$ ' = (V ) .
# !P & T ,n
Since ! can be negative for some systems, the entropy may not always decrease with the
pressure while the temperature is kept constant.
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Calculating an entropy difference !S between two equilibrium states
We can also calculate an entropy difference !S between two equilibrium states of a system
by using either of the two dS equations:
dS =
CV
!
dT +
dV
T
"T
dS =
CP
dT ! V"dP .
T
or
We calculate the entropy difference !S between the two states by calculating the change of
the entropy along a quasi-static process connecting these states. As the entropy of the system is a
state variable, this entropy change !S is process-independent so that we can use any quasi-static
process connecting these states.
To calculate !S between states, (T1 ,V 1 ,n) and (T2 ,V2 ,n) , we can use the following quasistatic process, (T1 ,V 1 ,n) ! (T2 ,V1 , n) ! (T2 ,V 2 , n) , so that
!S = S(T2 ,V2 ,n) " S(T1 ,V1 ,n) =
T2
= $T
1
$ dS
( T1 ,V1 ,n ) #( T 2 ,V1 ,n ) #( T2 ,V 2 ,n )
V 2 & (T , V %,n )
CV (T %,V1 ,n)
2
dT % + $V
dV %
1
T%
' T ( T2 , V %,n)
Example: For a low-density gas with a constant heat capacity at constant volume (i.e.,
CV = ncv = const ), we find
T2
!S = #T
1
,& ) cv
V2 & $ )
CV
T
dT " + #V (( ++
dV " = nRln ..(( 2 ++
1 ' % *
T"
T T = T2
-' T1 *
R
,& ) c v
& V )/
2
1
(( ++ = nR ln .(( T2 ++
.' T *
' V1 * 10
- 1
R
& v )/
(( 2 ++ 1
' v1 * 10
(HW#8.1.1: show this)
This entropy increases with both the final temperature T2 and the final molar volume v2
of the gas. As the molar internal energy of the low-density gas is given by u = cvT , the
entropy change can be also expressed as
(" % c v
u
!S = nRln ** $$ 2 ''
) # u1 &
R
( " % cv
" v %+
$$ 2 '' - = nR ln * $$ U2 ''
*# U &
# v1 & -,
) 1
R
" V %+
$$ 2 '' - .
# V 1 & -,
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To calculate !S between states, (T1 ,P1 ,n ) and (T2 ,P2 ,n) , we can use the following quasistatic process, (T1 ,P1 ,n ) ! ( T2 , P1 ,n) ! (T2 ,P2 ,n) , so that
!S = S(T2 ,P2 ,n) " S(T1 , P1 ,n ) =
T2
= $T
1
$ dS) (
( T1 , P1 , n )# ( T2 ,P1 ,n
# T 2 , P2 , n )
P2
CP (T %,P1 , n)
dT % " $P V (T2 , P %,n)& (T2 , P %,n)dP %
1
T%
Example: For a low-density gas with a constant heat capacity at constant pressure (i.e.,
CP = const ), we find
T2
!S = #T
1
, & ) cP
P2
CP
T
dT " $ #P (V % )T =T 2 dP " = nRln .. (( 2 ++
1
T"
-' T1 *
R
, & ) cv
& P )/
1
1
(( ++ = nRln . (( T2 ++
.' T *
' P2 * 10
- 1
R
& v )/
(( 2 ++ 1
' v1 * 10
(HW#8.1.2: show this)
This entropy increases with the final temperature T2 of the gas but decreases with its final
pressure P2 , which makes sense because higher pressures correspond to smaller molar
volumes.
Calculating quasi-static heat and quasi-static work
As noted above, using the equation ! Qq s = TdS , we can calculate a finite amount of heat Qq s
flowing quasi-statically into the system during a finite (as opposed to infinitesimal) quasi-static
process by integrating ! Qq s:
Qq s =
#!Q
qs
.
i" f
Using the dS equations,
dS =
CV
!
dT +
dV
T
"T
and
we can then express ! Qq s as
! Qq s = TdS = CV dT + T
"
dV
#T
dS =
CP
dT ! V"dP ,
T
10
and
! Qq s = TdS = CP dT " TV#dP ,
respectively, and calculate Qq s by
# !Q
Qq s =
=
qs
( Ti ,V i ,n ) "( T f ,V f , n )
#
( T i ,V i , n )" ( T f ,V f ,n )
or
# !Q
Qq s =
&(
*(
$ (T, V, n)
'CV ( T, V,n )dT + T
dV +
()
% T (T ,V,n) (,
qs
( Ti , Pi ,n ) "( T f ,P f ,n )
# {C (T ,P, n)dT $ TV (T ,P,n)% (T, P, n) dP}
=
(
P
( T i ,Pi , n )" T f , P f ,n
)
qs
We can also express an infinitesimal amount ! W of quasi-static work done on the system
as
! W q s = " P(T ,V,n)dV
or
! W q s = " PV (T ,P,n){# (T, P, n )dT " $ T (T ,P,n)dP} ,
where we have used dV = V!dT " V # T dP in ! W q s = " P(T ,V,n)dV . We can then calculate the
qs
qs
amount of quasi-static work W done on the system by integrating ! W along a finite process:
Wqs =
# !W
qs
#!W
qs
( T i , Vi , n )" (T f , V f ,n )
or
Wqs =
=$
# P (T, V, n)dV
( T i ,V i , n )" ( T f ,V f ,n )
( T i , Pi ,n ) "( T f , P f ,n )
=$
# [ PV (T, P, n ){% (T ,P,n)dT $ &
( Ti ,P i ,n ) "( T f ,P f ,n )
T
( T, P, n) dP}]
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Answers for the homework questions in Sec.8.1
HW#8.1.1
!S = #
T2
T1
V2 & $ )
T2 1
V2 P
CV
dT " + # (( ++
dV " = CV #
dT " + #
dV "
V1 ' % *
T1 T "
V1 T
T"
T T=T
2
2
= CV [ ln T "]TT21 + #
V2
V1
&T )
nR
dV " = CV ln (( 2 ++ + nR[ln V "]VV21
V"
' T1 *
,& T ) CV
&V )
CV &( T2 )+
2
= nR
ln
+ nRln (( ++ = nR ln .. (( 2 ++
nR (' T1 +*
' V1 *
-' T1 *
, & ) cv
T
= nRln .. (( 2 ++
-' T1 *
HW#8.1.2
T2
!S = #T
1
R
nR
&V )/
(( 2 ++ 1
' V1 * 10
& v )/
(( 2 ++ 1
' v1 * 10
P2
T2 1
P2 &
CP
dT " $ #P (V % )T =T 2 dP " = CP #T
dT " $ #P ( V
1
1 T"
1 '
T"
= CP [ ln T " ]T 21 $ #
T
P2
P1
&T )
nR
P
dP " = CP ln (( 2 ++ $ nR[ ln P "]P21
P"
' T1 *
,& ) c P
&P )
CP &( T2 )+
T
2+
(
= nR ln( + $ nR ln ( + = nRln ..(( 2 ++
nR ' T1 *
' P1 *
-' T1 *
,& ) ( c v + R ) R & ) /
,& T ) c v
T2
P1 1
.
(( ++ 1 = nR ln .(( 2 ++
= nRln . (( ++
.' T1 *
' P2 * 0
-' T1 *
, & ) cv
T
= nRln .. (( 2 ++
-' T1 *
R
, & ) cv
& Rv ) /
2
1
((
++ = nRln . (( T2 ++
.' T *
' v1 R * 10
- 1
R
R
R
&P )/
(( 1 ++ 1
' P2 * 10
& P1 T2 ) /
((
++ 1
' T1 P2 * 10
& v )/
(( 2 ++ 1
' v1 * 10
8.2 The dS equations from the fundamental equation
Our main goal in this section is to derive the following dS equations:
dS =
CV
!
dT +
dV
T
"T
dS =
CP
dT ! V"dP ,
T
and
1)
+
dP "
T * T =T 2
12
using dV = V!dT " V # T dP
introduced in Sec.5.6.1 and the fundamental equation of
thermodynamics,
dU = TdS ! PdV .
!
!
!
!
which we may call “ F = ma ” of thermodynamics because like “ F = ma ” in classical mechanics,
almost all the thermodynamic relations are consequences of this equation or its extended form,
which includes a term with a change dn in the mole number of a system so that
dU = TdS ! PdV + µ˜ dn ,
where µ˜ is called the chemical potential of the system. This extended form is useful when we
deal with a quasi-static process in which the mole number of the system also changes because the
system is connected with a particle reservoir that can exchange molecules or atoms with the
system. In this section, we assume that the mole number of a system stays constant or n = const .
Mathematically, the fundamental equation, dU = TdS ! PdV + µ˜ dn , indicates that U changes
when S, V, or n is changed so that in this equation U is regarded as a function of S, V, and n:
U = U( S, V,n) .
The first dS equation: dS = (CV T )dT + (! " T )dV ( n = const )
In this sub-section, we will derive the first dS equation,
dS =
CV
!
dT +
dV ,
T
"T
using the fundamental equation as well as a relation,
(! P ! T )v = " # T ,
derived from
dv = v !dT " v# T dP in Sec.5.6.1. We start with the following calculus equation for dS:
" !S %
" !S %
dS = $ ' dT + $ ' dV ,
# !T & V , n
# !V & T, n
where we regard S as a function of T, V, and n: S = S (T, V,n ) . Note that there is no term with dn
as we assume that n is kept constant. In Sec.8.1, we have already shown
" !S %
C
$ ' = V
# !T & V , n T
13
so that
dS =
" !S %
CV
dT + $
' dV .
# ! V & T ,n
T
We then need to show
" !S %
" ! P%
" !P %
(
$
' =$ ' = $ ' =
,
# !V & T ,n # !T & V , n # !T & v ) T
where the first equality is one of so-called Maxwell’s relations. The second equality holds
because the pressure P is an intensive variable so that it can be regarded as a function of two
intensive variables, the temperature T and the molar volume v, so that P = P (T, v ) and because
keeping V and n constant in (! P !T )V , n is equivalent to keeping v constant in (! P !T )v . As
mentioned above, in Sec.5.6.1, we have already derived the third equality, (! P !T )v = " # T ,
from dv = v !dT " v# T dP .
To show the first equality, we define a new state variable F called Helmholtz free energy by
F ! U " TS .
Using the fundamental equation of thermodynamics, dU = TdS ! PdV , we then find
dF = d(U ! TS) = dU ! d(TS ) = TdS ! PdV ! (TdS + SdT ) = !SdT ! PdV
or
dF = !SdT ! PdV ,
where we have used the following two calculus formulas (see Appendix 5):
d ( f ± g) = df ± dg
and
d ( fg ) = gdf + fdg ,
where both f and g are functions of two variables x and y. Note that since we assume that the
mole number n is kept constant, U and S can be regarded as functions of T and V so that x and y
in these calculus formulas correspond to T and V.
Comparing dF = !SdT ! PdV with the following calculus equation,
14
" !F %
" !F %
' dT + $
' dV ,
dF = $
# ! T & V ,n
# ! V & T ,n
we now find
# " F&
S = !% (
and
$ " T ' V ,n
# "F &
( .
P = !%
$ "V ' T, n
This procedure of creating the new variable F from the old variable U is called “the Legendre
transformation” and it creates the function F = F (T ,V,n) from the function U = U( S, V,n) by
subtracting TS from U so that the old independent variable S is replaced by the new independent
variable T.
We are now ready to show the above Maxwell’s relation:
" ! " ! F% %
" ! " !F % %
" !S %
" ! P%
$
' =$
$( ' ' = $
$(
' ' =$ ' ,
# !V & T ,n # !V # ! T & V ,n & T, n # !T # ! V & T ,n & V ,n # ! T & V ,n
where we have used a calculus formula for a function z = f ( x, y) :
( ! " !f % +
( ! " !f % +
* $ ' - =* $ ' - .
) ! x # !y & x , y ) !y # !x & y , x
There are many other Maxwell’s relations besides the one we have just derived:
" !S %
" ! P%
$
' =$ ' .
# !V & T ,n # !T & V , n
Many of these relations connect a partial derivative whose physical meaning is obscure (e.g., the
LHS of the above equation) with another partial derivative whose physical meaning is relatively
clear (e.g., the RHS of the above equation).
The second dS equation, dS = (CP T) dT ! V"dP ( n = const ) and c P = cv + Tv! 2 " T
In this sub-section, we will derive the second dS equation,
dS =
CP
dT ! V"dP ,
T
15
using the first dS equation and dV = V!dT " V # T dP . We start with the following calculus
equation for dS:
" !S %
" !S%
dS = $ ' dT + $ ' dP ,
# !T & P ,n
# !P & T, n
where we regard S as a function of T, P, and n: S = S (T, P, n ) . Note that there is no term with dn
as we assume that n is kept constant. In Sec.8.1, we have already shown
" !S %
C
$ ' = P
# !T & P,n T
so that
dS =
" !S %
CP
dT + $ ' dP .
# !P & T ,n
T
Using the first dS equation and dV = V!dT " V # T dP , we also find
dS =
2'
$
CV
!
C
!
(V!dT # V "T dP ) = 1 & CV + TV ! ) dT # V!dP .
dT +
dV = V dT +
T
"T
T
"T
T%
"T (
Comparing these two equations for dS in terms of dT and dP, we then arrive at the second dS
equation,
dS =
CP
dT ! V"dP ,
T
and the following relation we have derived in Ch.7:
CP = CV + TV
from which we obtain
c P = cv + Tv
!2
,
"T
!2
,
"T
which we have used to calculate cv from c P in Ch.7.
We also find
16
" !S %
" !V %
$ ' = (V ) = ( $
' ,
# !P & T ,n
# !T &P
which is another example of Maxwell’s relations and from which we obtain
" !v %
" !s %
$ ' = ($ ' ,
# !T & P
# !P & T
which we will later use in showing ! " 0 as T ! 0 :
The logical structure of equilibrium thermodynamics of single-component systems
As the dS equations lead to two equations for ! Qq s, which we have already used to derive
various equations in Ch.7, the equilibrium thermodynamics of single-component systems is
definitely built upon the fundamental equation of thermodynamics,
dU = TdS ! PdV ,
which is a direct consequence of the first law of thermodynamics,
dU = ! Qq s + ! W q s ,
where
! W q s = " PdV
and
! Qq s = TdS ,
which we will derive from the second law of thermodynamics in Ch.11.
Using the fundamental equation and dV = V!dT " V # T dP , we have derived the first dS
equation,
dS =
CV
!
dT +
dV ,
T
"T
from which we can also derive the second dS equation,
dS =
and
CP
dT ! V"dP
T
17
CP = CV + TV
!2
"T
using again dV = V!dT " V # T dP .
These dS equations then lead to the following equations for ! Qq s:
! Qq s = TdS = CV dT + T
"
dV
#T
and
! Qq s = TdS = CP dT " TV#dP .
In Ch.7, from these equations, we have derived
!2
CP = CV + TV
,
"T
and using the first equation for ! Qq s, we have derived
% "
(
dU = ! Qq s + ! W q s = CV dT + '' T
$ P** dV .
& #T
)
Using the second equation for ! Qq s and dV = V!dT " V # T dP , we have also derived
dU = ! Qq s " PdV = (CP " PV# )dT " V ( T# " P$ T )dP
and
dH = dU + d( PV ) = !Qq s + VdP = CP dT + V (1" T# )dP .
In the next chapter, we will derive
! T = ! S + Tv
"2
cP
from the second equation for ! Qq s and dV = V!dT " V # T dP .
18
SUMMARY FOR SEC.8.1 AND SEC.8.2
1. The entropy S of a system is an extensive state variable, whose infinitesimal change dS is
related to an infinitesimal quasi-static heat ! Qq s that flows into the system at temperature T
by
dS =
!Q q s
.
T
2. With the relation ! Qq s = TdS , we can rewrite the first law for an infinitesimal quasi-static
process, where the mole number n of a system is kept constant, as
dU = ! Qq s + ! W q s = TdS " PdV ,
which is called “the fundamental equation” of thermodynamics because almost all the
thermodynamic equations or formulas follow from this equation.
3. The entropy of a system is related to its heat capacities by
" !S %
CV = T $ '
# !T & V ,n
" !S %
CP = T $ ' .
# !T & P ,n
and
The corresponding equations for the molar quantities are
" !s %
cv = T $ '
# !T & v
" !s %
c P = T$ ' .
# !T& P
and
As CP ! CV > 0 , these equations imply that the entropy is an increasing function of
temperature while either the volume or the pressure of the system as well as its mole number
are kept constant.
4. A change in the entropy of a system can be calculated by
!S = S(T2 ,V2 ,n) " S(T1 ,V1 ,n) = $T
T2
1
V 2 % (T , V #,n )
CV (T #,V1 ,n)
2
dT # + $V
dV #
1 & (T , V #,n)
T#
T
2
or
!S = S(T2 ,P2 ,n) " S(T1 , P1 ,n ) = $T
T2
1
P2
CP (T #, P1 ,n)
dT # " $P V (T2 , P #,n )% (T2 , P #,n)dP# .
1
T#
5. The amount of heat Qq s that flows into a system quasi-statically can be calculated by
19
Qq s =
&
)
$
qs
(
++
!
Q
=
C
dT
+
TdV
( V
#
#
%
'
*
T
( Ti ,V i ,n ) "( T f ,V f , n ) ( Ti ,V i ,n ) "( T f ,V f , n )
or
Q =
# !Q
qs
qs
( Ti , Pi ,n ) "( T f ,P f ,n )
# ( C dT $ TV%dP ) .
=
P
( T i ,Pi ,n )" (T f , P f ,n )
6. The Helmholtz free energy F of a system defined by F ! U " TS satisfies
dF = !SdT ! PdV ,
which follows from the fundamental equation of thermodynamics, dU = TdS ! PdV , and
implies
# " F&
# "F &
S = !% (
P = !%
( ,
and
$ " T ' V ,n
$ "V ' T, n
which leads to
" ! " ! F% %
" ! " !F % %
" !S %
" ! P%
)
.
$
' = $$
$ ( ' '' = $$
$(
' '' = $ ' =
# !V & T ,n # !V # ! T & V ,n & T, n # !T # ! V & T ,n & V ,n # ! T & V ,n * T
so that
" !S %
" !S %
C
(
dS = $ ' dT + $ ' dV = V dT +
dV ,
# !T & V , n
# !V & T, n
T
)T
(the first dS equation)
where we have also used
" !S %
C
$ ' = V.
# !T & V , n T
7. The first dS equation and dV = V!dT " V # T dP lead to
dS =
1 #%
! 2 &(
C
C
+
TV
dT ) V!dP = P dT ) V!dP . (the second dS equation)
%
V
(
T$
"T '
T
from which we also obtain
!2
CP = CV + TV
"T
or
!2
c P = cv + Tv .
"T