?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
不可能把热从低温
物体传到高温物体，
而不引起其它变化
物理化学电子教案 —第 3章（下 )
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The corresponding coefficient relation
T
S
U
V
??
?
??
?
?
?
?
If the volume and
composition are constant,
the relation states that
the ratio of the change in thermodynamic
energy to the corresponding change in entropy
is equal to the temperature of the system,
whatever the latter’s nature.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The application of the …
TS
U
V
???
??
?
?
?
?
? ?? ???
T
US V
V ?? T
dTnC mV,
? ? ? ?TUS VV ???
If Cv,m is const.
△ Sv = n Cv,m ln (T2 / T1)
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
（ 1）求 U随 V的变化关系
Maxwell relation
( ) ( )TTUS TpVV??????在等温下，除以 dV,
证,dU = TdS - PdV
dA = - SdT - PdV
VT T
P
V
S ?
?
??
?
?
?
???
?
??
?
?
?
?
PTPTVU
VT
??????? ????????? ??
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Applications of Maxwell relations
() Vp nRTV? ??
解,对理想气体,/p V n R T p n R T V??
例 1 证明理想气体的热力学能只是温度的函数。
所以，理想气体的热力学能只是温度的函数。
() () VT pTpTUV ?? ?? ??
0nRTpV? ?? ?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Applications of Maxwell relations
（ 2）求 H 随 p 的变化关系
已知基本公式 d d dH T S V p??
等温下，除以 dP:
( ) ( )TTHS TVpp??????
( ) ( )TpSVpT????
( ) ( )TpHVVTpT????
所以
只要知道气体的状态方程，就可求得
值，即等温时焓随压力的变化值。 ()T
H
p
?
?
dG = VdP - SdT
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
,/p V n R T V n R T p??
Applications of Maxwell relations
解,
) (( )T pp VV TH T ??? ???
例 1 证明理想气体的焓只是温度的函数。
所以，理想气体的焓只是温度的函数。
对理想气体，
() pV nR
Tp
? ?
?
0nRVT
p
? ? ? ?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
表示 和 与温度的关系式都称为 Gibbs-
Helmholtz方程,用来从一个反应温度的 （或
）求另一反应温度时的 （或 ）。
它们有多种表示形式，例如：
rG? rA?
r1()AT?
r1()GT?
r2()GT? r2()AT?
2
()
( 4 ) [ ] V
A
UT
TT
??
???
?
()( 1 ) [ ]
p
G G H
TT
? ? ? ? ??
?
2
()
( 2 ) [ ] p
G
HT
TT
??
???
?
()( 3 ) [ ]
V
A A U
TT
? ? ? ? ??
?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
() pG ST? ???
所以 ()[]
p
G G H
TT
? ? ? ? ??
?
根据基本公式 d d dG S T V p? ? ?
()[]
p
G S
T
?? ? ? ?
?
根据定义式 G H T S??
在温度 T时,G H T S? ? ? ? ?
公式 的导出()
( 1 ) [ ] pG G HTT? ? ? ? ???
GHS
T
? ? ?? ? ?则
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
2
()
[] p
G
HT
T T
??
???
?
在公式 (1)等式两边各乘 得1
T 21 ( )[] pG G HTT T? ? ? ? ???
左边就是 对 T微商的结果，则()G
T
?
移项得
22
1 ( )[]
p
G G H
TT TT
? ? ? ?? ? ?
?
公式 的导出
2
()
( 2 ) [ ] p
G
HT
TT
??
???
?
移项积分得
2d ( ) dp
GH T
TT
??????
知道 与 T的关系式，就可从 求得 的值。,
pHC?
1
G
T
?
2
G
T
?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
根据基本公式
d d dA S T p V? ? ?
()( ) [ ]
VV
AA SS
TT
? ? ?? ? ? ? ?
??
根据定义式 A U T S??
在 T温度时 A U T S? ? ? ? ?
所以 ()[]
V
A A U
TT
? ? ? ? ??
?
公式 的导出()
( 3 ) [ ] VA A UTT? ? ? ? ???
AUS
T
? ? ?? ? ?则
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
在公式 (3)两边各乘 得1
T
Gibbs-Helmholtz equation
2
1 ( )[]
V
A A U
TT T
? ? ? ? ??
?
2
()
[] V
A
UT
T T
??
???
?
移项得
22
1 ( )[]
V
A A U
TT TT
? ? ? ?? ? ?
?
等式左边就是 对 T微商的结果，则()A
T
?
公式 的导出
2
()
( 4 ) [ ] V
A
UT
TT
??
???
?
移项积分得
2d ( ) dV
AU T
T T
??????
知道 与 T的关系式，就可从 求得 的值。,
VUC?
1
A
T
?
2
A
T
?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Derive and use the Clapeyron equation
Suppose both p and T are made to change by
an infinitesimal amount,but in such a way that the
two phase remain in equilibrium.
The new values of the chemical potentials
remain equal,and so the changes d?(?,P,T) and
d?(?,P,T) must be equal.
We know how to express an infinitesimal
change in ? in terms of changes in the pressure and
temperature and so
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
d? (?,P,T) = d? (?,P,T))))
implies
- Sm(?)dT + Vm(?)dP = - Sm(?)dT + Vm(?)dP
where Sm(?) and Sm(?) are the molar entropies of
the two phases and Vm(?) and Vm(?) their molar
volumes.
Rearranging this gives
{Vm(?)- Vm(?)}dP = {Sm(?) - Sm(?) }dT
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
This important result is called
the Clapeyron equation.
It is a thermodynamic result,it is exact,and it
applies to any phase change of pure materials.
dP / dT = △ Sm / △ Vm
△ Sm = △ Hm / T
m
m
VT
H
?
?
?dTdP
The ?S for a phase
transition under equilibrium conditions is
so that
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The P-V curve of the transition
S ? L S ? L
T,P T’,P’This equation is used for a
condensed system.
The integration of this equation gives an explicit
expression for ··,To perform the integration,we
need to know the dependence of ?H and ?V on T
and P.
Usually suitable approximations can be made.
Over short ranges of T and P,△ H and △ V can be
taken to be constant.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Clausius-Clapeyron equation
)/(g)(d
d mva p
m
mva p
pRTT
H
TV
H
T
p ????
v a p m
2
d l n
d
Hp
T RT
?
?
v a p m2
1 1 2
11l n ( )Hp
p R T T
?
??
For instance,if a liquid and its vapour are in
equilibrium,△ V can be replaced by Vm(g),If we
also assume that the vapour behaves perfectly
Vm(g) may be replaced by RT/P,
How vapor pressure
depends on temperature
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Trouton’s Rule
v a p m - 1 1
b
8 5 J K m o l
H
T
?? ? ? ?
Trouton proposed as an empirical generalization
in 1884 the rule that the ratio of the molar
enthalpy of vaporization to the normal boiling
point for all normal liquids is approximately
constant and equal to 92 J·K-1·mol-1,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Entropy change in irr,process
1mol C6H6(l)
268.2K,P?
1mol C6H6(s)
268.2K,P?
Example:
For C6H6(l)
at P?,
Tf=278.7K
?
1mol C6H6(l)
278.7K,P? ?
1mol C6H6(s)
278.7K,P?
?
?
(1)
(2)
(3)
dT
T
nC
S mP
T
T
,
1
2
1
???
t
t
T
H
S
?
?? 2
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Pay attention to △ S (surroundings)
321 SSSS s y s ???????
su rr
su rr
su rr T
Q
S ??
sy s
sy s
T
Q?
?
s y s
p
T
Q?
?
T
H??
?
321 HHHH ???????
??? 2
1
,1
T
T mP dTnCH
= -35.5 1?? KJ
183.36 ??? KJ
133.1 ???? KJS
i s o
The original
process is a
irreversible process.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Evaluating the entropy
The entropy of a substance at a temperature T can
be related to its entropy at absolute zero by
determining the heat capacity Cp as a function of
temperature and calculating the integral in eq.
??? f
i
T
T
P
if T
dTC
TSTS )()(
Where Ti = 0 K,then S(0K) =?.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The third law of thermodynamics
A basic law of thermodynamics was born from
the attempt to calculate equilibrium constants of
chemical reactions entirely from thermal data
(i.e.,enthalpies and heat capacities).
The Nernst heat theorem was later modified and
generalized by Planck,Simon,Lewis,Guggenheim,
and others.
One well-known version of the modified
theorem is as follows:
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The entropy change associated with
any isothermal reversible process of a condensed
system (i.e.,a solid or a liquid)approaches zero as
the temperature approaches absolute zero,
Symbolically we may write
? ? 0lim
0
??
?
S
T
Planck in 1911extended the Nernst theorem by
making the additional postulate that the absolute
value of the entropy of a pure solid or a pure liquid
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The Nernst heat theorem
approaches zero as the temperature
approaches absolute zero or symbolically,
0l i m 0??ST
The entropy change accompanying
transformations between condensed phases in
equilibrium,including chemical reactions,
approaches zero as the temperature approaches
zero.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Third Law:
If the entropy of every element in the state stable
at T=0K is taken as zero,every substance has a
positive entropy which at T=0K may become
zero,and does become zero for all perfect
crystalline substances,including compounds.
At T=0K A ? B
△ S = SB – SA = 0,then SB = 0,SA = 0
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The conventional (absolute) entropy
The choice S(0)=0 will henceforth be made,
and entropies reported on that basis will be
referred to as Third Law entropies.
已知
TTCS p d)/(d ?
0 0 ( / ) d
T
pTS S C T T?? ?
?? T p TC0 lnd
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
以 为纵坐标，
T为横坐标，求某物质
在 40K时的熵值。
/pCT
如图所示：
40
0
( / ) dpS C T T? ?
阴影下的面积，
就是所要求的该物质
的规定熵。
Calculate the entropy of B at T
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
图中阴影下的面积加上两个相变熵即为所求的熵值。
b
() d T p
T
C T
T? ?
气
如果要求某物质在沸点以上某温度 T时的熵变，
则积分不连续，要加上在熔点（ Tf） 和沸点（ Tb） 时
的相应熵，其积分公式可表示为：
f
0( ) ( 0 ) d
T pCS T S T
T?? ?
( 固)
m el t
f
H
T
??
b
f
()+dT p
T
C T
T?
液
v a p
b
H
T
??
Using integral method
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Conventional entropy
)( TS mr ??
),,( TBS
B
mB ??
???
The standard
molar entropy of
reaction
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
When the substance is in its standard state at T
the Third Law entropy is denoted S?(T).
The standard entropy of reaction
If the Third Law entropies of compounds are
known,it is a simple matter to calculate the
standard entropy of reaction:This is defined,like
the enthalpy of reaction,as the difference of
entropies between the pure,separated products
and the pure,separated reactants,all in their
standard states at the temperature of interest,end
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
RUDOLF JULIUS EMMANUEL CLAUSIUS
RUDOLF JULIUS EMMANUEL CLAUSIUS (1822-1888)
German mathematical physicist,is perhaps best
known for the statement of the second law of
thermodynamics in the form ―Heat cannot of itself pass
from a colder to a hotter body.‖which he presented to the
Berlin Academy in 1805.He also made fundamental
contributions to the field of the kinetic theory of gases and
anticipated Arrhenius by suggesting that molecules in
electrolytes continually exchange atoms.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
WILLIAM THOMSON,Lord Kelvin
WILLIAM THOMSON,Lord Kelvin (1824-1907)
Irish-born British physicist,proposed his absolute
scale of temperature,which is independent of the
thermometric substance in 1848.In one of his earliest papers
dealing with heat conduction of the earth,Thomson showed
that about 100 million years ago,the physical condition of
the earth must have been quite different from that of
today.He did fundamental work in telegraphy,and
navigation.For his services in trans-Atlantic
telegraphy,Thomson was raised to the peerage,with the title
Baron Kelvin of Larg.There was no heir to the title,and it is
now extinct.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
NICOLAS LEONHARD SADI CARNOT
NICOLAS LEONHARD SADI CARNOT (1796-1832)
a French military engineer.His only published work
was Reflexions Sur la Puissance Motrice du Feu et sur les
Machines Propres a Developer catte Puissance (1824),in
which he discussed the conversion of heat into work and
laid the foundation for the second law of thermodynamics,
He was the scion of a distinguished French family that was
very active in political and military affairs,His nephew,
Marie Francois Sadi Carnot (1837-1894),was the fourth
president of the Third French Republic.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
LUDWIG BOLTZMANN
LUDWIG BOLTZMANN (1844-1906),Austrian scientist,is
best known for his work in the kinetic theory of gases and
in thermodynamics and statistical mechanics,His suicide in
1906 is attributed by some to a state of depression resulting
from the intense scientific war between the atomists and the
energists at the turn of the century,On his tombstone is the
inscription S = k ln W.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
HERMANN LUDWIG FERDINAND von HELMHOLTZ
HERMANN LUDWIG FERDINAND von HELMHOLTZ
(1821-1894)
German scientist,worked in areas spanning the range
from physics to physiology,His paper Uber die Erhaltung
der Kraft (―On the Conservation of Force,‖1847) was one
of the epochal papers of the century,Along with Mayer,
Joule,and Kelvin,he is regarded as one of the founders of
the conservation of energy principle,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
HERMANN LUDWIG FERDINAND von HELMHOLTZ
His Physiological Optics was in its time the most important
publication ever to have appeared on the physiology of
ivsion.In connection with these studies he invented the
ophthalmoscope in 1851,still a fundamental tool of every
physician,His Sensations of Tone (1862) established many
of the basic principles of physiological acoustics.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
JOSIAH WILLARD GIBBS
JOSIAH WILLARD GIBBS (1839-1903),
American scientist,was professor of mathematical
physics at Yale University from 1871 until his death,His
series of papers ―On the Equilibrium of Heterogenous
Substances,‖ published in the Transactions of the
Connecticut Academy of Sciences (1876-1878) was one of
the most important series of statistical mechanics,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
JOSIAH WILLARD GIBBS
The Copley Medal of the Royal Society of London was
presented to him as ―the first to apply the second law of
thermodynamics to the exhaustive discussion of the relation
between chemical,electrical,and thermal energy and
capacity for external work.‖
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
JAMES CLERK MAXWELL
JAMES CLERK MAXWELL (1831-1879),
British physicist,presented his first scientific paper to
the Royal Society of Edihburgh at the age of 15.In
chemistry he is best known for his Maxwell distribution and
his contributions to the kinetic theory of gases,In physics
his name is most often associated with his Maxwell
equations for electromagnetic fields.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
BENOIT PIERRE EMILE CLAPEYRON
BENOIT PIERRE EMILE CLAPEYRON (1799-1864),
French scientist,was the first to appreciate the
importance of Carnot’s work on the conversion of heat into
work,In analyzing Carnot cycles,Clapeyron concluded that
―the work w produced by the passage of a certain quantity
of heat q from a body at temperature t1,to another body at
temperature t2 is the same for every gas or liquid … and is
the greatest which can be achieved‖ (B.P.E,Clapeyron,
Memoir sur la Puissance Motrice de la Chaleur
(Paris,1833)).
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
BENOIT PIERRE EMILE CLAPEYRON
Clapeyron was speaking of what we call a reversible
process,Kelvin’s establishment of the thermodynamic
temperature scale from a study of the Carnot cycle came not
from Carnot directly but from Carnot through Clapeyron,
since Carnot’s original work was not available to Kelvin.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Trouton
Trouton
英国物理学家，他提出了一个近似规则，被称为
楚顿规则。该规则说明了许多非极性液体的摩尔蒸发热
与其正常沸点之间的线性关系。
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
WALTHER NERNST
WALTHER NERNST (1864-1941),
German physical chemist,did much of the early
important work in electrochemistry,studying the
thermodynamics of galvanic cells and the diffusion of ions
in solution,Besides his scientific researches,he developed
the Nernst lamp,which used a ceramic body,This lamp
never achieved commercial importance since the tungsten
lamp was developed soon afterwards,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
WALTHER NERNST
His electrical piano,which used radio amplifiers instead of a
sounding board,was totally rejected by musicians,Nernst
was the first to enunciate the third law of thermodynamics,
and received the Nobel Prize in chemistry in 1920 for his
thermochemical work.
不可能把热从低温
物体传到高温物体，
而不引起其它变化
物理化学电子教案 —第 3章（下 )
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The corresponding coefficient relation
T
S
U
V
??
?
??
?
?
?
?
If the volume and
composition are constant,
the relation states that
the ratio of the change in thermodynamic
energy to the corresponding change in entropy
is equal to the temperature of the system,
whatever the latter’s nature.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The application of the …
TS
U
V
???
??
?
?
?
?
? ?? ???
T
US V
V ?? T
dTnC mV,
? ? ? ?TUS VV ???
If Cv,m is const.
△ Sv = n Cv,m ln (T2 / T1)
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
（ 1）求 U随 V的变化关系
Maxwell relation
( ) ( )TTUS TpVV??????在等温下，除以 dV,
证,dU = TdS - PdV
dA = - SdT - PdV
VT T
P
V
S ?
?
??
?
?
?
???
?
??
?
?
?
?
PTPTVU
VT
??????? ????????? ??
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Applications of Maxwell relations
() Vp nRTV? ??
解,对理想气体,/p V n R T p n R T V??
例 1 证明理想气体的热力学能只是温度的函数。
所以，理想气体的热力学能只是温度的函数。
() () VT pTpTUV ?? ?? ??
0nRTpV? ?? ?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Applications of Maxwell relations
（ 2）求 H 随 p 的变化关系
已知基本公式 d d dH T S V p??
等温下，除以 dP:
( ) ( )TTHS TVpp??????
( ) ( )TpSVpT????
( ) ( )TpHVVTpT????
所以
只要知道气体的状态方程，就可求得
值，即等温时焓随压力的变化值。 ()T
H
p
?
?
dG = VdP - SdT
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
,/p V n R T V n R T p??
Applications of Maxwell relations
解,
) (( )T pp VV TH T ??? ???
例 1 证明理想气体的焓只是温度的函数。
所以，理想气体的焓只是温度的函数。
对理想气体，
() pV nR
Tp
? ?
?
0nRVT
p
? ? ? ?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
表示 和 与温度的关系式都称为 Gibbs-
Helmholtz方程,用来从一个反应温度的 （或
）求另一反应温度时的 （或 ）。
它们有多种表示形式，例如：
rG? rA?
r1()AT?
r1()GT?
r2()GT? r2()AT?
2
()
( 4 ) [ ] V
A
UT
TT
??
???
?
()( 1 ) [ ]
p
G G H
TT
? ? ? ? ??
?
2
()
( 2 ) [ ] p
G
HT
TT
??
???
?
()( 3 ) [ ]
V
A A U
TT
? ? ? ? ??
?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
() pG ST? ???
所以 ()[]
p
G G H
TT
? ? ? ? ??
?
根据基本公式 d d dG S T V p? ? ?
()[]
p
G S
T
?? ? ? ?
?
根据定义式 G H T S??
在温度 T时,G H T S? ? ? ? ?
公式 的导出()
( 1 ) [ ] pG G HTT? ? ? ? ???
GHS
T
? ? ?? ? ?则
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
2
()
[] p
G
HT
T T
??
???
?
在公式 (1)等式两边各乘 得1
T 21 ( )[] pG G HTT T? ? ? ? ???
左边就是 对 T微商的结果，则()G
T
?
移项得
22
1 ( )[]
p
G G H
TT TT
? ? ? ?? ? ?
?
公式 的导出
2
()
( 2 ) [ ] p
G
HT
TT
??
???
?
移项积分得
2d ( ) dp
GH T
TT
??????
知道 与 T的关系式，就可从 求得 的值。,
pHC?
1
G
T
?
2
G
T
?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Gibbs-Helmholtz equation
根据基本公式
d d dA S T p V? ? ?
()( ) [ ]
VV
AA SS
TT
? ? ?? ? ? ? ?
??
根据定义式 A U T S??
在 T温度时 A U T S? ? ? ? ?
所以 ()[]
V
A A U
TT
? ? ? ? ??
?
公式 的导出()
( 3 ) [ ] VA A UTT? ? ? ? ???
AUS
T
? ? ?? ? ?则
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
在公式 (3)两边各乘 得1
T
Gibbs-Helmholtz equation
2
1 ( )[]
V
A A U
TT T
? ? ? ? ??
?
2
()
[] V
A
UT
T T
??
???
?
移项得
22
1 ( )[]
V
A A U
TT TT
? ? ? ?? ? ?
?
等式左边就是 对 T微商的结果，则()A
T
?
公式 的导出
2
()
( 4 ) [ ] V
A
UT
TT
??
???
?
移项积分得
2d ( ) dV
AU T
T T
??????
知道 与 T的关系式，就可从 求得 的值。,
VUC?
1
A
T
?
2
A
T
?
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Derive and use the Clapeyron equation
Suppose both p and T are made to change by
an infinitesimal amount,but in such a way that the
two phase remain in equilibrium.
The new values of the chemical potentials
remain equal,and so the changes d?(?,P,T) and
d?(?,P,T) must be equal.
We know how to express an infinitesimal
change in ? in terms of changes in the pressure and
temperature and so
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
d? (?,P,T) = d? (?,P,T))))
implies
- Sm(?)dT + Vm(?)dP = - Sm(?)dT + Vm(?)dP
where Sm(?) and Sm(?) are the molar entropies of
the two phases and Vm(?) and Vm(?) their molar
volumes.
Rearranging this gives
{Vm(?)- Vm(?)}dP = {Sm(?) - Sm(?) }dT
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
This important result is called
the Clapeyron equation.
It is a thermodynamic result,it is exact,and it
applies to any phase change of pure materials.
dP / dT = △ Sm / △ Vm
△ Sm = △ Hm / T
m
m
VT
H
?
?
?dTdP
The ?S for a phase
transition under equilibrium conditions is
so that
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The P-V curve of the transition
S ? L S ? L
T,P T’,P’This equation is used for a
condensed system.
The integration of this equation gives an explicit
expression for ··,To perform the integration,we
need to know the dependence of ?H and ?V on T
and P.
Usually suitable approximations can be made.
Over short ranges of T and P,△ H and △ V can be
taken to be constant.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Clausius-Clapeyron equation
)/(g)(d
d mva p
m
mva p
pRTT
H
TV
H
T
p ????
v a p m
2
d l n
d
Hp
T RT
?
?
v a p m2
1 1 2
11l n ( )Hp
p R T T
?
??
For instance,if a liquid and its vapour are in
equilibrium,△ V can be replaced by Vm(g),If we
also assume that the vapour behaves perfectly
Vm(g) may be replaced by RT/P,
How vapor pressure
depends on temperature
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Trouton’s Rule
v a p m - 1 1
b
8 5 J K m o l
H
T
?? ? ? ?
Trouton proposed as an empirical generalization
in 1884 the rule that the ratio of the molar
enthalpy of vaporization to the normal boiling
point for all normal liquids is approximately
constant and equal to 92 J·K-1·mol-1,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Entropy change in irr,process
1mol C6H6(l)
268.2K,P?
1mol C6H6(s)
268.2K,P?
Example:
For C6H6(l)
at P?,
Tf=278.7K
?
1mol C6H6(l)
278.7K,P? ?
1mol C6H6(s)
278.7K,P?
?
?
(1)
(2)
(3)
dT
T
nC
S mP
T
T
,
1
2
1
???
t
t
T
H
S
?
?? 2
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Pay attention to △ S (surroundings)
321 SSSS s y s ???????
su rr
su rr
su rr T
Q
S ??
sy s
sy s
T
Q?
?
s y s
p
T
Q?
?
T
H??
?
321 HHHH ???????
??? 2
1
,1
T
T mP dTnCH
= -35.5 1?? KJ
183.36 ??? KJ
133.1 ???? KJS
i s o
The original
process is a
irreversible process.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Evaluating the entropy
The entropy of a substance at a temperature T can
be related to its entropy at absolute zero by
determining the heat capacity Cp as a function of
temperature and calculating the integral in eq.
??? f
i
T
T
P
if T
dTC
TSTS )()(
Where Ti = 0 K,then S(0K) =?.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The third law of thermodynamics
A basic law of thermodynamics was born from
the attempt to calculate equilibrium constants of
chemical reactions entirely from thermal data
(i.e.,enthalpies and heat capacities).
The Nernst heat theorem was later modified and
generalized by Planck,Simon,Lewis,Guggenheim,
and others.
One well-known version of the modified
theorem is as follows:
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The entropy change associated with
any isothermal reversible process of a condensed
system (i.e.,a solid or a liquid)approaches zero as
the temperature approaches absolute zero,
Symbolically we may write
? ? 0lim
0
??
?
S
T
Planck in 1911extended the Nernst theorem by
making the additional postulate that the absolute
value of the entropy of a pure solid or a pure liquid
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The Nernst heat theorem
approaches zero as the temperature
approaches absolute zero or symbolically,
0l i m 0??ST
The entropy change accompanying
transformations between condensed phases in
equilibrium,including chemical reactions,
approaches zero as the temperature approaches
zero.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Third Law:
If the entropy of every element in the state stable
at T=0K is taken as zero,every substance has a
positive entropy which at T=0K may become
zero,and does become zero for all perfect
crystalline substances,including compounds.
At T=0K A ? B
△ S = SB – SA = 0,then SB = 0,SA = 0
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
The conventional (absolute) entropy
The choice S(0)=0 will henceforth be made,
and entropies reported on that basis will be
referred to as Third Law entropies.
已知
TTCS p d)/(d ?
0 0 ( / ) d
T
pTS S C T T?? ?
?? T p TC0 lnd
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
以 为纵坐标，
T为横坐标，求某物质
在 40K时的熵值。
/pCT
如图所示：
40
0
( / ) dpS C T T? ?
阴影下的面积，
就是所要求的该物质
的规定熵。
Calculate the entropy of B at T
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
图中阴影下的面积加上两个相变熵即为所求的熵值。
b
() d T p
T
C T
T? ?
气
如果要求某物质在沸点以上某温度 T时的熵变，
则积分不连续，要加上在熔点（ Tf） 和沸点（ Tb） 时
的相应熵，其积分公式可表示为：
f
0( ) ( 0 ) d
T pCS T S T
T?? ?
( 固)
m el t
f
H
T
??
b
f
()+dT p
T
C T
T?
液
v a p
b
H
T
??
Using integral method
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Conventional entropy
)( TS mr ??
),,( TBS
B
mB ??
???
The standard
molar entropy of
reaction
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
When the substance is in its standard state at T
the Third Law entropy is denoted S?(T).
The standard entropy of reaction
If the Third Law entropies of compounds are
known,it is a simple matter to calculate the
standard entropy of reaction:This is defined,like
the enthalpy of reaction,as the difference of
entropies between the pure,separated products
and the pure,separated reactants,all in their
standard states at the temperature of interest,end
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
RUDOLF JULIUS EMMANUEL CLAUSIUS
RUDOLF JULIUS EMMANUEL CLAUSIUS (1822-1888)
German mathematical physicist,is perhaps best
known for the statement of the second law of
thermodynamics in the form ―Heat cannot of itself pass
from a colder to a hotter body.‖which he presented to the
Berlin Academy in 1805.He also made fundamental
contributions to the field of the kinetic theory of gases and
anticipated Arrhenius by suggesting that molecules in
electrolytes continually exchange atoms.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
WILLIAM THOMSON,Lord Kelvin
WILLIAM THOMSON,Lord Kelvin (1824-1907)
Irish-born British physicist,proposed his absolute
scale of temperature,which is independent of the
thermometric substance in 1848.In one of his earliest papers
dealing with heat conduction of the earth,Thomson showed
that about 100 million years ago,the physical condition of
the earth must have been quite different from that of
today.He did fundamental work in telegraphy,and
navigation.For his services in trans-Atlantic
telegraphy,Thomson was raised to the peerage,with the title
Baron Kelvin of Larg.There was no heir to the title,and it is
now extinct.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
NICOLAS LEONHARD SADI CARNOT
NICOLAS LEONHARD SADI CARNOT (1796-1832)
a French military engineer.His only published work
was Reflexions Sur la Puissance Motrice du Feu et sur les
Machines Propres a Developer catte Puissance (1824),in
which he discussed the conversion of heat into work and
laid the foundation for the second law of thermodynamics,
He was the scion of a distinguished French family that was
very active in political and military affairs,His nephew,
Marie Francois Sadi Carnot (1837-1894),was the fourth
president of the Third French Republic.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
LUDWIG BOLTZMANN
LUDWIG BOLTZMANN (1844-1906),Austrian scientist,is
best known for his work in the kinetic theory of gases and
in thermodynamics and statistical mechanics,His suicide in
1906 is attributed by some to a state of depression resulting
from the intense scientific war between the atomists and the
energists at the turn of the century,On his tombstone is the
inscription S = k ln W.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
HERMANN LUDWIG FERDINAND von HELMHOLTZ
HERMANN LUDWIG FERDINAND von HELMHOLTZ
(1821-1894)
German scientist,worked in areas spanning the range
from physics to physiology,His paper Uber die Erhaltung
der Kraft (―On the Conservation of Force,‖1847) was one
of the epochal papers of the century,Along with Mayer,
Joule,and Kelvin,he is regarded as one of the founders of
the conservation of energy principle,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
HERMANN LUDWIG FERDINAND von HELMHOLTZ
His Physiological Optics was in its time the most important
publication ever to have appeared on the physiology of
ivsion.In connection with these studies he invented the
ophthalmoscope in 1851,still a fundamental tool of every
physician,His Sensations of Tone (1862) established many
of the basic principles of physiological acoustics.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
JOSIAH WILLARD GIBBS
JOSIAH WILLARD GIBBS (1839-1903),
American scientist,was professor of mathematical
physics at Yale University from 1871 until his death,His
series of papers ―On the Equilibrium of Heterogenous
Substances,‖ published in the Transactions of the
Connecticut Academy of Sciences (1876-1878) was one of
the most important series of statistical mechanics,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
JOSIAH WILLARD GIBBS
The Copley Medal of the Royal Society of London was
presented to him as ―the first to apply the second law of
thermodynamics to the exhaustive discussion of the relation
between chemical,electrical,and thermal energy and
capacity for external work.‖
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
JAMES CLERK MAXWELL
JAMES CLERK MAXWELL (1831-1879),
British physicist,presented his first scientific paper to
the Royal Society of Edihburgh at the age of 15.In
chemistry he is best known for his Maxwell distribution and
his contributions to the kinetic theory of gases,In physics
his name is most often associated with his Maxwell
equations for electromagnetic fields.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
BENOIT PIERRE EMILE CLAPEYRON
BENOIT PIERRE EMILE CLAPEYRON (1799-1864),
French scientist,was the first to appreciate the
importance of Carnot’s work on the conversion of heat into
work,In analyzing Carnot cycles,Clapeyron concluded that
―the work w produced by the passage of a certain quantity
of heat q from a body at temperature t1,to another body at
temperature t2 is the same for every gas or liquid … and is
the greatest which can be achieved‖ (B.P.E,Clapeyron,
Memoir sur la Puissance Motrice de la Chaleur
(Paris,1833)).
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
BENOIT PIERRE EMILE CLAPEYRON
Clapeyron was speaking of what we call a reversible
process,Kelvin’s establishment of the thermodynamic
temperature scale from a study of the Carnot cycle came not
from Carnot directly but from Carnot through Clapeyron,
since Carnot’s original work was not available to Kelvin.
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
Trouton
Trouton
英国物理学家，他提出了一个近似规则，被称为
楚顿规则。该规则说明了许多非极性液体的摩尔蒸发热
与其正常沸点之间的线性关系。
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
WALTHER NERNST
WALTHER NERNST (1864-1941),
German physical chemist,did much of the early
important work in electrochemistry,studying the
thermodynamics of galvanic cells and the diffusion of ions
in solution,Besides his scientific researches,he developed
the Nernst lamp,which used a ceramic body,This lamp
never achieved commercial importance since the tungsten
lamp was developed soon afterwards,
?上一内容 ?下一内容 ?回主目录 ?返回 2010-5-14
WALTHER NERNST
His electrical piano,which used radio amplifiers instead of a
sounding board,was totally rejected by musicians,Nernst
was the first to enunciate the third law of thermodynamics,
and received the Nobel Prize in chemistry in 1920 for his
thermochemical work.
