O F P R I N C I P L E S F O R G E N E R A L S Y S T E M S
I2.I GENERAL SYSTEMS
Throughout the first eleven chapters the principles of thermodynamics have been so stated that their generalization is evident. The fundamental equation of a simple system is of the form
( 1 2 . 1 ) The volume and the mole numbers play symmetric roles throughout, and we can rewrite equation 12.1 in the symmetric form
U : U ( X s , X 1 , X 2 , X 3 , . . . , X , )
(r2.2)
For the convenience of the reader we recapitulate briefly the main theorems of the first eleven chapters, using a language appropriate to general systems.
I2-2 THE POSTULATES
Postulate I. There exist particular states (called equilibrium states) that, macroscopically, are characterized completely by the specification of the internal energ/ U and a set of extensiue parameters X1, Xr,. . . , X, later to be specifcally enumerated.
283
284 Summary of Principles for General Systems
Posfulate II. There exists a function (called the entropy) of the extensiDe parameters, defined for all equilibrium states, and hauing the following property. The ualues assumed by the extensiue parameters in the absence of a constraint are those that maximize the entropy ouer the manifutd of con- strained equilibrium states.
Postulate lll. The entropy of a composite system is additiue ouer the constituent subsystems (whence the entropy of each constituent system is a homogeneous firstorder function of the extensiue parameters). The entropy is continuous and differentiable and is a monotonically inueasing function of the energt.
Posfulats Tt/. The entropy of any system uanishes in the state for which T = ( A U / A S ) x , , X , , . . , : 0 .
I2.3 THE INTENSIVE PARAMETERS
The differential form of the fundamental equation is
d u : r d s + f r o a x o
I
(72.3)
in which: lrodxo
0
(r2.4)
The term Tds is the flux of heat and ElPo dxo is the work. The intensive parameters are functions of the extensive parameters, the functional relations being the equations of state. Furthermore, the conditions of equilibrium with respect to a transfer of xo between two subsystems is the equality of the intensive parameters Po.
The Euler relation, which follows from the homogeneous first-order property, is
(12.s)
and the Gibbs-Duhem relation is
t
l x o d r o : o
0
Similar relations hold in the entropy representation.
U : l P o x o
0
(r2.6)
I2-4 LEGENDRETRANSFORMS
A partial Legendre transformation can be made by replacing the variables Xs, X1, Xr,..., X" by Po, Pr,..., P". The Legendre transformed function is
s
U l . P o , P r , . . . , P " ] : U - L P k X k
0
(72.7) The natural variables of this function Lta Ps,. . ., Pr, Xr+r,. . . , X,, and the natural derivatives are
- X o , ( k : 0 , 1 , . . . , s ) ( 1 2 . 8 )
oPo
0 u 1 n o , . . . , P " l oXo
and consequently
: P * ( k : s + 1 , . . . , r ) ( L 2 . 9 )
d U l p o , . . . , p " l : D( - x) dpk + L pkdxk ( 1 2 . 1 0 )
0 s + l
The equilibrium values of any unconstrained extensive parameters in a system in contact with reservoirs of constant Ps, P1,..., P, minimize
U l P o , . . . , P " l a t c o n s t a n t P o , . . . , P , , X r * r . . . X , .
I2.5 MAXWELL RELATIONS
The mixed partial derivatives of the potential UlPo,..., P,] are equal, whence, from equation 12.10,
0 X , | x o t . i
f i : n , - ( i f 7'k < s)
0X, - AP,.
74,: -iF; (if
"r < s and /r > s)
aPj _ aPk
?xo 0x,
(12.u)
(r2.t2)
( i f 7 , k > s )
( 1 2 . 1 3 )
286 Summary of Principles for General Systems
xk
u t . . . l
xj
u I " . P j l
u I . . . P h l
Pj FIGURE 12.1
u l - P j , P h l
Pk
The general thermodyramic mnemonic diagram. The potential Ut.. I is a gen- eral Legendre transform of U. The p o t e n t i a l u l . . . , P j l i s u [ . . ] - 4xi.
That is, Ul,. . . ,41 is transformed with respect to P, in addition to all the vari- ables of Ut. . .1. The other functions are similarly defined.
In each of these partial derivatives the variables to be held constant are all those of the set P0,... t Pst Xs+r,..., X,, except the variable with respect to which the derivative is taken.
These relations can be read from the mnemonic diagram of Fig. 12.1.
12.6 STABILITY AND PHASE TRANSITIONS
The criteria of stability are the convexity of the thermodynamic poten- tials with respect to their extensive parameters and concavity with respect to their intensive parameters (at constant mole numbers). Specifically this requires
c o > c u > 0 r c r > r " > 0 and analogous relations for more general systems.
If the criteria of stability are not satisfied a system breaks up or more phases. The molar Gibbs potential of each component equal in each phase
(r2.14)
into two 7 is then
(tz.rs)
t"l,: ptj: pllt
The dimensionality / of the thermodynamic "space" in which a given number M of phases can exist, for a system with r components, is given by the Gibbs phase rule
f : r - M + 2 (r2.16)
The slope, in the P-T planq of the coexistence curve of two phases is given by the Clapeyron equation
d P A s /
dT Lu TA,u
(72.r7)
I2.7 CRITICALPHENOMENA
Near a critical point the minimum of the Gibbs potential becomes shallow and possibly asymmetric. Fluctuations diverge, and the most probable values, which are the subject of thermodynamic theory, differ from the average values which are measured by experiment. Thermody- namic behavior near the critical point is governed by a set of "critical exponents." These are interrelated by "scaling relations." The numerical values of the critical exponents are determined by the physical dimen- sionality and by the dimensionality of the order parameter; these two dimensionalities define " universality classes" of systems with equal criti- cal exponents.
I2.8 PROPERTIES AT ZERO TEMPERATURE
For a general system the specific heats vanish at zero temperature.
c rr, t2,. . . - - + Q a s Z - 0 ( 1 2 . 1 8 )
C t r , , r p - r , P 1 , , x 1 a a 1 , . . .
the four following
+ Q a s Z - - + 0
(L2.re)
Furthermore,
temperature. types of derivatives vanish at zero
= r(#)",,,
i d s \
l , - | + Q
\ o * o J r , x 1 , - . , x 1 , _ , , x p * 1 .
I a P , \
l-i-l --+ Q
\ w r / x 1 , x 2 , . .
/ d s l
l . - - l - - + Q
\ a P o J r , x 1 , . . , x p - 1 , x 2 1 y .
as T--+ 0 (12.20)
as T-+ 0 (72.27)
as Z--+ 0 (12.22)
| 0 x , \
| -=--3 | -+ Q
\ A f 1 1 1 , . . , x 1 1 . P , , x y . s 1 . . . .
a s Z + 0 ( 7 2 . 2 3 )
