Clausius versus Sackur-Tetrode entropies

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by  Thomas Oikonomou

Clausius vs. Sackur-Tetrode ideal gas entropy in open systems
Thomas Oikonomou∗
Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, 15310 Athens, Greece
(Dated: November23, 2011)
Based on the properties of homogeneous functions of first degree, we derive in a mathematically
consistent manner the explicit expressions of the chemical potential µ and the Clausius entropy S
for the case of monoatomic ideal gases in open systems within phenomenological thermodynam-
1
ics. Neither information theoretic nor quantum mechanical statistical concepts are invoked in this
1
0 derivation. Considering a specific expression/value of the constant term of S, the derived entropy
2 coincides with the Sackur-Tetrode entropy in the thermodynamic limit. We demonstrate however,
that the former limit is not contained in the classical thermodynamic relations, implying that the
v
two aforementioned entropies are distinct in finitesystems.
o
N
PACSnumbers: 05.70.-a;51.30.+i;05.70.Ce
Keywords: idealgasentropy, chemicalpotential, homogeneity, secondlawofthermodynamics
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2

]
h
c I. INTRODUCTION
e
m
The first law of thermodynamics presents an expression of the energy conservation principle. In terms of total
- differentials the former law is given as
t
a
t T dS =dU +P dV −µdN, (1)
s
.
t
a whereT, S, U, P andV denotethetemperature,the(Clausiusdefinitionof)entropy,theinternalenergy,thepressure
m and the volume, respectively. N denotes the number of particles and µ the chemical potential associated with the
- particle exchange (for one species of particles). A possible solutionof Eq. (1) for the case of monoatomic ideal gases,
d with respect to the entropy as a function of U,V and N, has been initially computed within quantum statistical
n
mechanics, with a Hamiltonian consisting only of kinetic energy contributions, presenting a solution of Boltzmann-
o
Gibbs statistics as well. The obtained result is known as the Sackur-Tetrode (ST) entropy and it has the following
c
[ expression

3 Φ V 3 U 5 3 3h2
v SST(U,V,N)=Nl→im∞kBln(cid:18)h3NN!(cid:19)=kBN(cid:20)ln(cid:18)N(cid:19)+ 2ln(cid:18)N(cid:19)+ 2 − 2ln(cid:18)4πm(cid:19)(cid:21), (2)
2
4
where k and h are the Boltzmann and Planck constants respectively, and Φ is the 6N-dimensional phase-space
4 B
5 volume of the region enclosed by the constant energy hyper surface [1]. The term m denotes the mass of the
. particles. The factor N! stems from the traditional way of resolving the Gibbs paradox [2], interpreted by means of
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the quantum mechanical concept of the particles indistinguishability, taking in this way into account the invariance
0
under permutations. An information theoretical derivation of the ST-entropy has been also demonstrated by A.
1
1 Ben-Naim in his textbook [3], where the former function is the result of the combination of the following four terms:
: positional uncertainty, momenta uncertainty, quantum mechanical uncertainty principle and the indistinguishability
v
of the particles. In the same book one may also find an alternative derivation of the factor N!, with the Schr¨odinger
i
X equation being the point of departure. S is then expressed in terms of the de Broglie wavelength.
ST
r From the mathematical point of view, the ST-entropy in Eq. (2) presents a homogeneous function of first degree.
a We remind that a function f : Rq → R of class Cn≥2 is said to be homogeneous of τth degree (τd) generally, when
+
it satisfies the following relation
f(αx)=ατ f(x), (3)
where α ∈ R and x ∈ Rq. Then, homogeneity of degree one (1d) is characterized by Eq. (3) for τ = 1. Requiring
+ +
the aforementioned mathematical property as a fundamental property of the thermodynamic entropy S, we aim in
this paper to derive a solution of Eq. (1) for monoatomic ideal gases in open systems within the context of the
∗Electronicaddress: [email protected]
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phenomenological thermodynamics. It is demonstrated that the entropy function S obtained from our homogeneity
analysis coincides with the ST-entropy for a properly chosen constant term of S. In the current derivation however,
we donotexplicitly invokethe N →∞limit, incontrasttothe S derivation,ascanbe seeninEq. (2). We further
ST
showthatthefirstlawofthermodynamicsdoesnotcontainthethermodynamiclimit,implyingthatif1d-homogeneity
is afundamentalentropicproperty,thenthe monoatomicidealgasentropyisnotgivenby S inEq. (2) butis given
ST
by the Clausius entropy S. This result is of greattheoreticalinterest, since it indicates that, under the requestof the
entropic 1d-homogeneity, the statistical backgroundof S does not succeed to derive the correct entropy expression
ST
of monoatomic ideal gases.
Thepaperisorganizedasfollows. SectionII reviewsthe basicrelationscharacterizingthe homogenousfunctionsof
firstdegree. Afterhavingoutlinedtheformalismoftheformerfunctions,inSectionIII,weapplyittothermodynamic
functions deriving the chemical potential and the entropy expression for monoatomic ideal gases in open systems.
Conclusions are presented in Section IV.
II. HOMOGENEOUS FUNCTIONS OF FIRST DEGREE
Let us consider a 1d-homogeneous (1dH) function f : Rq → R of class Cn≥2. Then, according to Eq. (3) (for
+
τ =1), f(x) satisfies the relation
f(αx)=αf(x). (4)
An example of the former function for the case q = 2 can be provided as f(x,y) = xexp(x/y), where x = x and
1
x =y. The differentiation of Eq. (4) with respect to α yields
2
q ∂f(αx)
x =f(x). (5)
i
∂(αx )
Xi=1 i
Eq. (5) is valid for any α∈R , thus we can choose α=1. We then obtain
+
q
f(x)= x A (x), (6)
i i
Xi=1
where A (x) := ∂f(x)/∂x . Regarding our example above, we have A (x) = A (x,y) = (1+x/y)exp(x/y) and
i i 1 x
A (x)=A (x,y)=−exp(x/y)x2/y2. Indeed then, f(x,y) can be expressedas in Eq. (6), i.e., f(x,y)=xA (x,y)+
2 y x
yA (x,y). The aforementioned equation is referred to as Euler’s 1d-homogeneous function theorem, stating that a
y
1dH-function f(x) can always be expressed in the form of Eq. (6). It can easily be shown that the relation between
Eq. (4)andEq. (6)is bijective. Fromthe latterequationweseethatthe functionsA (x)are0d-homogeneous(0dH),
i
i.e., A (αx) = A (x), which is verified also from our example, A (αx,αy) = A (x,y) and A (αx,αy) = A (x,y).
i i x x y y
Computing the partial derivative of f(x) in Eq. (6) with respect to x , we obtain
j
∂f(x) q ∂A (x)
=A (x)+ x i , (7)
j i
∂x ∂x
j Xi=1 j
implying
q ∂A (x)
i
x =0. (8)
i
∂x
Xi=1 j
As can be seen, Eq. (8) is satisfied by any 1dH-function and thus it presents another characteristic relation of the
former class of functions. In fact, Eq. (8) presents a set of q conditions, since j =1,...,q. Due to f(x,y), the former
conditions take the form
∂A (x,y) ∂A (x,y) x 2 x x2 2x x
x y
x +y =x + exp +y − − exp =0, (9)
∂x ∂x (cid:18)y2 y(cid:19) (cid:18)y(cid:19) (cid:18) y3 y2(cid:19) (cid:18)y(cid:19)
∂A (x,y) ∂A (x,y) x2 2x x x3 2x2 x
x y
x +y =x − − exp +y + exp =0. (10)
∂y ∂y (cid:18) y3 y2(cid:19) (cid:18)y(cid:19) (cid:18)y4 y3 (cid:19) (cid:18)y(cid:19)
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Now, we want to prove the reverse,namely if Eq. (8) is true, then f(x) is 1dH. Eq. (8) is valid for any translation
f(x) of f(x), i.e., f(x) = f(x) + c, where c is a constant, so that A (x) := ∂f(x)/∂x = ∂f(x)/∂x = A (x).
i i i i
Multiplying Eq. (8) by dx and then summing over all j’s in the aforementioned equation, yields
j
e e e e
q q ∂A (x) q q ∂A (x) q
x i dx = x i dx = x dA (x)=0. (11)
i j i j i i
∂x ∂x
Xj=1Xi=1 e j Xi=1Xj=1 e j Xi=1
e
The changeintheorderofthe summationinEq. (11)canbe performed,since q ( q x ∂A (x)/∂x dx =0<
j=1 i=1 i i j j
∞)<∞. Adding the term q A (x)dx in Eq. (11), we obtain P P (cid:2) (cid:3)
i=1 i i e
P
q q e q q
A (x)dx = A (x)dx + x dA (x) ⇒ df(x)=d x A (x) ⇒
i i i i i i i i
(cid:18) (cid:19)
Xi=1 Xi=1 Xi=1 Xi=1
e e e e e (12)
q q
f(x)= x A (x)+a ⇒ f(x)= x A (x)+(a−c),
i i i i
Xi=1 Xi=1
e e
whereaisanintegrationconstant. We canalwayschoosec=a,sothatEq. (12)isidentifiedwithEq. (6). Herewith,
we have proven that the conditions in Eq. (8) are necessary and sufficient conditions for having 1d-homogeneity.
Thus, they can be considered as an alternative definition of 1dH-functions. An important consequence of Eq. (8) (or
equivalently Eq. (11) ) is that the differential of the function f(x) is exact
df(x)= q A (x)dx + q x dA (x)(1=1) q A (x)dx = q ∂f(x)dx , (13)
i i i i i i i
∂x
Xi=1 Xi=1 Xi=1 Xi=1 i
implying simply, that there are indeed functions satisfying Eq. (4).
It is worth remarking that Eq. (8), because of the equality ∂A (x)/∂x = ∂A (x)/∂x which holds true for any
i j j i
Cn≥2-function, can be rewritten as q x ∂A (x)/∂x = 0. One can easily verify that the former equation is the
i=1 i j i
respective result of Eq. (6) for 0dHPfunctio(cid:2)ns. It becom(cid:3)es obvious then that Eq. (8) fully characterizes both the
1dH-function f(x) and the 0dH-functions A (x) simultaneously.
i
III. MONOATOMIC IDEAL GAS ENTROPY OF THREE INDEPENDENT VARIABLES
We now make use of the results presented in the previous section on 1dH- and 0dH-functions to model thermo-
dynamic quantities. Considering three independent variables, i.e, U, V and N, representing the internal energy, the
volume and the number of the particles of a system under scrutiny, respectively. Further, we assume a 1dH-function
S =S(U,V,N) as the entropy. Then, according to Eq. (6), S can be expressed as follows
1 P(U,V,N) µ(U,V,N)
S(U,V,N)= U + V − N, (14)
T(U,V,N) T(U,V,N) T(U,V,N)
where
1 ∂S(U,V,N) P(U,V,N) ∂S(U,V,N) µ(U,V,N) ∂S(U,V,N)
:= , := , :=− . (15)
T(U,V,N) ∂U T(U,V,N) ∂V T(U,V,N) ∂N
Inthermodynamics,Eq. (14) isreferredto asthe Euler equation. Theleft handsideofthe definitions inEq. (15)are
concrete physicalquantities. T,P and µ denote the temperature, pressure andchemical potential, respectively. They
suppose to be 0dH-functions, otherwise the assumption of S being 1dH cannot hold true. As shown in the previous
section, S has to satisfy the conditions in Eq. (8). In the thermodynamic notation the former conditions take the
form
∂ 1 ∂ P ∂ µ
U +V −N =0, (16a)
∂U T ∂U T ∂U T
∂ 1 ∂ P ∂ µ
U +V −N =0, (16b)
∂V T ∂V T ∂V T
∂ 1 ∂ P ∂ µ
U +V −N =0. (16c)
∂N T ∂N T ∂N T
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If the 0dH-functions P,T and µ satisfy Eq. (16), then the differential of S is exact yielding the first law of thermo-
dynamics, given in Eq. (1). Considering the relations in Eq. (16) simultaneously, they can be written as
1 P µ
Ud +Vd −Nd =0. (17)
(cid:18)T(cid:19) (cid:18)T(cid:19) (cid:18)T(cid:19)
We identify Eq. (17) with the well known Gibbs-Duhem relation [4].
Let us now explore the conditions in Eq. (16) for the case of an ideal monoatomic gas in an open system, dN 6=0.
The expressions of pressure and temperature are given by
2U 2U
P(U,V,N)=P(U,V)= , T(U,V,N)=T(U,N)= . (18)
3V 3k N
B
ThepressureinEq. (18)hasbeenderivedwithinthekineticgastheory. ThetemperatureinEq. (18),withinclassical
thermodynamics, presents an ad hoc definition, being compatible with the equation of state for the ideal gases, i.e.,
PV = k NT. Indeed, combining P and T in Eq. (18) one obtains the former equation of state. Apparently, the
B
functionsP andT are0dH.Inthemathematicallytrivialcaseofconstantpressureandtemperature,dP =0=dT ⇒
U ∼N and V ∼N, Eq. (16) yields a constant chemical potential µ. In this case, the respective entropy function in
Eq. (14) is 1dH and linear with respect to N, i.e., S(U,V,N)=S(N)∼N. For the general case of varying pressure
and temperature i.e. dP 6=0 and dT 6=0, from Eq. (16), we read
∂ 1 1
µ(U,V,N)= µ(U,V,N)− , (19a)
∂U U N
∂ 2U
µ(U,V,N)=− , (19b)
∂V 3VN
∂ 1 5U
µ(U,V,N)=− µ(U,V,N)+ , (19c)
∂N N 3N2
respectively. Solving this system of differential equations, we are able to determine the expression of the chemical
potential compatible with Eqs. (1) and (18). The aforementioned system can be solved stepwise. We first solve Eq.
(19a). TheresultwilldependonanintegrationconstantZ(V,N),withrespecttoU. Then,pluggingtheformerresult
into Eq. (19b), Z(V,N) reduces to an integration constant Z′(N). Finally, following the same procedure with Eq.
(19c), Z′(N) reduces to a constant, which cannot be further reduced. Thus, for varying pressure and temperature,
the monoatomic ideal gas chemical potential is determined apart from a constant ξ, having the expression
U N 2 N
µ(U,V,N)= ξ+ln + ln . (20)
N(cid:20) (cid:18)U(cid:19) 3 (cid:18)V (cid:19)(cid:21)
As expected, the µ-function is 0dH. The respective entropy function S in Eq. (14) is then 1dH, given as
V 3 U 5 3
S(U,V,N)=k N ln + ln + − ξ . (21)
B (cid:20) (cid:18)N(cid:19) 2 (cid:18)N(cid:19) 2 2 (cid:21)
StudyingthevalidityofS,weobservethatthe functionS inEq. (21)canbe derivedbymeansofthe1d-homogeneity
formalismwhenP,T,µ6={0,const.,±∞},beforehand,andwhendU 6=0,dV 6=0anddN 6=0,simultaneously. Ifone
oftheformerdifferentialsisequaltozero,implyingthattherespectivevariableisaconstant,thenthe1d-homogeneity
formalism is not applicable, since in this case not all of the partial derivatives of S, i.e., 1/T, P/T or µ/T are 0dH.
Comparing now the Clausius ideal gas entropy S in Eq. (21) with the Sackur-Tetrode entropy S in Eq. (2), we
ST
observethat they both have the same form when ξ takes the value ξ =ln 3h2 . This specific expressionof ξ cannot
4πm
be obtained by virtue of the 1d-homogeneity analysis but only within qua(cid:0)ntum(cid:1)statistical mechanics. However, even
then, it is merely a constant and has no physical impact on the entropy expression. So, we observe that the physical
part of Sackur-Tetrode entropy coincides with the one of the ideal gas entropy S in the N → ∞ limit. We stress
that the analytical expression of ST-entropy is obtained after the explicit computation of the former limit, as can be
seen in Eq. (2), while S has been derived in practice for any N > 0. Thus, the two aforementioned entropies may
be identified only under the assumption that the thermodynamic limit is implicitly included in the thermodynamic
relations. Thisinturnwouldimply takingintoaccountthe restrictionsmentionedaboveregardingthe 0dH-functions
T,P andµ, that as muchthe energyU as the volume V are consideredin the same limit. If this is the case,then one
has to be very careful in discussing phenomenological thermodynamics of finite systems. In different case, studying
ideal gases in open systems for finite N we have S 6=S , yielding merely the same picture in the N →∞ limit.
ST
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A possible way to shed light on the aforementioned issue, namely whether phenomenological thermodynamics
implicitly contains the thermodynamic limit, or equivalently whether S and S are identical, is by considering the
ST
general solution (gs) of the differential FLT in Eq. (1), regarding the ideal gas entropy function. The latter is given
as in Eq. (21), with the ξ-term being a function of N, i.e., ξ →Θ(N), namely
V 3 U 5 3
S (U,V,N)=k N ln + ln + − Θ(N) . (22)
gs B (cid:20) (cid:18)N(cid:19) 2 (cid:18)N(cid:19) 2 2 (cid:21)
Inthis generalcase,the chemicalpotential, µ =−T(∂S /∂N),maygenerallynotpreservethe 0dHproperty. From
gs gs
Eqs. (2) and (22), we observe that in the case of ST-entropy the Θ-function in finite systems is inhomogeneous and
has the explicit expression
2 ln(N!)
Θ (N)=ξ+ 1−ln(N)+ , (23)
ST 3 N
h i
satisfying the lim Θ (N) = ξ limit. Let us then explore which is the most general expression of Θ(N) being
N→∞ ST
compatible with 1d-homogeneity, S →S1dH (and µ →µ0dH). Therefore, we compute again the conditions in Eq.
gs gs gs gs
(16), intermsofS now. Fromthe firsttwoconditionswe donotobtainanynew information,yetthe lastoneyields
gs
∂2Θ(N) ∂Θ(N) a
2
N +2 =0 ⇒ Θ(N)=a − , (24)
∂N2 ∂N 1 N
where a and a are integration constants. Plugging the result in Eq. (24) into Eq. (22) we obtain
1 2
1 P(U,V,N) µ0dH(U,V,N) 3
S1dH(U,V,N)= U + V − gs N + k a
gs T(U,V,N) T(U,V,N) T(U,V,N) 2 B 2
(25)
V 3 U 5 3 3
=k N ln + ln + − a + k a ,
B (cid:20) (cid:18)N(cid:19) 2 (cid:18)N(cid:19) 2 2 1(cid:21) 2 B 2
where µ0dH = µ in Eq. (20) with a ≡ ξ. Comparing Eqs. (21) and (25), it becomes obvious that S1dH contains an
gs 1 gs
extraadditiveconstant. However,inEq. (12)weshowedthatsuchaconstantcanalwaysbetransformedaway. Thus,
the 3k a -term has neither physical nor mathematical impact on S1dH so that the former is identified with S in Eq.
2 B 2 gs
(21). Eq. (24) then, unveils that the thermodynamic limit is not included in the phenomenological thermodynamic
equations. If it wouldbe included, then the a -termwouldnot appear,lim a /N =0. Accordingly,the Clausius
2 N→∞ 2
1dH entropy function S in Eq. (21), derived by means of purely thermodynamic relations, is distinct from the 1dH
ST-entropy, derived within statistical mechanics in Eq. (2).
IV. CONCLUSIONS
Studying homogenous functions of first degree (1d), we demonstrated that they can be defined through a set of
q-specificconditionsgiveninEq. (8),whereqdenotesthenumberofindependentvariablesoftheaforementionedfunc-
tions. Ifthe1d-homogeneousfunctionunderconsiderationisthethermodynamicentropy,thentheformerconditions,
expressed compactly in a single equation, are identified with the well-known Gibbs-Duhem relation.
Considering particularly the pressure P and the temperature T for the case of monoatomic ideal gases in open
systems, where the number of particles N, the internal energy U and the volume V are the independent variables,
we derived by means of the 1d-homogeneity conditions, the analytical formulas of the associated chemical potential
µ, related to the particle exchange,and the associatedthermodynamicalentropy S. Interestingly enough,for dP 6=0
and dT 6= 0 (⇒ dµ 6= 0), the former entropy coincides with the well known Sackur-Tetrode entropy S , apart
ST
from a non-physicalN-translation. In practice however,distinct physical conditions are requiredfor their derivation.
In fact, the two aforementioned entropies can be identified to each other only under the assumption of a “hidden”
thermodynamic limit in the phenomenological thermodynamic relations. We explicitly showed yet, that the former
assumptiondoesnotholdtrue. Accordingly,theClausiusentropyS,derivedonthe groundofhomogeneousfunctions
of degree one within phenomenological thermodynamics differs from the Sackur-Tetrode entropy S , derived within
ST
statistical mechanics, sharing merely the same expressions in the thermodynamic limit.
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Acknowledgments
The author would like to thank G. B. Bagci and S. Abe for fruitful discussions.
[1] K. Huang, Statistical mechanics, 2nd Edition, Singapore: John Wiley & Sons, 1963.
[2] M. A. M. Versteegh, D. Dieks, The Gibbs paradox and the distinguishability of identical particles, American Journal of
Physics: 79 (2011) 741.
[3] A.Ben-Naim,A Farewell toEntropy: Statistical Thermodynamics BasedonInformation,WorldScientific,Singapore,2008.
[4] T. Fließbach, Statistische Physik, Heidelberg: Spectrum-Verlag,1999.