Table Of ContentDedicated to: Antonino Loiacono Carmela Mazzuti
WEINHOLD’LENGTH IN AN ISOCHORIC THERMODYNAMIC
SYSTEM WITH CONSTANT HEAT CAPACITY
MANUEL SANTORO
5
0
0 Abstract. The purpose of this paper is to study thermodynamic length of an
2
isochoric two dimensional thermodynamic system with constant heat capacity. We
n
a find that length is related to the heat flow into the substance. We give examples of
J
Ideal gas and Van der Waals gas.
2
1
2
v 1. Introduction
3
6
Weinholdproposedametric12 η inthermodynamicstatespaceassecondderivative
0 ij
9 of internal energy with respect to extensive variables X and X , namely η = ∂2U
0 i j ij ∂Xi∂Xj
4 with i,j = 1,...,n. Such a thermodynamic state space is the energy surface geo-
0
/ metrically described by constitutive relation U = U(X ,...,X ) where, for example,
h 1 n
p X = S, X = V,etc, with S and V being, respectively, the entropy and the volume
- 1 2
h
of the thermodynamic system.
t
a
m Weinholdmetricwas,then,consideredbyP.Salamon5,6,7,8,R.S.Berry5,7,8,J.Nulton6,8,
: E. Ihrig6, and others7,9 to study the physical significance of thermodynamic length. It
v
Xi wasfoundthatthelocal meaningofηij isthedistance7 betweentheenergysurfaceand
r the linear space tangent to this surface at some point where η is evaluated. Indeed,
a ij
consider a particular energy state denoted by coordinates (X0,...,X0). The tangent
1 n
space is attached to the energy surface at point (U ,X0,...,X0). Consider, now, a
0 1 n
newenergy state(X ,...,X )close enoughto(X0,...,X0). Thenthedistance between
1 n 1 n
the point on the surface (U,X ,...,X ) and the tangent space is the availability5,7 or
1 n
the available work5 of the system. This is a local interpretation since it requires just
small displacements, like for fluctuations, from the given point (U ,X0,...,X0) on the
0 1 n
surface.
On the other hand, we could consider thermodynamic length taking the metric η
ij
globally. With this approach, we look at a path φ on the energy surface between two
states a and a and consider the length of the path
0 1
1
2 MANUEL SANTORO
a1
1
L = [ η dX dX ]2
a0a1 ij i j
Za0 i,j
X
Such a length, in general, represents7 the change in mean molecular velocity de-
pending on the particular nature of the thermodynamic process defining the path φ
and its dimension is square root of energy. But thermodynamic length was explicitly
studied7 just in the Ideal case. In particular, it was found that, for a reversible or
endoreversible Ideal process at constant volume, thermodynamic length is given by7
θ
Lv = 2 C [ T T ] = [ v∗ v∗]
v 1 − 0 3 1 − 0
r
where v∗ and v∗ are, resppectivpely, thepmean of thepsquarespof the gas-kinetic veloc-
1 0
ities at T and T , and θ = mCv, where m is the particle mass.
1 0 R
In this paper we shall generalize the physical interpretation of thermodynamic
length to a two dimensional thermodynamic system with constant heat capacity and
we shall see that length is strictly related to the concept of heat flux.
2. Thermodynamic length with Weinhold metric
The thermodynamic length between two points a and a in thermodynamic state
0 1
space is given by the following equation
a1
1
L = [ η dX dX ]2 (2.1)
a0a1 ij i j
Za0 i,j
X
where η are elements of the thermodynamic metric and X represent independent
ij i
coordinatesinthermodynamicstatespace. Wearegoingtoshowthatthermodynamic
length in an isochoric thermodynamic system with two degrees of freedom with heat
capacity constant is related to the heat flux of a quasi-static process at constant mole
number.
In this manuscript, we shall give, first, a general discussion about thermodynamic
length and, then, we shall focus our attention to the relation between length and heat
flow.
WEINHOLD’LENGTH IN AN ISOCHORIC THERMODYNAMIC SYSTEM WITH CONSTANT HEAT CAPACITY3
Let’s consider constitutive relation u = u(s,v) where u is the molar internal energy,
s is the molar entropy and v is the molar volume. s and v are the two independent
variables. Then, it is known that Weinhold metric is given by3
1 T Tα
η = −kT (2.2)
ij cv Tα cp !
−kT vkT
where
(1) c is the molar heat capacity at constant volume:
v
∂s
c = T( ) ,
v v
∂T
(2) c is the molar heat capacity at constant pressure:
p
∂s
c = T( ) ,
p p
∂T
(3) α is the thermal coefficient of expansion:
1 ∂v
α = ( ) ,
p
v ∂T
(4) κ is the isothermal compressibility:
T
1 ∂v
κ = ( ) .
T T
−v ∂p
Thermodynamic length with such a metric is given by7
T Tα c
L = [ (ds)2 2 dsdv + p (dv)2]12 (2.3)
c − c κ vc κ
v v T v T
Z
and, if molar entropy and molar volume are given parametrically as s = s(ξ),
v = v(ξ), then we have7
L = ξf[T (ds)2 2 Tα dsdv + cp (dv)2]21dξ (2.4)
c dξ − c κ dξ dξ vc κ dξ
Zξi v v T v T
Our study is going to focus on length at constant volume which is given by9
T
Lv = ds = √η ds (2.5)
c 11
Z r v Z
4 MANUEL SANTORO
3. Thermodynamic length in an isochoric TD system with constant
heat capacity
3.1. General discussion. Considerconstantmolarheatcapacityatconstantvolume
c = T(∂s) . Then (∂T) = T and since T = (∂u) , we get the following equation11
v ∂T v ∂s v cv ∂s v
∂2u 1 ∂u
= 0. (3.1)
∂s2 − c ∂s
v
Integrating once we get
∂u u
= +f (v),
2
∂s c
v
with an arbitrary function f (v). Solving this equation we get the fundamental con-
2
stitutive law in the form11
s−s0
u(s,v) = f1(v)e cv cvf2(v), (3.2)
−
with another arbitrary function f (v).
1
Example 1. Ideal Gas
It is known1 that for an Ideal Gas
u v
s = s +c ln( )+Rln( )
0 v u v
0 0
Let u = v = 1 for simplicity. Then, solving for the internal energy, we get
0 0
−R s−s0
u = v cve cv (3.3)
−R
Therefore, considering (3.2), we get the Ideal Gas case if we set f1(v) = v cv and
f (v) = 0.11
2
Example 2. Van der Waals Gas
The entropy function of the Van der Waals Gas is given by1
a
s = s0 +Rln[(v −b)(u+ v)cRv]
where a and b are positive constants.
Then, solving for u, we get
−R s−s0 a
u = (v b) cve cv (3.4)
− − v
WEINHOLD’LENGTH IN AN ISOCHORIC THERMODYNAMIC SYSTEM WITH CONSTANT HEAT CAPACITY5
Therefore, considering the general case with heat capacity constant (3.2), we get
the Van der Waals Gas case if we set f1(v) = (v−b)−cRv and f2(v) = cvav.11
We have realized, so far, that equation (3.2) is very general as long as heat capacity
is constant. Therefore, our first goal is to try to express thermodynamic length at
constant volume in the same way.
Considering (2.5), we have the following relation
∂2Lv 1 ∂Lv
= (3.5)
∂s2 2c ∂s
v
Integrating once we get
∂Lv Lv
= +h (v)
2
∂s 2c
v
where h (v) is some arbitrary function of v.
2
Integrating once more we get the interesting result
Lv = 2[h1(v)es2−csv0 cvh2(v)] (3.6)
−
with h (v) also arbitrary function.
1
In this case, h and h are completely unknown functions of molar volume. Never-
1 2
theless, we can still recover some information just about h in the following way.
1
Since
∂u u
T = ( ) = +f (v)
∂s v c 2
v
then, it is easy to see that cTv = f1c(2vv)es−cvs0 and , therefore,
T f1(v) s−s0
= e 2cv (3.7)
c c
r v p v
This implies that thermodynamic length at constant volume is given by
Lv = fc1(v) es2−csv0ds = 2 f1(v)es2−csv0 +k(v) (3.8)
p v Z
p
where k(v) is any function of the molar volume.
6 MANUEL SANTORO
This result implies that h is known for specific examples being h (v) = f (v),
1 1 1
while h is not. Therefore, we exactly know what thermodynamic length of a two
2 p
dimensional isochoric system at constant heat capacity is up to a constant.
Let’srewritethermodynamiclengthinthefollowingwaybysettingk(v) = 2c h (v),
v 2
−
Lv = 2[ f1(v)es2−csv0 cvh2(v)] (3.9)
−
p
Remark 1. It seems reasonable to think that h would be somehow related to f as
2 2
h is to f . But, at this point, such a statement cannot be made.
1 1
Now, since for an isochoric system dq = du = Tds = f1(v)es−cvs0ds, where dq is the
cv
infinitesimal heat or heat flux per unit mole, then we have the following result
Lemma 1.
dLv 1 dq
( )2 = (3.10)
ds c ds
v
3.2. Relation between ”isochoric” length and heat. Here we’ll derive a relation
between the heat flux q as difference in molar internal energy and thermodynamic
length of a quasi-static isochoric process.
Since the molar volume is constant, consider u = u(s ,v) and u = u(s ,v).
2 2 1 1
Let both s and s being greater than the initial entropy s . It is important to
1 2 0
realize that conditions on the molar entropy s are equivalent to conditions on the
temperature since T = f1(v)es−cvs0 with both heat capacity and volume constant. Since
cv
q = ∆u = u u , then, considering (3.2), we have
2 1
−
s2−s0 s1−s0
q = u2 u1 = f1(v)[e cv e cv ] = cv(T2 T1) (3.11)
− − −
with the mass of the substance normalized to 1.
Similarly, considering the general expression of thermodynamic length (3.9) we
have
Lv = 2 f1(v)[es22−cvs0 es12−cvs0] (3.12)
−
Let’s set s < s or, equivalentply, T < T . Then, requiring length to be always
1 2 1 2
positive or equal to zero, we will consider
WEINHOLD’LENGTH IN AN ISOCHORIC THERMODYNAMIC SYSTEM WITH CONSTANT HEAT CAPACITY7
s2 T
1)Lv = ds
c
Zs1 r v
s1 T
2) Lv = ds
− c
Zs2 r v
Let’s now, for simplicity, set s = s . Then we can rewrite both equations (3.11)
0 1
and (3.12) in the following way
s2−s1 s2−s1
qin = f1(v)[e cv 1] > 0 qout = f1(v)[1 e cv ] < 0 (3.13)
− −
and
Lv = 2 f1(v)[es22−cvs1 1] Lv = 2 f1(v)[1 es22−cvs1] (3.14)
− − −
From these importpant expression of heat and lengtph it follows,
Theorem 1.
Lv
q = [Lv +4 f (v)] (3.15)
in 1
4
or, equivalently, p
(Lv)2 +4 f (v)Lv 4q = 0 (3.16)
1 in
−
p
It easily follows that
Lv
q = q = − [Lv +4 f (v)] (3.17)
out in 1
− 4
or, equivalently, p
(Lv)2 +4 f (v)Lv +4q = 0 (3.18)
1 out
We also have the following two coprollaries,
Corollary 1.
Lv = 2[ f (v)+q f (v)] = 2[√u √u ] (3.19)
1 in 1 2 1
− −
p p
8 MANUEL SANTORO
which is equivalent to
Lv = 2[ f (v) f (v) q ] = 2[√u √u ] (3.20)
1 1 out 1 2
− − − −
p p
Corollary 2.
Lv = 0 iff q = 0 (3.21)
±
Considering equation (3.11), q = 0 iff T = T . Therefore, it does make sense that,
2 1
at thermal equilibrium, thermodynamic length Lv = 0.
4. Examples
−R
4.1. Ideal Gas. In example 1, we have seen that f1(v) = v cv. Therefore,
Lv = 2[ v−cRv +qin v−cRv] (4.1)
−
q q
or
Lv = 2[ v−cRv v−cRv qout] (4.2)
− − −
q q
4.2. Van der Waals Gas. In case of Van der Waals gas, we have shown in example
−R
2 that f1(v) = (v b) cv. Therefore,
−
Lv = 2[ (v b)−cRv +qin (v b)−cRv] (4.3)
− − −
q q
or
Lv = 2[ (v b)−cRv (v b)−cRv qout] (4.4)
− − − − −
q q
5. Conclusions
We have tried to achieve a physical interpretation of thermodynamic length in a
simple isochoric thermodynamic system with two degree of freedom with constant
heat capacity. We also gave examples like Ideal Gas and Van der Waals Gas.
6. Comments
I am aware of the fact that reference [11] has not been submitted yet. But all the
steps needed to the reader for a full understanding are entirely reported.
WEINHOLD’LENGTH IN AN ISOCHORIC THERMODYNAMIC SYSTEM WITH CONSTANT HEAT CAPACITY9
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Department of Mathematics and Statistics, Portland State University, PO Box
751, Portland, OR 97207-0751, USA
E-mail address: [email protected]
