Indian Journal of Science and Technology, Vol 10(15), DOI: 10.17485/ijst/2017/v10i15/71242, April 2017
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Joule – Thomson Inversion Curves for Van Der Waals
Gas from a Mathematical Point of View
J. Venetis*
Department of Applied Mathematics and Physical Sciences, National Technical University of Athens (NTUA),
Zografou Campus 9, Iroon Polytechniou street, Zografou – 15780, Greece; johnvenetis4@gmail.com
Abstract
A continuation of our ongoing investigation into Joule – Thomson inversion curves for van der Waals gas, is performed
from a mathematical viewpoint. The methodology basis of our analysis is the quadratic polynomial theory. In this context,
focusing on the parametric equation of inversion curves in a P – V frame of reference we obtain a qualitative illustration of
variables T, V by means of two inequality relations. However, we should elucidate that these inequalities are valid only for
the intersection points between the family of Joule – Thomson inversion curves and the isothermal spinodal lines, provided
that they are both sketched in a common P – V coordinate system. The mathematical treatment of the parametric equation of these curves has been carried out in a rigorous manner and no further restriction is introduced for the variables
T, V. Thus, the proposed inequalities have a wider range of validity when compared with those that had been previously
presented by the author and therefore their possible applications to P – V – T surfaces of van der Waals gas, are also wider.
Keywords: J – T Inversion Curves, P – V System, Quadratic Polynomial, Spinodal Lines, Van Der Waals Gas
1. Introduction
The Joule – Thomson (J – T) inversion curve is determined as the set of thermodynamic states, in which the
temperature of an ideal or real gas does not vary with
isenthalpic expansion.
Yet, a direct measurement of these curves is many
times a troublesome experimental procedure and unfortunately may result in unreliable conclusions1. At near
– inversion conditions, the vanishing of Joule – Thomson
coefficient implies that even very large pressure changes
will result in small temperature differences and therefore
extremely accurate measurements of temperature are
needed for the reliable determination of inversion pressures1,2.
Meanwhile, the Joule – Thomson coefficient which
depends on the volume, specific heat capacity, temperature, and thermal expansion coefficient of the gas is
estimated as follows2,3.
m JT =
V
(aT − 1) (1)
CP
*Author for correspondence
Where a is the thermal expansion coefficient of the gas
which arises from the following formula
a=
1 Ê ∂V ˆ
(2)
V ÁË ∂T ˜¯ P
Moreover, the enthalpy depends on the specific heat
capacity, along with the temperature and pressure of the
gas before expansion. For any real gas the coefficient
μJT vanishes at some point, (inversion point). If the gas
temperature is below its inversion point temperature,
this coefficient is positive and if the gas temperature
is above its inversion point temperature, it is negative.
Furthermore, the variation of pressure is always negative
when a gas expands. Hence, one may infer the following
conclusions3,4.
i) If the gas temperature is below its inversion temperature the coefficient μJT is positive and given that the
change of P is always negative it implies that the gas
must cool since the change of T must be negative as
well.
Joule – Thomson Inversion Curves for Van Der Waals Gas from a Mathematical Point of View
ii) If the gas temperature is above its inversion temperature the coefficient is now negative and given that the
change of P is always negative, the gas heats since the
change of T must be positive.
Depending on state conditions, this coefficient may be
positive or negative. Positive values induce a cooling of
the gas, since it passes through an adiabatic throttle.
The curve linking all state points where μJT vanishes is
the Joule –Thomson inversion curve. Apparently, this is
an equivalent definition of this curve.
On the other hand, it is generally accepted that the
prediction of the Joule – Thomson inversion curve
consists in a very trustworthy test of an equation of
state1,4.
In the past years, a considerable amount of recent
research work has been made towards the evaluation
of Joule – Thomson inversion curves for many types of
ideal or real gases. Particularly, remarkable molecular
simulation analyses were presented either for pure fluids or for mixtures in several valuable investigations5–9.
Concurrently, this family of curves was determined by
means of other approaches in References10–13.
Finally, the influence of spinodal curve condition on J
– T inversion curves for van der Waals gas, was discussed
in Reference14.
In the present work, we extend some previous results
of our ongoing research by obtaining some further qualitative illustration with regard to this family of curves, for
real gases of van der Waals type.
1.1 Towards a Qualitative Illustration of the
Variables T, V
The thermodynamic behavior of any van der Waals gas is
outlined by the following constitutive law
Ê
n2 a ˆ
+
P
Á
˜ (V − nb) = nRT (3)
V2 ¯
Ë
Where the constants a,b are interconnected with the
coordinates of the critical point (P0, V0, T0) as follows:
V0 = 3b ; 27b2 P0 = a ; 27bRT0 = 8a (4 a, b, c)
In continuing, equation (3) when n = 1 i.e. for one
mole of the gas, is reduced to the following expression
a ˆ
Ê
ÁË P + 2 ˜¯ (V − b) = RT (5)
V
2
Vol 10 (15) | April 2017 | www.indjst.org
Then, one may also report that after the necessary
algebraic manipulation the following equivalent third
degree polynomial equation arises4.
RT ˆ 2 a
a
Ê
V3 −Áb+
V + V − b = 0 (6)
˜
Ë
P ¯
P
P
In the sequel, let us concentrate on equation (3),
which can be solved for T to give
1Ê
a ˆ
P + 2 ˜ (V − b) (7)
R ÁË
V ¯
Also, a differentiation of equation (7) with respect to
V yields
T=
1 Ê 2a
aˆ
Ê ∂T ˆ
ÁË ∂V ˜¯ = R ÁË − 3 (V − b) + P + 2 ˜¯ ⇔
V
b
P
Ê ∂V ˆ
ÁË ∂T ˜¯ =
P
R
⇔
2ab a
P+ 3 − 2
V
V
1 Ê ∂V ˆ
=
V ÁË ∂T ˜¯ P
R
2ab a (8)
PV + 2 − 2
V
V
Consequently, the thermal expansion coefficient is
given as
R
2ab a (9)
PV + 2 −
V
V
In this context, the Joule – Thomson coefficient μJT
can be calculated as
a=
m JT =
m JT
V
(
CP
RT
− 1) ⇔
2ab a
PV + 2 −
V
V
2ab a
Ê
RT − PV − 2 +
Á
V
V
V
=
Á
2ab a
CP Á
PV + 2 −
Ë
V
V
ˆ
˜
˜ (10)
˜
¯
The latter can be combined with equation (5) to yield
1 Ê 2aV 2 − PbV 3 − 3abV ˆ
(11)
CP ÁË PV 3 − aV + 2ab ˜¯
Hence, the parametric equation of inversion curves
arises
m JT =
PbV 3 − 2aV 2 + 3abV = 0 ⇔
Indian Journal of Science and Technology
J. Venetis
PbV 2 − 2aV + 3ab = 0 (12)
In addition, it is known from Equilibrium
Thermodynamics15 that the spinodal points of any isothermal curve in a P – V coordinate system emerge from
the following formula
∂P
RT
2a
=−
+ 3 = 0 (13)
2
∂V
(V − b) V
2
x1 ≤ B x1 + C
∧
2
x2 ≤ B x2 + C Thus, since equation (12) is equivalently expressed as
2a
3a
V − V+
= 0 we can write out
Pb
P
2
2
r1 ≤
At this point, we should emphasize that the critical
points defined by equations (3a, b, c) also satisfy equation
(13), although it is not equivalent to the group of equations (3a, b, c).
Hence, we deduce that
Ê (V − b)2 RT ˆ
−
= 0 ⇔ (V − b)2 = RT V 3 ⇔
Á
2a ˜¯
(V − b)2 Ë V 3
2a
2a
a ab ˆ
Ê
ÁË PV − Pb + V − 2 ˜¯
V
(V − b)2 =
V3 ⇔
2a
Thus, referring to the isothermal spinodal points,
which evidently are defined as the intersection points
between the inversion curves and the isothermal spinodal
ones, one may regard the pressure P as a parameter. Also,
the same hypothesis holds for the intersection points of
the inversion curves with the family of curves arising
from equation (14). In this context, one may suppose that
the terms in the left side of equation (12) constitute a single – valued quadratic polynomial.
Next, without violating the generality, let us assume
that equation (12) has two distinct roots r1, r2 (real or
complex), the values of which are nonzero.
Obviously, the following expressions hold,
2a
(15a)
r1 + r2 =
Pb
3a
(15b)
P
Here, we shall take into account that given a quadratic
equation in the form x2 + Bx + C = 0 having two distinct nonzero roots x1;x2 (real or complex) the following
inequalities hold16,17.
r1 ⋅ r2 =
Vol 10 (15) | April 2017 | www.indjst.org
2a
3a
r1 +
Pb
P
∧
2
r2 ≤
2a
3a
r +
Pb 2
P (17 a, b)
Hence, adding by members one finds
2
2
r1 + r2 ≤
2a
6a
(18)
r1 + r2 +
Pb
P
(
)
Inequality (18) can be modified as
(r
1
+ r2
)
2
≤
2aV 2 + 2ab2 − 4abV = PV 4 − PbV 3 + aV 2 − abV ⇔
PV 4 − PbV 3 − aV 2 + 3abV − 2ab2 = 0 (14)
(16 a, b)
r1 − r2 ≤
2a
6a
r1 + r2 +
+ 2 r1 ⋅ r2 ⇒
Pb
P
(
)
r1 ⋅ r2
2a 6a
1
+
+2
⇔ (19)
Pb
P r1 + r2
r1 + r2
Since r1 − r2 ≤ r1 + r2 ≤ r1 + r2 it follows
r1 − r2 ≤
r1 ⋅ r2
2a 6a
1
+
+2
⇔
Pb
P r1 + r2
r1 + r2
3a
2 a2 − 3aPb2
2a 6a 1
P
≤
+
+2
2a
bP
Pb
P 2a
bP
bP (20)
After some algebra, we obtain
a2 − 3aPb2 ≤ a + 3b2 P (21)
and therefore
b2 P 2 + aP ≥ 0 (22)
Meanwhile, given that for one mole of a gas of van der
RT
a
Waals type the pressure is written as P =
− 2 ,
V − b 3V
one infers
b2 RT ab2
−
+a≥0⇔
V − b 3V 2
Indian Journal of Science and Technology
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Joule – Thomson Inversion Curves for Van Der Waals Gas from a Mathematical Point of View
b2 RT ab2
−
+ 27b2 P0 ≥ 0 ⇔
V − b 3V 2
RT
a
− 2 + 27 P0 ≥ 0 (23)
V − b 3V
a2 − 3aPb2 ≤ a2
On the other hand, when the discriminant of the quadratic polynomial occurring in equation (12) is strictly
negative, i.e. 4a2 –12Pb2 a < 0 the roots r1; r2 are complex
conjugates and hence it implies that r1 − r2 = 0 .
Thus inequality (18a) or inequality (20) yields
a
3b
2
+
RT
a
+
≥ 0 (24)
V − b 3V 2
Here, one may point out that inequalities (23) and (24) are
valid only for the intersection points between the inversion curves and the set of curves defined by equation (14).
2. Discussion
Inequalities (23) and (24) can be applied to P – V – T thermodynamic systems on the premise that the designation of
isothermal spinodal curves, where obviously the pressure P
may be supposed to be a parameter, was a priori executed.
Moreover, the fact that inequalities (23) and (24) hold
solely on a grid constructed by the intersection between
the isothermal spinodal lines and the family of inversion
curves indeed may constitute a solid assumption.
On the other hand, one may observe that the proposed
inequalities (23) and (24) have a very important advantage when compared with those presented in Reference14,
since they performed mathematical derivation leading
to (23) and (25) was carried out without assuming that
equation (12) has real roots.
Consequently, the discriminant of the quadratic polynomial appearing in the left side of equation (12) is not
supposed to be strictly positive, fact that would lead to
a further constraint between the variables T, V and the
other involved parameters.
Also, it can be claimed that the results obtained here
may be regarded as basic ones and therefore they can be
combined with theoretical formulas and experimental or
numerical results concerning more complicated thermodynamic systems of practical importance appeared in the
literature18,19.
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Vol 10 (15) | April 2017 | www.indjst.org
Οn the other hand, the performed inequalities (23)
and (24) due to their rigorous mathematical character
could be evaluated and assessed in parallel with straightforward approaches that shed light on issues of Equations
of state for P – V – T systems20.
3. Conclusions
In this paper, the author obtained a further mathematical
analysis of some recent results of an ongoing research in
the matter of J – T inversion curves for the class of gases
described by van der Waals Eos. The objective of this work
was not to perform an analytical or numerical prediction
of this family of curves, but to derive inequality relations
concerning the variables T, V and the parameters occurring in this constitutive law. In this context, the author
concentrated on the intersection points between the family of J – T inversion curves and the isothermal spinodal
lines, as long as they are drawn in a common P – V coordinate system. Thus, the pressure P was considered as a
parameter and then two inequalities were extracted which
could concern any grid being motivated by the possible
family of inversion curves. Obviously, these inequalities
may hold whenever the circumstantial P – V – T system
enables us to assume this variable as a parameter or as a
sequence of distinct values.
4. References
1. Colazo A, da Silva F, Müller E, Olivera F. Joule-Thomson
inversion curves and the supercritical cohesion parameters
of cubic equations of state. 1992; 22:135.
2. Smith EB. Basic chemical thermodynamics. 3rd Edition,
Clarendon Press, Oxford; 1982. p. 119.
3. Caldin EF. An introduction to chemical thermodynamics.
Clarendon Press, Oxford; 1958. p. 424.
4. Mc Glashan ML. Chemical thermodynamics. Academic
Press, London; 1979. p. 94.
5. Vrabec J, Kedia GK, Hasse H. Prediction of Joule–Thomson
inversion curves for pure fluids and one mixture by molecular simulation. Cryogenics. 2005; 45(4):253–8. Crossref
6. Vrabec J, Kumar A, Hasse H. Joule–Thomson inversion
curves of mixtures by molecular simulation in comparison
to advanced equations of state: natural gas as an example.
Fluid Phase Equilibria. 2007; 258:34–40.
7. Escobedo FA, Chen Z. Simulation of isoenthalps and
Joule-Thomson inversion curves of pure fluids and mixtures. Molecular Simulation. 2006 Sep 23; 26(6):395–416.
Crossref
Indian Journal of Science and Technology
J. Venetis
8. Colina C, Lisal M, Siperstein F, Gubbins K. Accurate CO2
Joule–Thomson inversion curve by molecular simulations.
Fluid Phase Equilibria. 2002 Nov 15; 202(2):253–62.
9. Castro-Marcano F, Olivera-Fuentes CG, Colina CM.
Joule Thomson inversion curves and third virial coefficients for pure fluids from molecular-based models.
Industrial Engineering Chemical Research. 2008 Sep 27;
47(22):8894–905.
10. Matin NS, Haghighi B. Calculation of the Joule-Thomson
inversion curves from cubic equations of state. Fluid Phase
Equilibria. 2000 Oct 1; 175(1–2):273–84. Crossref.
11. Haghighi B, Laee MR, Husseindokht MR, Matin NS.
Prediction of Joule-Thomson inversion curves by the use
of equation of state. Journal of Industrial and Engineering
Chemistry. 2004; 10(2):316–20.
12. Nichita DV, Leibovici CF. Calculation of Joule–Thomson
inversion curves for two-phase mixtures. Fluid Phase
Equilibria. 2006; 246(1–2):167–76.
13. Bessieres D, Randzio SL, Pineiro M, Lafitte T, Daridon
J. A combined pressure-controlled scanning calorimetry
and monte carlo determination of the Joule−Thomson
inversion curve: application to methane. Journal of
Physics and Chemistry B. 2006 Feb 23; 110(11):5659–64.
Crossref
Vol 10 (15) | April 2017 | www.indjst.org
14. Venetis J. The effect of the spinodal curve condition on J
– T inversion curves for van der Waals real gas. Scientific
Research and Essays. 2015 Oct; 10(19):610–4.
15. Adkins CJ. Equilibrium thermodynamics. Cambridge
University Press, United Kingdom; 1968.
16. Prasolov VV. Polynomials. Algorithms and Computation in
Mathematics, Springer –Verlag, Berlin, Heidelberg. 2004;
11:301.
17. Borwein P, Erdelyi T. Polynomials and polynomial inequalities. Graduate Texts in Mathematics, Spinger. 1995;
161:1–482. Crossref
18. Jafari M, Salarian H, Bazrafshan J. Study on entropy generation of multi – stream plate fin heat exchanger with
use of changing variables thermodynamic and fluids flow
rate between plates and provide an optimal model. Indian
Journal of Science and Technology. 2016 Feb; 9(7):1–7.
DOI: 10.17485/ijst/2016/v9i7/87736.
19. He X, Doolen G. Thermodynamic foundations of kinetic theory and lattice boltzmann models for multiphase flows. Journal
of Statistical Physics. 2002 Apr; 107(1):309–28. Crossref
20. Kalanov TZ. The correct theoretical analysis of the foundations of classical thermodynamics. Indian Journal
of Science and Technology. 2009 Jan; 2(1):12–7. DOI:
10.17485/ijst/2009/v2i1/29364.
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