CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
HIGH-PRESSURE VAPORIZATION AND BOILING OF CONDENSED
MATERIAL: A GENERALIZED CLAUSIUS-CLAPEYRON
EQUATION
Alexander L. Conor1
Applied Science and Engineering Consulting, 624-523 Finch Avenue West, Toronto, ON, Canada
M2R 1N4
Abstract In this study a generalized Clausius-Clapeyron equation, based on the equilibrium
thermodynamics and including the pressure effect, is derived. It is assumed that the vapor and the liquid
(or solid) phases of the material are in equilibrium, whereas liquid phase is under total pressure equaling
the sum of vapor pressure and external pressure. New equations with high external pressure give the
values of partial vapor pressure significantly higher and boiling points lower compared with data
obtained by Clausius-Clayperon equation.
pressure on vapor pressure is presented below.
INTRODUCTION
To determine the partial vapor pressure, p , on
the surface of the particle (molten metal) or the
drop of the liquid, the following well-known
Clausius-Clapeyron equation is applied
PHASE EQUILIBRIUM FOR MIXTURE:
VAPOR/SURROUNDING GAS AND
CONDENSED PHASE.
We consider a closed system with given volume
V=Vi+V2, where Vi and V2 are volume of gaseous
mixture and condensed phase, respectively. Let Ni
(i= 1,2,3) be mole number of vapor, condensed
phase and surrounding gas, respectively, whereas
N3=constant, since gas is impermeable for
condensed phase, and N1+N2=const
Free energy (Helmgoltz function) for such
system can be represented in the following form,
(1)
In Eq. (1), p* and 1 are constants. T, 1 and R
denote the temperature, the latent heat of
vaporization of material and universal gas constant.
In the case of detonation of the mixture with
metallic particles where the external pressure, p , in
10
8
the reaction zone is of the order of -10 Pa, the
vapor phase fraction of the metal particles,
calculated using Eq. (1), was found to be negligible
even at high temperatures. Possibly, ClausiusClapeyron equation, determining partial pressure of
saturated vapor on the condensed surface in
vacuum (p =0), is responsible in this case for the
g
low vapor pressure.
Due to this, the derivation of the corresponding
equation taking into account the effect of external
A = A! (T, V1? Nb N3) +A2 (T, V2,N2),
where A* (i=l,2) is free energy of the gaseous
mixture and condensed phase, respectively. It is
known1 that for equilibrium system free energy is
minimum with constant T and V. Particularly, it
means that partial derivatives of A at variables Vi
and NI, equal zero.
Since dA} IdN^ = //j and dA2 I dN2 = ju2 we
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(similar to the approach used in deriving ClausiusClapeyron equation). Therefore, Eq. (3) reduces to
the following form
have equality of the corresponding chemical
potentials,ju^ = // 2 .It is important to note, that
within the framework of the model of ideal gas free energy of gaseous mixture is represented as
additive function of mole number NI (i=l,3), i.e.
Aj=Av (T, Vi, NO+Ag (T, V1? N3). Differentiating
this expression we obtain that a chemical potential,
//!, does not depend on the mole number N3 and
coincides with the chemical potential of the pure
vapor, occupying volume Vi. Therefore, the
chemical potentials for the vapor ^ and the liquid
phase jj,2 must be equal in the following form
^)=exp
(4)
where pv represents the partial vapor pressure
corresponding to the case when p=0, i.e. it is
g
defined by Eq. (1), j^ is molecular mass of the
vapor and p is constant density of condensed
material. The implicit Eq. (4) can be used to
calculate the partial vapor pressure, p , as a
function of temperature and pressure. Analysis of
Eq. (4) showed that its solution lies in the range
0<pg<pgm where pgm can be determined from the
correspondent equation. It is important to notice
that for p >0 the vapor pressure, PV, predicted by
g
Eq. (4) is found to be greater than the
corresponding value predicted by ClausiusClapeyron equation. Note, that Eq. (4) was derived
without assumption of smallness for molar volume
of the condensed phase, (V2/N2)« (Vi/NO.
(2)
where p = PV + pg is total pressure.
The equilibrium condition with different
pressures was also used in the derivation of the
well-known Van't Hoff equation for osmotic
pressure. The equality (2) implies that
EFFECT OF EXTERNAL PRESSURE ON
PARTIAL VAPOR PRESSURE AT PHASE
TRANSITION WITH CONSTANT
TEMPERATURE
DEPENDENCE OF BOILING POINT OF THE
CONDENSED MATERIAL ON PRESSURE
To start with we find function f (T, p ) for phase
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transition at constant value of T.
Since, as per2,
Eq. (4) can also be used to determine the
dependence of the boiling point of the molten
particle on the pressure of the surrounding gas.
Under boiling conditions, the dependence of the
boiling point, Tb, on the pressure could be
expressed in the form of an implicit equation
similar to Eq. (4). However, if p* is negligible
compared to pg then the equation for Tb reduces to
the following explicit equation:
N
which means that the differentiation of Eq. (2)
leads to the following equation,
(3)
„ i-v»mpKip)
Here Si, Vi and Ni denote entropy, volume and
mole numbers of each phase, respectively.
Equation (3) can be integrated assuming a
constant value of T and using EOS for ideal gas
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Hence, for function p , (6), the following partial
derivatives take place,
Table 1. T as a Function of pg
l.OOE+08 5.00E+08 7.00E+08
pg(Pa)
Tb°K:Eq.(5)
5956
7315
7217
Tb°K:(Ref. 3) 6125
9072
8376
It is interesting to note that the equation to
determine the boiling point, derived based on the
Clausius-Clapeyron approach, is different from Eq.
(5). The second term in the numerator of Eq. (5) is
missing in that equation. Therefore, the values of
Tb predicted by Eq. (5) will be lower than the
corresponding values obtained based on Clausius Clapeyron equation. Comparison of the Tb results
for Al calculated from Eq. (5) and those obtained,
based on the Clausius-Clapeyron equation, from
(Ref 3) is shown in Table 1. Since pressure, pg, has
to be less than pressure in critical point (C.P.),
then for aluminum maximum pressure, shown in
Table 1, is ~ IGpa. At this point in vicinity of
C.P., aluminum density contained in Eq. (5) will
be4 p = pc = 0.64 g/cm3. Overall, determining the
boiling points near CR demands caution due to the
gas ionization that can change the final results.
Analysis showed that the function T (p^,
represented by Eq. (5), contains ascending and
descending branches in the range 0<p <oo. It makes
g
physical sense when the boiling point increases
with pressure in the range 0<pg<pgmb where pgmb
can be calculated from the correspondent equation.
(7)
Since, the volume per mole of a condensed
phase, (V /N ), is much less than that of a vapor
phase, (V /N), then these smaller terms were
neglected in the denominators of Eq. (7) (similar to
the approach used in deriving Clausius-Clapeyron
equation).
An unknown function, p , has to satisfy two
partial differential equations (7). It is known that
solution to the problem exists if mixed derivatives
are equal to each other:
dpgdT'
(8)
Using for vapor EOS for ideal gas, it is not hard
to reduce condition (8) for mixed derivatives to
equality,
(9)
CONDENSED MATERIAL VAPORIZATION
WITH GENERAL EQUILIBRIUM
CONDITION
where F (T) is an arbitrary function of
temperature. As it is known, at small values of
pressure, a latent heat of the phase transition is
considered as constant, i.e. F (T) = 1. In the
general case the heat of transition is 1 = HI/NI H2/N2, where HI andH2 are enthalpies of vapor and
condensed phases, respectively. In the problem
under consideration this expression reduces to the
correlation matched with Eq. (9).
Now we will turn to an integration of the
equations (7), preliminary transforming them to the
following form
The equality 111=^2 obtained above is correct for
general case including nonisothermal and
nonisobaric processes2. In this section we consider
general case of the phase transition when both
parameters (T, pg) could be changed. Therefore,
Eq. (2) represents a function of two variables. To
discover this functionality, the equation (2) is used
in the following form:
(6)
= Pv+Pg>
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2
dT
RT
conditions, partial vapor pressure increases by
10-80 times compared with data obtained by
Clausius-Clapeyron equation.
(10)
pRT
CONCLUSIONS
A general solution of the Eq. (10)-2 can be
represented as
/> v =/(F)exp
\pRT
The primary results, Eq. (4), Eq. (5) and Eq.
(12), were obtained based on equilibrium
thermodynamics. In relation to this it should be
noted that L. Landau5 discussed the impact of
surrounding gas on vaporization of condensed
materials. He admitted that vaporization process in
this case might be also considered as equilibrium,
since vapor penetrates into surrounding space
slowly (by diffusion).
The increase of partial vapor pressure and
decrease of the boiling point with rise of the
external pressure appear to be unexpected results.
Qualitatively they may be explained if we take
into account that external pressure increases
enthalpy of condensed phase and does not change
enthalpy of vapor at given temperature.
(11)
where f(T) is an arbitrary function. Substituting
expression (11) in Eq. (10)-1, we find that
Constant A=p*, since with p=0 the vapor
g
pressure p = p^. As a result, the vapor pressure is
defined by the following equation,
(12)
ACKNOWLEDGMENTS
The author is grateful to J. Pogotto for his
support in origination this paper and to F. Zhang
for the valuable comments. Acknowledgment is
expressed to the Department of National Defence
for funding the work related to the paper.
The Eq. (12) is the simplest form of generalized
Clausius-Clapeyron equation, obtained with
standard assumptions that (V/N) « ( V / N )
and F (T) = 10 The above inequality reduces to
condition that ratio npv/(pRT) « 1. Analysis has
shown that for aluminum with T > 1000 °K and pg
< lOGpa the latter inequality is fulfilled with high
accuracy.
The solution of equations (7) - (8) was also
obtained without assumption of smallness of the
above ratio. Then ODE and implicit finite equation,
instead formula (12) determine vapor pressure.
The equations (4) and (12) are correct if the
density of material is slightly changed under high
pressure and EOS for ideal gas properly describes
the vapor behavior. The method may also be
extended for more complicated EOS, allowing for
fulfillment of equality (2). In this case equations
(7) - (8) are to be solved by numerical integration.
New correlations, (4) and (12), were used for
calculations of the Al particle vaporization with
10
external pressure up to -10 Pa. At these
REFERENCES
1. Kuo, K. K., Principles of Combustion, John Wiley &
Sons, New York, 1986, Ch. 1.
2. WiUiamsJF. A.,CombustionTheory, Benjamin
Cummings Publ. Comp., Inc. CA, 1985, pp. 522.
3. Glassman, L, Combustion of Metals. Physical
Considerations, Progress in Astronautics and Rocketry,
1 Academic Press, New York, 1960, pp. 253.
4. Fortov, E. V. and lakubov, I. T., The Physics of NonIdeal Plasma, World Scientific Publishing CO.,
Singapore, 2000, pp. 403
5. Landau, L. D., Akhiezer, A. L and Lifshitz, E. M.,
General Physics, Mechanics and Molecular Physics,
Pergamon Press, London , 1967, pp.206.
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