Internal pressure of liquids from the ca

Journal of Molecular Liquids 216 (2016) 862–873

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Journal of Molecular Liquids
journal homepage: www.elsevier.com/locate/molliq

Internal pressure of liquids from the calorimetric measurements near the
critical point
Ilmutdin M. Abdulagatov a,b,⁎, Nikolai G. Polikhronidi a, Rabiyat G. Batyrova a
a
b

Institute of Physics of the Dagestan Scientific Center of the Russian Academy of Sciences, Makhachkala, Dagestan, Russia
Thermophysical Properties Division, Geothermal Research Institute of the Russian Academy of Sciences, 39 A Shamil Ave., Makhachkala, Dagestan, 367030, Russia

a r t i c l e

i n f o

Article history:

Received 30 December 2015
Accepted 3 February 2016
Available online xxxx
Keywords:
Coexistence curve
Critical point
Equation of state
Internal pressure
Internal energy
Isochoric heat capacity
Thermal-pressure coefficient
Vapor–pressure

a b s t r a c t
A new relation between the internal pressure and isochoric heat capacity jump of liquids along the coexistence
curve near the critical point was found. Our previously reported one- and two-phase isochoric heat capacities
and specific volumes at saturation were used to calculate internal pressure of molecular liquids (water, carbon
dioxide, alcohols, n-alkanes, DEE, etc.).The internal pressure derived from the calorimetric measurements was
compared with the values calculated from the reference (NIST, REFPROP) and crossover equations of state.
Locus of the isothermal and isochoric internal pressure maxima and minima was studied using calorimetric

data and the reference and crossover equations of state near the critical point. The maximum of the internal
pressure of light and heavy water around the temperature of 460 K along the liquid saturation curve was
sat
, and isochoric heat capacity,
found. We also found very simple relation between the internal pressure, ΔPint
ΔCV, jumps near the critical point.
© 2016 Elsevier B.V. All rights reserved.

1. Introduction
In liquids, cohesion is extremely important because the intermolecular interaction is intense in liquid phase. The internal pressure provides
the estimate of magnitude of cohesive forces. Cohesion (or intermolecular
forces) creates a pressure within a liquid of value between 102 to 103 MPa.
For example, if the liquid compressed more, internal pressure decreases
and becomes large and negative. This means that the repulsive forces
become predominant. Internal pressure provides very useful information
for understanding the nature of the intermolecular interaction and
the structure of molecular liquids. For example, according to statistical
mechanical definition the internal pressure is

P int ¼ −

2π 2
ρ KT
3

Z∞

r3

∂ϕ ∂g
dr;
∂r ∂r

ð1Þ

0

where ϕ(r) = ϕA(r) + ϕR(r), is the potential energy between a pair
molecules separated by a distance r, where ϕA(r)is the smoothly varying
long-range attraction energy and ϕR(r) is the steep short-range repulsion;

g(r) is the radial distribution function. Eq. (1) allows splitting the
⁎ Corresponding author at: Thermophysical Properties Division, Geothermal Research
Institute of the Russian Academy of Sciences, 39 A Shamil Ave., Makhachkala, Dagestan
367030, Russia.
E-mail address: ilmutdina@gmail.com (I.M. Abdulagatov).

http://dx.doi.org/10.1016/j.molliq.2016.02.009
0167-7322/© 2016 Elsevier B.V. All rights reserved.

repulsive PA and attractive PR contributions of the total experimentally
observed internal pressure. Therefore, internal pressure is a very valuable
thermodynamic quantity (instrument) to directly study intermolecular
interactions through the macroscopic properties. Internal pressure is a
part of the total pressure, which is caused by the intermolecular interactions, i.e., describes the effect of intermolecular interactions on the
measuring total external pressure. The total measured external pressure
and internal pressures are related as
P int ¼

 
 

∂U
∂P
¼T
−P:
∂V T
∂T V

ð2Þ

Internal pressure is important also in development equation of state
of liquids for physically correct selection of the attractive and repulsive
parts of the external pressure. The internal pressure is useful in understanding the properties of the liquids and in the study of structural
changes (internal structure), and ordered structure in binary mixtures.
As follows from Eq. (2), the internal pressure describes the sensitivity
of internal energy U(T, V) to a change in volume (isothermal expansion
or compression) at isothermal condition, i.e., isothermal influence of
volume on the intermolecular interaction energy.
2. Brief review of the internal pressure measurement methods
All of the available methods of internal pressure measurements can
be divided into two groups: the calculation of the internal pressure from

I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

the direct measurement of thermal-pressure coefficient, (∂P/∂T)V, using
Eq. (2) and an indirect determination of the internal pressure through
thermal expansion, αp, and isothermal compressibility,KT, using Eq. (3).
P int ¼

 
∂U
αP
−P
¼T
KT
∂V T

ð3Þ

measurements of the thermal pressure coefficient (∂ P/∂ T) V of
neopentane near the coexistence curve were reported by Few and

Rigby [11]. These data together with reported PVT data were used
to determine the P int along the coexistence curve from the triple
point to the critical point.
2.2. Acoustic method

Most reported internal pressure data were derived using indirect
method, based on the αpandKTdata derived from PVT or speed of sound
measurements. Most cases the thermal-pressure coefficient (∂P/∂T)V
is calculated from the equation of state, for example, Tait-type, cubic,
or multiparametric equations of state. The direct measurements of the
derivative (∂P/∂T)V are very rare.
2.1. Direct methods
Dack [1,2] and Barton [3] performed direct measurements of the
derivative (∂P/∂T)V from experimentally determined heat of vaporization
from the equation


P int ¼ ΔH vap −RT ðρ=M Þ;

863

ð4Þ

where M is the molecular weight, ρis the density, and ΔHvap is the
enthalpy of vaporization. This is a most frequently used method.
Dack [1] used constant volume apparatus to determine the internal
$ pressure of several liquids at 298 K. The uncertainty in Pintdetermination
in this method is about 2%. Grant-Taylor and Macdonald [4] determined
thermal-pressure coefficient of acetonitrile + water mixture at temperatures between 298 and 328 K using 25 ml glass constant volume cell. The
measured P–T isochores were fitted with the linear equation of P=(∂P/
∂T)VT + C. The derived values of (∂P/∂T)V were corrected for the finite
expansion and compression of glass. The measured values of (∂ P/
∂ T)V were used to calculate the energy-volume (∂ U/∂ V)T (internal
pressure) coefficient and other thermodynamic quantities. The
uncertainty of the derived values of (∂ P/∂ T)V is 2%. Macdonald and
Hyne [5] reported thermal-pressure and energy-volume coefficients
measurements for dimethyl sulfoxide + water mixtures at temperatures between 286 and 328 K and at atmospheric pressure using the
same technique. Westwater et al. [6] and Smith and Hildebrand [7]
directly measured (∂ P/∂ T)Vusing the constant-volume thermometer
apparatus. The liquid was confined to a glass bulb with a capillary
neck in which its level may be maintained at a fixed point by an

electrical contact to a mercury interface. The coefficient (∂ P/∂ T) V
was found directly as the slope of a plot of pressure against temperature.
McLure and Arruaga-Colina [8] reported thermal-pressure coefficient
measurements for ethanenitrile, propanenitrile, and butanenitrile from
297 to 398 K. Measurements were made with an apparatus consisting
of a constant-volume thermometer in which the pressure is controlled
and measured for a series of temperatures at a series of different
constant densities. McLure et al. [9] also measured the thermalpressure coefficient for five dimethylsilioxane oligomers in the
temperature range from 298 to 413 K. Thermal-pressure coefficients
were measured in Pyrex cells (dilatometers). The sample was confined
by mercury, and changes in its volume were monitored by weighing the
amount of mercury expelled from or drawn into the dilatometer.
Measured isochore slopes, (∂P/∂T)V, were corrected using the effect of
the thermal expansion and compression of both quartz and mercury.
The uncertainty in thermal-pressure coefficient measurements is
about 1.0%. Bianchi et al. [10] determined the internal pressure for
carbon tetrachloride, benzene, and cyclohexane by direct measurements of (∂ P/∂ T)V. The measured values of (∂ P/∂ T)V were corrected
for thermal expansion and compressibility of the Pyrex cell. They
studied temperature dependence of Pint from 293 to 333 K. Direct

Some authors [12,13] derived the values of internal-pressure coefficient from the direct measurements of the isothermal compressibility KT
and isobaric coefficient of thermal expansion αP. Then Eq. (3) can be
used to calculate the internal pressure. Usually, in most cases, the values
of KTand αP are calculating from density measurements. Zorębski [14,
15] studied different approaches to calculate the internal pressure.
Zorębski and Gepper-Rybczyńska [16] reported density, kinematic
viscosity, and speed of sound of binary (1-butanol + 1,4-butanediol)
mixture over the temperature range from 298.15 to 318.15 K. The
measured values together with literature isobaric heat capacity data
were used to calculated internal pressure using Eq. (3). Kumar et al.
[17] used measured ultrasonic velocity and density data to study internal pressure of binary mixtures (acetone-CCl4 and acetone-benzene).
The measured data were used to study of the molecular interactions in
binary liquid mixtures. Vadamalar et al. [18] also used acoustic and
viscometric parameters to accurately calculate the internal pressure
for binary mixtures of tert-butanol and isobutanol with methyl methacrylate. Sachdeva and Nanda [19] used measured ultrasonoic wave velocity and density measurements of normal paraffins to calculate internal
pressure. The acoustic method was used by Dzida [20] to calculate the
internal pressure of cyclopentanol at pressures up to 100 MPa and at
temperatures from 293 to 318 K. Allen et al. [25] also determined the
internal pressure for some compounds using the speed of sound data.
2.3. Volumetric method

Verdier and Anderson [21] used indirect method to estimate
the values of internal pressure of mixtures, using thermal expansivity
(determined by microcalorimeter) and isothermal compressibility
(determined by density measurements). Korolev [22] studied internal
pressure of alcohols using the values of volumetric coefficient (thermal
expansion and isothermal compressibility coefficients). Shukla et al.
[23] studied the internal-pressure coefficient and its correlation with
solubility and pseudo-Gruneisen parameters for binary and multicomponent liquid mixtures over a wide range of concentration at 298 K
using the measured values of viscosity, density, and ultrasonic velocity.
Singh and Kumar [24] measured density, speed of sound, and refraction
index of IL [C8mim][Cl], [C4mim][C1PSO3], and [C4mim][C8OSO3] over
the temperature range from 283 to 343 K. The measured density and
speed of sound data were used to calculate the internal pressure from
relation (3) where thermal expansion coefficient and isothermal
compressibility were calculated using the measured values of density
and speed of sound. Piekarski et al. [26] reported density, heat capacity,
and speed of sound data for binary acetonitrile + 2-methoxyethanol
mixture at 298.15 K in the whole composition range. The measured
data were used to calculate the internal pressure of the mixture using
relation (3). Almost linear dependence of the internal pressure as
a function of concentration was observed. Kannappan et al. [27]
measured the speed of sound, density, and viscosity of ternary
mixture of alcohols with DMF and cyclohexane at three temperatures
303, 308, and 313 K. The measured data were used to calculate excess
internal pressure.
2.4. Determination of the internal pressure from calorimetric measurements
In our several previous publications [28–31], we have proposed a
new technique of internal pressure determination in calorimetric
experiment by simultaneously measurements of the thermal-pressure

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Fig. 1. Measured thermal-pressure coefficient (left) and internal pressure (right) of DEE as a function of temperature near the critical point including saturation curve [63].

coefficient, (∂ P/∂ T)V, external pressure, P, and heat capacity, CV. This
method provides accurate measurements of the total external pressures,
P, and the temperature derivatives, (∂ P/∂ T)V, therefore and internal
pressure using Eq. (2). Thus, in the same apparatus, we can simultaneously measure the temperature derivatives of the internal energy
(energy-temperature coefficient), (∂U/∂T)V = CV, isochoric heat capacity and the volume derivatives of the internal energy (energy-volume
coefficient), (∂U/∂V)T =Pint, i.e., the internal pressure. The experimental
details have been described previously in publications [28–31]. Only essential information will be given briefly here. The direct measurements
of the thermal-pressure coefficient, (∂P/∂T)V, were performed using the
high-temperature and high-pressure nearly constant-volume adiabatic
piezo-calorimeter. The pressure (P) and temperature derivatives of the
pressure at constant volume,(∂P/∂T)V, were measured with a calibrated
pressure transducer. The synchronously recording both temperature
changes (thermograms, T − τ, reading of the resistance platinum

thermometer PRT) and pressure changes (barograms, P − τ, readings
of the tenzotransducer) was performed with a strip-chart recorders.
Using the records of the thermo-barograms (T − τ and P − τ) the
changes in temperature ΔTand in pressuresΔP, therefore, the derivative
(∂ P/∂ T)V = lim ðΔP=ΔTÞV , at any fixed time can be calculated. This

Fig. 2. Measured and calculated isochoric heat capacities of argon as a function of density
in the supercritical region. Symbols are reported by Anisimov et al. [54,55]; solid lines are
calculated from Tegeler et al. [56].

Fig. 3. Measured and calculated isochoric heat capacities of carbon dioxide as a function of
density in the supercritical region. Symbols are reported by Abdulagatov et al. [57,58] and
Amirkhanov et al. [59–61]; solid lines are calculated from crossover model [62].

ΔT→0

technique was successfully applied for various molecular fluids
and fluid mixtures such as DEE, water + ammonia, and carbon
dioxide + n-decane [31,63,64]. The measured thermal-pressure
coefficient and internal pressure data for DEE, as an example, are
depicted in Fig. 1.
Therefore, simultaneously measured values of CV = (∂ U/∂ T)V and
thermal-pressure coefficient, (∂P/∂T)V, together with PVT measurement
in the same experiment can be used [52] to determine the values of internal pressure using relation (2). The uncertainties of temperature
ΔTand in pressureΔPchange measurements are much more accurate

I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

Fig. 4. Measured and calculated isochoric heat capacities of light water as a function of
density in the critical and supercritical regions. Symbols are reported by Abdulagatov
et al. [65,66]; dashed lines are calculated from the crossover model [41].

than the measurements of their absolute values (T and P). Therefore, the
uncertainty in (∂P/∂T)Vmeasurements is within 0.12% to 1.5% depending on the temperature increment (ΔTchanges within 0.02 to 0.10 K).
The expanded uncertainty of the isochoric heat-capacity measurements
at the 95% confidence level with a coverage factor of k = 2 is estimated
to be (2 to 3) % in the near-critical and supercritical regions, (1.0 to 1.5)
% in the liquid phase, and (3 to 4) % in the vapor phase. Uncertainties of
the density and temperature measurements are estimated to be 0.05%
and 15 mK, respectively.

865

Fig. 6. Second temperature derivative of external pressure (or first temperature derivative
of the internal pressure, (∂Pint/∂T)V) as a function of density along the near- and
supercritical isotherms of pure water calculated from crossover equation of state [41].
Symbols are derived from direct isochoric heat capacity measurements [65,66,72].

of Pint(T, V)=(∂U/∂V)T and CV(T, V)=(∂U/∂T)V, (see above Section 2.4)
provides the caloric equation of state U(T,V),
dU ¼ P int ðT; V ÞdV þ C V ðT; V ÞdT;

ð5Þ

or the internal pressure can be expressed as

3. Internal pressure and one- and two-phase isochoric heat capacities
P int ðT; V Þ ¼ ðdU=dV Þ−C V ðT; V ÞðdT=dV Þ;
3.1. One-phase isochoric heat capacity and internal pressure
The internal pressure, (∂U/∂V)T =Pint, and isochoric heat capacity,
(∂U/∂T)V = CV, i.e., volume and temperature derivatives of the internal
energy U(T, V), describes the sensitivity of internal energy to a change
in specific volume V and temperature T at the isothermal and isochoric
processes, respectively. Therefore, direct simultaneous measurements

which is significantly accurate than thermal P(T,V) equation of state.
Thus, the total differential of internal energy defines through Pint and CV
(see Eq. 5), which are can be measured in the same experiment as described above in Section 2.4. The caloric equation of state can be derived
directly by integrating relation (5).

Fig. 5. Measured and calculated isochoric heat capacities of heavy water as a function of
density in the critical and supercritical regions. Symbols are reported by Polikhronidi
et al. [49]; solid lines are calculated from crossover model [42]; dashed line is the
isothermal CV maximum loci.

as a function of density along the selected supercritical isotherms.

Fig. 7. Calculated from crossover equation of state [62] values of ð∂P∂Tint ÞV for carbon dioxide

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I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

capacity exhibits isothermal maximum in the near- and supercritical
regions (along the near- and supercritical isotherms, see Figs. 2–5)
and isothermal minimum [33,36] at high densities (≈ 2ρC), where
2

∂P int
∂ P
(∂ CV/∂ V)T = 0. The density dependence of ð∂T
2 Þ , therefore ð ∂T ÞV ,
V

along the near-critical isotherms for pure H 2O and CO2 calculated
from the crossover model together with the values derived from

Fig. 8. Temperature dependence of the derivative ð∂P∂Tint ÞV for pure water along the nearcritical and supercritical isobars calculated from crossover model [41].

The temperature derivate of the internal pressure (internal thermalpressure coefficient) can be calculated directly from Eq. (2) as
!


2
∂P int
∂ P
¼T
;
∂T V
∂T 2 V

ð6Þ

or




∂P int
∂C V
¼
:
∂T V
∂V T

ð7Þ

As one can be noted, the measured internal pressure as a function of
temperature is providing information of second temperature derivative
2

∂ P
(curvature) of the real external pressure properties of the liquids, ð∂T
2Þ

V

or isothermal specific volume behavior of the isochoric heat capacity,
V
ð∂C
Þ . As well-known (see, for example [32–39]), the isochoric heat
∂V T

C Vmeasurements is depicted in Figs. 6 and 7. The values of ð∂P∂Tint ÞV
calculated from the crossover model for light water along the various
near-critical isobars are depicted in Fig. 8. Therefore, the locus of isothermal extreme of C V (or inflection point of the P–T isochores,
where (∂2P/∂ T2)V = 0) is determined by the isochoric temperature
extreme of the internal pressure, i.e., (∂ P int /∂ T)V = 0. The locus of
isochoric temperature maximum and minimum of the internal pressure
or isothermal extreme of CVfor some selected molecular liquids in ρ − T
and P − Tprojection is shown in Figs. 9 and 10. The isothermal maxima
and minima loci of CVand isochoric maxima and minimums of the internal pressure Pintare associated with the inflection points in P–T
isochores where (∂2P/∂ T2)V = 0 (see Fig. 9). The curvature of P–T
isochores is positive within the isochore inflection curve and negative
outside (see Figs. 6 and 7). As Figs. 5 and 8 show, the isothermal CV maxima or isochoric temperature maxima of Pintcurves in the supercritical
region are not exactly at the critical isochore. The isothermal CV maxima
curve in the supercritical region starts at the critical point and runs
along a density minimum (ρmin ≈0.95ρC) to higher temperatures (see
Fig. 5). This curve intersects twice with the critical isochore, namely at
T = TC and approximately T= 1.1TC. Abdulagatov et al. [32] reported a
detail discussion of the loci of isochoric heat capacity maxima and
minima behavior for molecular liquids along the supercritical isotherms.
The locus of isochoric heat capacity or internal pressure extreme as
it also is the locus along which P–T isochores have an inflection, (∂2P/
∂T2)V =0 is one of the important characteristic curves of the PVT surface.
The analysis of the extremum properties of isochoric heat capacity and
the internal pressure is one way to study the qualitative behavior of the
thermodynamic surface of fluids, especially near the critical point where
the PVT surface shows anomalous behavior. Comparison experimental
CV and Pint extreme with calculations is a stringent test of the accuracy
and reliability of the equation of state. Loci of CV extreme are very sensitive to the structure of an equation of state and the molecular structure

Fig. 9. Isothermal CVmaximum and minimum loci (or isochoric temperature maximum and minimum of the internal pressure) of water and carbon dioxide calculated from crossover
equation of state [41,62].Dashed lines are isothermal maximum (CA) and minimum (AB) of CVand isochoric maximum and minimum of the internal pressure, where (∂Pint/∂T)V =
(∂CV/∂V)T =0.

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I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

Fig. 10. Isothermal maxima and minima loci of CV for n-pentane calculated from crossover model [67] in T−ρ and P−Tplanes. CB-isothermal CVmaxima loci; BD-isothermal CVminima loci.

of fluids; therefore, they can be used to test the quality of the equation of
state and molecular theory of thermodynamic property. Figs. 11 and 12
are demonstrate temperature behavior of the internal pressure of CO2,
H2O, C3H8, and n-C5H12 calculated from crossover equation of state [41,
62,67] along the various liquid and vapor isochores near the coexistence
curve. As one can see from Figs. 11 and 12 the crossover model
correctly represented scaling behavior of the CVnear the critical and the
supercritical regions and temperature behavior of the internal pressure.
(See Fig. 9.)
3.2. Two-phase isochoric heat capacity and internal pressure
Two-phase isochoric heat capacity also directly related with the
internal pressure along the saturation curve. The Yang and Yang [47]
relation for two-phase isochoric heat capacity is

where second temperature derivative of chemical potential,
vapor pressure,

2

d PS
,
dT 2

d2 μ
,
dT 2

and

are function of temperature only. By integrating

relation (8), using the Eq. (9)


∂U 2
C V2 ¼
∂T V

ð9Þ

we can derive the Yang and Yang caloric equation of state for twophase system as
ZT
ZT
2
2
d μ
d PS
ΔU 2 ¼ − T 2 dT þ V T
dT;
dT 2
dT
T0

ð10Þ

T0

where ΔU2 = U(T) − U(T0) and the second temperature derivatives of
2

C V2 ¼ −T

d μ
dT

2

2

2

þ VT

d PS
dT

2

;

ð8Þ

2

chemical potential, ddTμ2 , and vapor–pressure,ddTP2S, can be directly calculated
from two-phase isochoric heat capacity measurements as a function

Fig. 11. Internal pressure of carbon dioxide and light water as a function of temperature along the various liquid and vapor isochores near the phase transition curve calculated from the
crossover equation of state [41,62]. Dashed-dotted line is the isochoric temperature maxima loci of the internal pressure, (∂Pint/∂T)V =0.

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I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

Fig. 12. Detailed view of the internal pressure of propane and n-pentane as a function of temperature along the critical, liquid, and vapor isochores in the immediate vicinity of the coexistence curve together with the values of Pint along the vapor–pressure curve (dashed-dotted line). Dashed and solid lines are the vapor and liquid internal pressures along the coexistence
curve, respectively, calculated from crossover model [62,67].

of temperature or using measured values of saturated properties
(TS,V',C'V2,V ' ',C ' 'V2) [43,48,49] as
2

d PS
dT 2

2

C ″ −C 0V2
d μ V ″ C 0V2 −V 0 C ″V2
 and 2 ¼

 :
¼ V2
″
0
dT
T V −V
T V 0 −V ″

ð11Þ

Internal pressure from the Yang and Yang two-phase caloric equation state (Eq. (10)) is

P sat2
int ¼



ZT
2
∂ΔU 2
d PS
dP S
−P S
¼ T
dT ¼ T
dT
∂V T
dT 2
T0

ð12Þ

Thus, from caloric equation of state (Eq. (10)), based on Yang–Yang
two-phase isochoric heat capacity Eq. (8), we derived the equation for
internal pressure along the vapor–pressure curve, Eq. (12), through
S
. Relation (12) is the definition
the slope of the vapor–pressure curve dP
dT
of the internal pressure through the vapor–pressure equation by analogy to equation of state Eq. (2) for one-phase region.
Therefore, as follows from Eq. (12), the internal pressure at saturation can be directly calculated from two-phase isochoric heat capacity
measurements at saturation as

P sat2
int ¼

ZT

C 00 V2 −C 0 V2
dT:
V 00 −V 0

ð13Þ

T0

Fig. 13. Temperature derivatives of the internal pressure along the vapor–pressure saturation curve directly derived from calorimetric measurements [49,65,66,68,69,72] (Eq. 15) together
with the values calculated from vapor pressure equation (REFPROP [46], IAPWS standard) for light and heavy water.

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I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

Fig. 14. Temperature derivatives of the internal pressure along the vapor–pressure
saturation curve directly derived from calorimetric measurements [58–61] (Eq. 15)
together with the values calculated from vapor pressure equation (REFPROP [46]) for
carbon dioxide.

As follows from Eq. (12), the first temperature derivative of the internal pressure at saturation diverge at the critical point as
2

dP sat2
d P S −α
int
¼T
∝t
dT
dT 2

ð14Þ

∂P
∂T

¼

V

dP S 1 dT
þ
ΔC V ;
dT T dV

ð16Þ

or
T

Eqs. (14) and (15) are the analogy of the Eqs. (6) and (7) for ð∂P∂Tint ÞV in the
one-phase region. Figs. 13 to 15 demonstrate the comparison between
dP sint
derived
dT

sat

ð15Þ

The slope of the saturated internal pressure equation Psint-T directly
related with the two-phase isochoric heat capacity difference (C″V2 −CV2′)
between the vapor (C″V2) and the liquid (CV2′) heat capacities at saturation.

the values of

measurements and calculated from the vapor–pressure data for
some selected very well studied liquids and gasses.
As follows from well-known thermodynamic relation (Abdulagatov
et al. [50–52], Polikhronidi et al. [53]) the one-phase partial temperature
derivative (thermal-pressure coefficient) of pressure at saturation is


or
dP sat2
C ″ −C 0V2
int
:
¼  V2
dT
V ″ −V 0

Fig. 16. Measured ΔPsat
int verses ΔCVplot for DEE. Solid line is calculated from experimental
coexistence curve (V–T) data.

∂P
∂T



sat

¼T

V

dP S dT
þ
ΔC V ;
dT dV

ð17Þ

sat

∂P
ÞV one-phase partial temperature derivative of external
where ð∂T
pressure at saturation (or initial slope of the P–T curves (isochores)

at saturation curve;

dP S
dT

is the slope of the two-phase saturation

from the direct isochoric heat capacity

Fig. 15. Temperature derivatives of the internal pressure along the vapor–pressure
saturation curve directly derived from calorimetric measurements [50,51,70,71] (Eq. 15)
together with the values calculated from vapor pressure equation (REFPROP [46]) for nalkanes.

Fig. 17. Internal pressure of light water at the liquid and vapor one-phase saturation curve
derived from isochoric heat capacity measurements together with the values calculated
(solid and dashed lines) from reference equation of state (REFPROP [46], IAPWS
formulation). Dashed line is the internal pressure along the two-phase vapor–pressure
curve. CP-the critical point. Symbols are derived from calorimetric measurements
[65,66,69,72] (Eq. 15).

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I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

Fig. 18. a–c. Internal pressure of propane (a), n-pentane (b), and n-hexane (c) at the liquid and vapor one-phase saturation curve derived from isochoric heat capacity measurements together with the values calculated (solid and dashed lines) from reference equation of state (REFPROP [46]). Dashed-dotted lines are the internal pressure along the two-phase vapor–pressure curve. CP-the critical point. The symbols are derived from calorimetric measurements [50,51,70] (Eq. 15).

Fig. 19. Internal pressure of ethanol (left) and DEE (right) at the liquid and vapor one-phase saturation curve derived from isochoric heat capacity measurements together with the values
calculated (solid and dashed lines) from reference equation of state (REFPROP [46]). Dashed lines are the internal pressure along the two-phase vapor–pressure curve. CP-the critical point.
Symbols are derived from calorimetric measurements [53,74] (Eq. 15).

I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

871

Fig. 20. a,b. Internal pressure jumps of heavy water (a), methanol (b, left) and CO2 (b, right) at the liquid (solid lines) and vapor (dashed liens) one-phase saturation curve derived from
isochoric heat capacity measurements together with the values calculated (solid and dashed lines) from reference equation of state (REFPROP [46]). CP-the critical point. Symbols are
derived from calorimetric measurements [49,58–61,68,73] (Eq. 20).

dT
curve (vapor–pressure curve); dV
is the temperature derivative of the
specific volume at saturation curve; and ΔCV = CV2 − CV1 isochoric
heat capacity jump at the saturation. As follows from Eq. (17), the
internal pressure in the one-phase region at the saturation is

P sat1
int

dT
¼ P sat2
ΔC V
int þ
dV

ZT

C 00 V2 −C 0V2
dT
dT þ
ΔC V ; i:e: ; ; ;C 0 V2” to ; ;C 0V2 ”:
V 00 −V 0
dV

ΔP sat
int ¼

dT
ΔC V ;
dV

ð20Þ

ð18Þ

or

P sat1
int ¼

sat1 sat2
jump, ΔPsat
int =Pint -Pint , when the phase transition curve is crossing can
be calculated as

ð19Þ

T0

As one can see from Eq. (18), the internal pressure at saturation in
the one-phase (liquid and vapor) region Psat1
int defined from two-phase
internal pressurePsint (Eq. (13)) and isochoric heat capacity jump ΔCV
or only liquid (C'V2) and vapor (C"V2) two-phase heat capacities and
ΔCV (Eq. (19)). As was mentioned above (Section 2.4) the saturated
liquid (V') and vapor (V' ') specific volumes can be accurately measured
in the same calorimetric experiment. Other words, the internal pressure

i.e., completely defined by two-phase through the isochoric heat
capacity jump at saturationΔC V , and the slope of the coexistence
dT
curve, dV
. Thus, the internal pressure jump at the saturation ΔPsat
int is
the linear function of isochoric heat capacity jump at saturation
ΔCVwhere the slope is defined as a coexistence curve temperature
dT
derivative, dV
. Fig. 16 shows the plot of measured ΔP sat
int verses of
ΔCV for DEE. This figure also contains the values of ΔPsat
int as a function
dT
of ΔCVderived using the calculated values of dV
from equation of state
(REFPROP [46]). As one can note, the agreement is good, which is confirm the thermodynamic consistency of our previous calorimetric
(ΔCV) and thermal,(dT/dV), (∂P/∂T)Vsat, and (dPS/dT) measurements at
saturation. The values of one liquid and vapor one-phase saturated insat
ternal pressure,Psat1
int , and difference (or jump) ΔPint , calculated from
our previous measured isochoric heat capacity data for some selected

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I.M. Abdulagatov et al. / Journal of Molecular Liquids 216 (2016) 862–873

molecular liquids and gasses are presented in Figs. 17 to 20. The values
of Psat2
int can be calculated directly from two-phase isochoric heat capacities (Eq. (13)). As follows from Eq. (20), near the critical point internal
1−α−β
. The derived from the caloripressure jump ΔPsat
int behave as ∝t
metric measurements values of liquid and vapor saturated ΔPsat
int as a
function of temperature near the critical point for heavy water, methanol, and carbon dioxide as an example shown in Fig. 20a,b. For vapor
}
"sat
dT
phase the values of ΔPsat
int was calculated as ΔPint =dV } ΔC V .

In addition, another useful relation for the vapor–pressure equation
can be presented through the two-phase internal pressure as
P S ¼ ðC S −C V2 Þ

dT
−P sat2
int ;
dV

ð21Þ

where CS is the saturated heat capacity.
4. Conclusions
A new method of calculation of the internal pressure of molecular
fluids based on the calorimetric (one- and two-isochoric heat capacity)
measurements near the critical and supercritical regions have been
proposed. Previously reported one- and two-phase isochoric heat
capacities and specific volumes at saturation were used to calculate
internal pressure of molecular liquids (water, carbon dioxide, alcohols,
n-alkanes, DEE, etc.). The results were compared with the values calculated from the reference and crossover equations of states. We demonstrated good agreement between the calorimetric method of calculation
of the internal pressure and direct method of calculation from the thermal equation of state. We also showed that the isothermal CV extreme is
the isochoric temperature extreme of the internal pressure. Using our
previously published CVdata we studied locus of the CVmaxima and
minima (or isochoric internal pressure extreme) for various molecular
fluids. For light and heavy water we found temperature maximum of
the internal pressure at the saturation curve around 460 K. Based on
our previous isochoric heat capacity measurements we have calculated
internal pressure of molecular fluids at saturation curve near the critical
point. These data were compared with the values directly calculated
from reference and crossover equations of state. The temperature derivative of the internal pressuredPsat
int /dT at saturation was directly calculated from the measured two-phase isochoric heat capacity measurements
at saturation. It was illustrate that (dPsat
int /dT) and (∂Pint/∂ T)Vhave the
same singularity at the critical point as the second temperature derivatives of (d2PS/dT2) and (∂2P/∂ T2)V.We found a very simple relation
between the internal pressure, ΔPsat
int, and isochoric heat capacity, ΔCV,
dT
jumps, ΔPsat
int =dV ΔC V .
Acknowledgments
This work was supported by a Russian Foundation of Basic Research
(RFBR) grants NHK13-08-00114/15 and 15-08-01030. The authors
thank Dr. Magee for his interest to the work and useful discussion.
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