Research Journal of Recent Sciences ________________________________________________ ISSN 2277-2502
Vol. 1 (ISC-2011), 365-367 (2012)
Res.J.Recent Sci.

Short Communication

Hartmann’s equation of state for Materials at Extreme Compression
Vijay A.
Department of Applied Sciences, GLNA Institute of Technology, Mathura (U.P.)-281406, INDIA

Available online at, www.isca.in
(Received 27th September 2011, revised 16th January 2012, accepted 28th January 2012)

Abstract
Hartmann’s equations of state formulated for liquids, polymers and nanomaterials have been demonstrated in the present study to
satisfy the thermodynamic constraints at extreme compressions. This reveals the applicability of Hartmann’s equation for materials at
very high pressures. We have also derived expressions for the pressure derivatives of bulk modulus up to third order. The expressions
thus derived have been verified with the help of identities which are valid at extreme compressions. An application of the Hartmann
equation has been presented here to predict the pressure-volume-temperature relationships for NaCl crystal and CaSiO3 perovskite
mineral. The results obtained in the present study are found to compare well with the experimental data.
Keywords

Introduction
An equation of state (EOS) formulation is an important tool
for investigating the pressure P- volume V- temperature T
relationship1. Volumes of a material at high pressures and
high temperatures are needed for understanding its
thermoelastic behaviour2, 3. The thermoelastic properties can
be described in terms of pressure derivatives of the bulk
modulus4-6. An equation of state must satisfy the boundary
conditions at zero – pressure as well as in the limit of infinite
pressure. The Grüneisen parameter provides a useful link
between thermal and elastic properties of materials2, 3, 6, 7.
There exists an equation of state due to Hartmann8 which is
based on the fundamental thermodynamic principles9, 10. This
EOS is capable of predicting the changes in pressure as well
as in temperature. An application of the Hartmann equation
has been presented here to predict the pressure-volumetemperature relationships for NaCl crystal and CaSiO 3
perovskite mineral. The results obtained in the present study
are found to compare well with the experimental data.

The Hartmann EOS representing the relationship between P,
V and T as follows8
n

An expression for the bulk modulus

K  V dP dV T is

obtained by differentiating Eq. (1) with respect to pressure at
constant temperature. Thus we find
n

V 
K  K 0    nP
( 2)
 V0 
At P=0, V  V0 , we have K  K 0 , the zero-pressure
value of bulk modulus. With the increase in pressure P, the
volume ratio V/V0 decreases, and both the terms on right side
of Eq. (2) increase rapidly, and become infinitely large. Thus
the bulk modulus K tends to infinity in the limit of infinite
pressure. At infinite pressure, Eq. (2) with the help of Eq. (1)
gives
(3)
1
 P 

 
n
 K 

Material and Methods

P V  T 
   
K 0  V0   T0 

of P and T. At very high pressures in the limit of extreme
compression, we have the volume V tends to be fulfilled by
any EOS in order to be physically acceptable. In case of the
Hartmann EOS, these conditions are satisfied.

32

V
 ln
V0

(1)

K 0 is the bulk modulus at P=0. The exponent n is a
material-dependent constant. T0 is the temperature
characteristic of the material. Equation (1) gives V  V0 at
where

P=0 and T=0. This is the boundary condition at initial values

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Equation (2) gives the following expression for the pressure
derivative of bulk modulus
dK
n
(4)
K  nP 
K' 
 n
dP

K

At P=0, Eq. (4) yields
n

K 0'
2

(5)

At infinite pressure, we use Eq. (3) in Eq. (4) to obtain

n  K '

(6)

Equation (5) and (6) then yield
K ' 

K 0'
2

(7)

365

Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502
Vol. 1 (ISC-2011), 365-367 (2012)
Res.J.Recent Sci.
Thus values of n and K ' are different for different materials
since K 0' is a material-dependent parameter. This is
consistent with the earlier findings due to Stacey
derived the following identity

11, 12

1
 P

 
'
K
 K 

who

(8)

Equation (8) is satisfied by equations (3) and (6).
Expressions for higher pressure derivatives of bulk modulus
are obtained from eq (4) by differentiating it with respect to
pressure
(9)
KK ' '  n 2 1  K ' P / K 
where K ' '  d 2 K dP 2 . We multiply K ' ' by K so that
KK ' ' is dimensionless. Eq (9) reveals that KK ' ' is
negative at zero-pressure and at finite pressures, and it
becomes zero at extreme compression (V→0, and P→∞)
because Eq (8) becomes valid. eq (9) on differentiating with
respect to pressure yields
P
(10)


K 2 K ' ' '  KK ' ' n 2
 2K '



K




Where K ' ' '  d 3 K dP 3 . We multiply K ' ' ' by K 2 so
that K 2 K ' ' ' is dimensionless. At extreme compression
K 2 K ' ' ' tends to zero, since KK ' ' also tends to zero. But
their ratio remains finite at extreme compression, as eq (10)
gives
 K 2K ''' 
'

 KK ' ' 
  K



At zero pressure KK ' ' and
K '2
K 0 K 0''   0
4
K 0'3
2
'''
K0 K0 
2

 K 2 K ''' 
1  KK ' ' 

  2 K '  ' 

K   1  K ' P K  
 KK ' '  

(17)

Equation (9) based on the Hartmann EOS gives

 KK ' ' 

   K '2
1

K
'
P
K



(18)

Equations (11) and (18) satisfy the identity given by Eq.
(17). We have thus found that the Hartmann EOS8-10, 14, 15 is
consistent with the infinite pressure behaviour of materials1113
.

Results and Discussion
The results for isothermal compressions derived from the
Hartmann EOS are given in table 1 for CaSiO3 and in Figure
1 for NaCl. In both the cases the experimental data 16,17 have
been included in the Table as well as Figure for the sake of
comparison. It is found that the results obtained in the
present study using the Hartmann equation of state (EOS) are
in good agreement for NaCl crystal as well as CaSiO3
perovskite mineral. The pressure-volume-temperature
relationships provide useful informations regarding various
thermodynamic processes.

(11)

At constant temperature, i.e. isothermal conditions by
studying pressure-volume relationships we can determine the
isothermal bulk modulus K  V dP dV T of the materials.

(12)

At constant pressure, i.e. isobaric conditions the volumetemperature relationships yield the relationship for the
thermal expansivity    1 V dV dT P .At constant

K 2 K ' ' ' both are finite

(13)







volume, i.e. isochoric thermal conditions pressuretemperature relationships are useful for predicting the

The Grüneisen parameter γ provides useful link between
thermal and elastic properties of materials. We have the
following relationship
1
1
(14)
  K '

thermal pressures. In this case we make use of identity

so that, at infinite pressure we have

We have found that the Hartmann equation of state which
has been widely applicable for liquids, polymers and
nanomaterials, is consistent with the infinite pressure
behavior of solids. The Grüneisen parameter γ plays central
role in explaining thermal and elastic properties of materials.
The identity between the pressure derivatives of bulk
modulus at infinite pressure used here satisfy by the
Hartmann equation of state (EOS). The pressure-volumetemperature relationships, as discussed at length by Stacey
and Davis, in the present study are found to compare well
with the experimental data.

2



6

1
1

K ' 
2
6

(15)

According to the Hartmann EOS K '  K 0' / 2 , we have

 

1 ' 1
K0 
4
6

(16)

Since K 0' is always greater than one,   is always greater
than zero, i.e. the Grüneisen parameter γ remains positive
and finite at extreme compression. There exists an identity
between the pressure derivatives of bulk modulus at infinite
pressure given as follows13
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dP

dT V  KT .

Conclusion

366

Research Journal of Recent Sciences _____________________________________________________________ ISSN 2277-2502
Vol. 1 (ISC-2011), 365-367 (2012)
Res.J.Recent Sci.

Table 1
Results for CaSiO3, volumes (Å3) calculated from the Hartmann EOS (eq. 1), and experimental values from Wang et al. 16
T = 4.8,
K0 = 232 GPa
KT = 7.2 x 10-3 GPa K-1
V(Å3)
T(K)
P(GPa)
Calculated
Experimental
301
2.66
45.08
45.03
303
4.15
44.81
44.87
303
6.54
44.40
44.38
303
7.94
44.17
44.14
302
8.97
44.00
44.01
304
9.63
43.90
43.93
306
10.07
43.83
43.83
570
4.68
45.06
45.12
572
7.09
44.63
44.63
570
8.48
44.39
44.37
575
9.50
44.23
44.22
572
10.15
44.12
44.12
572
10.58
44.05
44.07
771
5.04
45.26
45.25
772
7.52
44.81
44.78
774
8.97
44.56
44.51
770
9.98
44.38
44.38
769
10.58
44.28
44.28
769
10.92
44.23
44.23
980
5.51
45.46
45.44
976
9.37
44.74
44.71
980
10.49
44.55
44.53
977
11.06
44.45
44.45
970
11.35
44.40
44.41
1172
11.69
44.59
44.54
1368
12.04
44.77
44.76

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