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On the theory of liquids
Physica
VII,
no 8
October
ON THE THEORY
OF LIQUIDS
by-H. C. BRINKMAN Laboratory
of N.V.
De Bataafsche
Petroleum
19.40
Maatscbappij
. (Royal
Dutch
Shell
Group)
Summary Using E y r i n g s concept of free volume a difference-equation for the free energy of .liquids is derived. This equation yields a formula for the compressibility of liquids, which contains only one constant, viz. the volume of the molecules in the closely packed state b. A satisfactory agreement between theory and esperiment is obtained, while b has nearly the same value as deduced from V a n W ij k’s and S e e d e r’s theory of viscosity.
For the calculation of the thermodynamic properties of a liquid it is sufficient to evaluate its partition function. For spherically symmetrical molecules this function is defined by: e-FlkT
=
&f
. . . . [e--wg(g,.*-qN)lkT dql . . . . dqnr
(1)
where F is the H e 1 m h o 1 t,z free energy per mol, Nthe Avogadro number, IV, the potential energy, q1 . . . . qN the coordinates of the centres of the molecules.
For liquids mathematical difficulties prevent our performing the exact integration of the integral occurring in (1). If we integrate over the coordinates q, of the first molecule the result will depend in a complicated way on the coordinates of the other molecules and it will hardly be possible to make any further integration. E y r i n g and collaborators ‘) have tried to make anapproximate -
747 -
748
;
H..C.
BRINKMAN
evaluation of (1). They introduce write for (1) d-F/kT
=
_
the concept of free volume --
’
N!?,3N
and
(2)
v,N
where V, is the total free volume. The free volume per molecule v1 is taken to be equal to the effective volume in which the centre of a molecule can move when all its neighbours are held fixed in their mean positions. In types of packing where the free volume is similar in shape to the unit cell of the lattice, E y r i n g ‘) proves that the molecular free volume nzllis equal to Vf = 8{ (I/‘/N)“,
-
v$}”
(3)
where V is the volume of the liquid and 21~ is the volume per molecule when the substance is in the closely packed state. ZJ~is supposed to be constant. From the following it will appear that this supposition only holds good in a limited region where v is larger than about 1.1 NzJ,. In (2) the total free volume is introduced V, = NV, (4 as E y r i n g supposes that the total free volume in a liquid is shared by all molecules. This distinguishes the liquid sharply from the solid, where no sharing of free volume is supposed to occur. From (2) and (3) one immediately finds the equation of state
RT aF fi = - w = JT.--(NV,)%J7% By introducing obtains
Van
fi++=
der
Waals’
V-
constant
a Eyring
RT
(6)
(Nv,)“a V’s
an equation which by a suitable choice of a and v. represents. the experimental results for liquids. However, in our opinion.it is preferable to use the concept of free volume in a way differing somewhat from E y r i n g’s. From the way in which the free volume is introduced it is clear that it represents an approximation to the integration over the coordinates of the first molecule in (l), as discussed above *). If this *) Formula (3) for the free volume is obtained from a very simplified lar potential holes). Our results therefore are only valid in a restricted represent the critical phenomena. *.
model region
(rectanguand do not
ON
THE
THEORY
749
OF LIQUIDS
.
is supposed to be the case we find.by performing one integration (1) and introducing (3) and (4) the following difference equation the free energy : e-WWT = (N - l)! 8 {F/‘/s _ (N,,,,)%}3 ,+W--1)IkT N!A3 Thisisdifferent from (2) and also leads to a different state “). From (7) it follows immediately: aF -= aN
- kT 3 III {V”s - (NTI,,)“~} + In &] [ Now aF/aN is equal to the thermodynamic potential:
in for
(7)
equation
.
of
.
(8)
By differentiating (8) with respect to V and using (9) a differential equation for fi is found :
J/&=av
where 0: is the compressibility. Integration of this equation
$ = C-3RT
1 RT (Nv,,)“a V”a = ii *
V-
(10)
yields:
ln v”‘-v{~)‘~
+
(NV,;%VW+
2(NV&7%}
(l ‘)
where C is an integration constant. Because of this constant, the absolute value of the pressure is not determined, but only the pressure difference- of two states of different volume (compressibility, cf. (10)). This is a consequence of the fact that we’ have not given a formula for the total free energy, but have introduced the difference equation (7) for this quantity. It would not be correct to make use of a formula for the free volume like (3) in order to approximate all integrations in ( 1). This would lead to: e-F/IT
=
1 h’lhj~
BNk!, {V’l*
-
(~~o)“9~
However, for the last integrations it is not permissible to use (3). A formula for the free volume like V-b, as used by Van der W a a 1 s, would be indicated there. It may be remarked that for V --t 00 (11) changes into B o y le’s
750
H.
C. BRINKMAN
law +V = RT, when C is taken to be zero. This, however, has no physical significance, but only results from the mathematical fact that according to (3) V, + 8V for V + co. The last two terms in (11) prove to be unimportant numerically when V is of the same order of magnitude as NV, and so, introducing NV, = b, we find the following approximate formula
In fig. 1 we have compared our formula (12) with experiments. The compressibilities of ethyl ether and ethyl alcohol at 20°C were measured in the laboratory of the Bataafsche Petroleum Maatschappij by W. A. S e e d e r “). The other experimental values are taken from Bridgman4). We have plotted the relative volume (VIId. = 1 for p = 1) against the pressure. The points represent the experimental values. The curves were calculated from (12), using the best values for b. We have
ON
THE
THEORY
OF
7.51
LIQUIDS
taken 2, to be constant for different temperatures. By letting b increase slightly for higher temperatures a still closer fit could bc obtained. TACLE \.nlurs
ethyl ether ethyl nlcohol. iso-pentme
.
I
of 0 in cc/m01
06 45 90
90.5 50.5 100
In table I the values of b used by us are compared with the values of this constant as determined by V a n W ij k and S e e d e r from their theory of viscosity 5). A closer agreement between the values of b cannot be expected as they are determined from quite different phenomena. From figure 1 we conclude that the agreement between experiment and theory is satisfactory, the more so as WC had only one constant, viz. b, which could be adapted in order to obtain a fit between experimental and theoretical values. The agreement of B r i d g m a n’s measurements with theory is somewhat less than S e e d e r’s. As he himself states, B r i d g m a n’s technique is not so accurate in the region of pressures considered. For pressures higher than 1000-1500 atmospheres formula (12) no longer holds good. This was to be expected, as for these pressures the liquids under consideration are so much compressed that b can no longer be treated as a constant. The author wishes to thank the management of N.V. De Bataafsche Petroleum Maatschappij for their permission to publish this paper. Amsterdam, Received
August
Sth,
1940.
June lOth, 1940.
752
H.
C. BRINKMAN,
ON THE
THEORY
OF
LIQUIDS
REFERENCES 1) H. E yring, J. them. Phys. 4, 283 (1936); R. Newton and H. Eyring, Trans. Farad. Sot. 33,73 ( 1936) ; H. E y r i n g and J. H i r s c h f e I d e r, J. Phys. Chem.d1,249(1937);J. Hirschfelder, D. Stevenson andH. Eyring, J. them. Phys. 5, 896 (1937). 2)
A difference-equation for the free energy analogous to our equation (7) was introduced by L. S. 0 r n s t e i n in his derivation of the equation of state for gases cf. L. S. 0 r n s t e i n, Arch. neerl., s&ieIIIA, 4, 203 (1918). 3) The results of these measurements were published in Nederl. Tijdschr. N’atuurk. 7, 2 I6 ( 1940) ; a detailed account will be given elsewhere. 4) I’. W. B r i d g m a n, The Physics of high Pressure, London 1931. 5) W. R. v a n W ij 1~ and 1%‘. A. S e e d e r, Physica “L, 1073 (1937) ; G, 129 (1939).
VII,
no 8
October
ON THE THEORY
OF LIQUIDS
by-H. C. BRINKMAN Laboratory
of N.V.
De Bataafsche
Petroleum
19.40
Maatscbappij
. (Royal
Dutch
Shell
Group)
Summary Using E y r i n g s concept of free volume a difference-equation for the free energy of .liquids is derived. This equation yields a formula for the compressibility of liquids, which contains only one constant, viz. the volume of the molecules in the closely packed state b. A satisfactory agreement between theory and esperiment is obtained, while b has nearly the same value as deduced from V a n W ij k’s and S e e d e r’s theory of viscosity.
For the calculation of the thermodynamic properties of a liquid it is sufficient to evaluate its partition function. For spherically symmetrical molecules this function is defined by: e-FlkT
=
&f
. . . . [e--wg(g,.*-qN)lkT dql . . . . dqnr
(1)
where F is the H e 1 m h o 1 t,z free energy per mol, Nthe Avogadro number, IV, the potential energy, q1 . . . . qN the coordinates of the centres of the molecules.
For liquids mathematical difficulties prevent our performing the exact integration of the integral occurring in (1). If we integrate over the coordinates q, of the first molecule the result will depend in a complicated way on the coordinates of the other molecules and it will hardly be possible to make any further integration. E y r i n g and collaborators ‘) have tried to make anapproximate -
747 -
748
;
H..C.
BRINKMAN
evaluation of (1). They introduce write for (1) d-F/kT
=
_
the concept of free volume --
’
N!?,3N
and
(2)
v,N
where V, is the total free volume. The free volume per molecule v1 is taken to be equal to the effective volume in which the centre of a molecule can move when all its neighbours are held fixed in their mean positions. In types of packing where the free volume is similar in shape to the unit cell of the lattice, E y r i n g ‘) proves that the molecular free volume nzllis equal to Vf = 8{ (I/‘/N)“,
-
v$}”
(3)
where V is the volume of the liquid and 21~ is the volume per molecule when the substance is in the closely packed state. ZJ~is supposed to be constant. From the following it will appear that this supposition only holds good in a limited region where v is larger than about 1.1 NzJ,. In (2) the total free volume is introduced V, = NV, (4 as E y r i n g supposes that the total free volume in a liquid is shared by all molecules. This distinguishes the liquid sharply from the solid, where no sharing of free volume is supposed to occur. From (2) and (3) one immediately finds the equation of state
RT aF fi = - w = JT.--(NV,)%J7% By introducing obtains
Van
fi++=
der
Waals’
V-
constant
a Eyring
RT
(6)
(Nv,)“a V’s
an equation which by a suitable choice of a and v. represents. the experimental results for liquids. However, in our opinion.it is preferable to use the concept of free volume in a way differing somewhat from E y r i n g’s. From the way in which the free volume is introduced it is clear that it represents an approximation to the integration over the coordinates of the first molecule in (l), as discussed above *). If this *) Formula (3) for the free volume is obtained from a very simplified lar potential holes). Our results therefore are only valid in a restricted represent the critical phenomena. *.
model region
(rectanguand do not
ON
THE
THEORY
749
OF LIQUIDS
.
is supposed to be the case we find.by performing one integration (1) and introducing (3) and (4) the following difference equation the free energy : e-WWT = (N - l)! 8 {F/‘/s _ (N,,,,)%}3 ,+W--1)IkT N!A3 Thisisdifferent from (2) and also leads to a different state “). From (7) it follows immediately: aF -= aN
- kT 3 III {V”s - (NTI,,)“~} + In &] [ Now aF/aN is equal to the thermodynamic potential:
in for
(7)
equation
.
of
.
(8)
By differentiating (8) with respect to V and using (9) a differential equation for fi is found :
J/&=av
where 0: is the compressibility. Integration of this equation
$ = C-3RT
1 RT (Nv,,)“a V”a = ii *
V-
(10)
yields:
ln v”‘-v{~)‘~
+
(NV,;%VW+
2(NV&7%}
(l ‘)
where C is an integration constant. Because of this constant, the absolute value of the pressure is not determined, but only the pressure difference- of two states of different volume (compressibility, cf. (10)). This is a consequence of the fact that we’ have not given a formula for the total free energy, but have introduced the difference equation (7) for this quantity. It would not be correct to make use of a formula for the free volume like (3) in order to approximate all integrations in ( 1). This would lead to: e-F/IT
=
1 h’lhj~
BNk!, {V’l*
-
(~~o)“9~
However, for the last integrations it is not permissible to use (3). A formula for the free volume like V-b, as used by Van der W a a 1 s, would be indicated there. It may be remarked that for V --t 00 (11) changes into B o y le’s
750
H.
C. BRINKMAN
law +V = RT, when C is taken to be zero. This, however, has no physical significance, but only results from the mathematical fact that according to (3) V, + 8V for V + co. The last two terms in (11) prove to be unimportant numerically when V is of the same order of magnitude as NV, and so, introducing NV, = b, we find the following approximate formula
In fig. 1 we have compared our formula (12) with experiments. The compressibilities of ethyl ether and ethyl alcohol at 20°C were measured in the laboratory of the Bataafsche Petroleum Maatschappij by W. A. S e e d e r “). The other experimental values are taken from Bridgman4). We have plotted the relative volume (VIId. = 1 for p = 1) against the pressure. The points represent the experimental values. The curves were calculated from (12), using the best values for b. We have
ON
THE
THEORY
OF
7.51
LIQUIDS
taken 2, to be constant for different temperatures. By letting b increase slightly for higher temperatures a still closer fit could bc obtained. TACLE \.nlurs
ethyl ether ethyl nlcohol. iso-pentme
.
I
of 0 in cc/m01
06 45 90
90.5 50.5 100
In table I the values of b used by us are compared with the values of this constant as determined by V a n W ij k and S e e d e r from their theory of viscosity 5). A closer agreement between the values of b cannot be expected as they are determined from quite different phenomena. From figure 1 we conclude that the agreement between experiment and theory is satisfactory, the more so as WC had only one constant, viz. b, which could be adapted in order to obtain a fit between experimental and theoretical values. The agreement of B r i d g m a n’s measurements with theory is somewhat less than S e e d e r’s. As he himself states, B r i d g m a n’s technique is not so accurate in the region of pressures considered. For pressures higher than 1000-1500 atmospheres formula (12) no longer holds good. This was to be expected, as for these pressures the liquids under consideration are so much compressed that b can no longer be treated as a constant. The author wishes to thank the management of N.V. De Bataafsche Petroleum Maatschappij for their permission to publish this paper. Amsterdam, Received
August
Sth,
1940.
June lOth, 1940.
752
H.
C. BRINKMAN,
ON THE
THEORY
OF
LIQUIDS
REFERENCES 1) H. E yring, J. them. Phys. 4, 283 (1936); R. Newton and H. Eyring, Trans. Farad. Sot. 33,73 ( 1936) ; H. E y r i n g and J. H i r s c h f e I d e r, J. Phys. Chem.d1,249(1937);J. Hirschfelder, D. Stevenson andH. Eyring, J. them. Phys. 5, 896 (1937). 2)
A difference-equation for the free energy analogous to our equation (7) was introduced by L. S. 0 r n s t e i n in his derivation of the equation of state for gases cf. L. S. 0 r n s t e i n, Arch. neerl., s&ieIIIA, 4, 203 (1918). 3) The results of these measurements were published in Nederl. Tijdschr. N’atuurk. 7, 2 I6 ( 1940) ; a detailed account will be given elsewhere. 4) I’. W. B r i d g m a n, The Physics of high Pressure, London 1931. 5) W. R. v a n W ij 1~ and 1%‘. A. S e e d e r, Physica “L, 1073 (1937) ; G, 129 (1939).