960
A.
K.
is at least as good as other potentials when the condition of spherical symmetry is not strictly satisfied.
BARUA
potential (considering also the values given by Corner!)
for the case {J= 0, we may conclude that the effect of
considering the particular form of the exp 6-8 potential
for
m is to cause a lowering of the value of a from
what would have been obtained if Eq. (1) alone were
considered.
,$,
CONCLUSIONS
It will be seen, as has also been observed by Corner,!
for A and Ne that the second virial coefficient is not
able to throw light on the relative importance of ,-6
and
contributions. From a comparison of the potential parameters for the exp 6-8 and the exp-six
,-8
THE JOURNAL OF CHEMICAL PHYSICS
ACKNOWLEDGMENT
The author wishes to thank Professor B. N.
Srivastava for his valuable guidance.
VOLUME 31,
NUMBER 4
OCTOBER,
1959
Equation of State and Thermodynamic Properties of Gases at High Temperatures.
I. Diatomic Molecules
O.
SINANO~LU* AND
K. S.
PITZER
Department of Chemistry and Lawrence Radiation Laboratory, University of California, Berkeley 4, California
(Received April 7, 1959)
Methods for evaluating the thermodynamic properties of assemblies of chemically reacting unionized
atoms are discussed. The desirability of using the virial coefficients at high temperatures instead of the
customary use of the molecular partition functions with anharmonicity corrections is emphasized. The
most realistic three-parameter diatomic potential energy function that is available at present, i.e., the
Rydberg potential, U IU.= - (l+b/~) exp( -b/~) with ~= (rlr.) -1 is selected for the evaluation of the
classical second virial coefficient. B(T), T(dBldT) and P(d2BldT2) are obtained as linear combinations
involving the five functions:
.!h(lI) = 2;[(lIe) "Inn+k],
"-1
[k= -1,0, 1, 2, and 3, and 11= (U.lkT)] with only the coefficients that mUltiply Ak depending on b'. Ak
is tabulated for 0.05,0,10. A simple expression for estimating the quantum correction to B (T) is given.
The inclusion of the contribution of the higher diatomic electronic states to B (T) is considered. The treatment is applied to sodium (including the 32: repulsive state), and B(T) and thermodynamic properties
calculated at two temperatures by several methods are compared.
INTRODUCTION
imperfections given by
THE
thermodynamic properties and equation of
state of a dissociating but unionized gas could in
principle be obtained by the use of two different
formalisms. In the first of these, the gas is treated as a
mixture of certain significant molecular species, e.g.,
diatomic, triatomic, ... molecules, i.e., it is assumed
that only those portions of the phase space that are in
the vicinity of some highly probable configurations are
important and the equilibrium composition of the gas
in terms of these species and the contribution of each
to the thermodynamic properties are calculated from
the ordinary partition functions of the "molecules."
It is apparent that this method is most appropriate
for an assembly of chemically reacting atoms and at
such temperatures that certain molecular species are
unambiguously definable. In the second method, the
entire gas is treated as a monatomic assembly with
* University of
California Predoctoral Fellow, 1958-1959.
or
PV=RT+B(T)P+C'(T)P2+ ... ,
(lb)
and then all the thermodynamic properties are obtainable1 as corrections to those of the monatomic gas
in terms of the virial coefficients B(T), C(T)'" and
their first and second temperature derivatives.
Of course the two formalisms are, in principle, quite
equivalent as can be seen best from the most general
derivation of the virial coefficients by the use of the
quantum grand canonical partition function which is
not restricted to a classical gas with pairwise additive
1 See, for example, Hirschfelder, Curtiss, and Bird, Molecular
Theory of Gases and Liquids (John Wiley & Sons, Inc., New
York, 1954), p. 230.
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E QUA T ION 0 F
S TAT E, THE R MOD Y N A M I CPR 0 PER TIE S 0 F
interactions.2 For instance, the second virial coefficien t
for a general gas is given by
B( T) = - (No/2V) (V/Ql)2(2Q2-Q12),
(2)
where No is Avogadro's number and Ql and Q2 are the
partition functions for monomer and dimer in the
volume V.
The equivalence between the two formalisms has
been discussed in great detail by HilP and others.4
Differences arise for the most part from the ambiguity
of defining "physical clusters" and in the manner the
two methods are conventionally applied; for instance,
the chemical equilibrium method will not give the
correct pressure unless interactions between unbound
molecules are introduced amounting to the use of
activities rather than concentrations in the equilibrium quotient. 3 ,5
In practice, for atoms with strong chemical interactions and at ordinary temperatures, i.e., when appreciable fractions of molecules are present, the chemical equilibrium treatment is the convenient one, since
even with a gas composed of only diatomic molecules
and atoms, in the virial treatment not only B(T), but
also higher coefficients would be important. On the
other hand, the virial coefficients formalism is most
useful when interatomic potential energies are comparable to kT. At these temperatures the virial coefficients can usually be taken in their classical limit; for
instance, B (T) for atoms interacting on only one
potential curve reduces to
GAS E S
virial coefficient, taking care all of the binary interactions of the atoms including "chemical association,"
is the most important term in contributing to the
thermodynamic properties. We expect to extend this
treatment in the near future to the third virial coefficient of an assembly of chemically reacting atoms
which have strongly nonpairwise additive interatomic
potentials.
DIATOMIC MOLECULES
At quite low temperatures, the partition function,
(p.f.) of a diatomic molecule, Q2, is approximated by
the harmonic oscillator-rigid rotator p.f. At higher
temperatures, the population of levels corresponding
to the nonparabolic regions of the potential energy
curve U(r) is no longer negligible and it becomes
necessary to modify the simple p.f. by the introduction
of "anharmonicity corrections" into the energy levels
of the molecule in the form
(Ev;/hc) = (v+!)we- (v+!)w e x.+j(j+l)B e
-P(j+l)2D e - (v+!)j(j+1)a e" ' ,
(') _
Q2.int,' - ~ exp (
E ,(i»)
--,;r ,
-V,J
(Sa)
V,l
and
\Nt
2 J. E. Kilpatrick, J. Chern. Phys. 21, 274 (1953).
• T. L. Hill, Statistical Mechanics (McGraw-Hill Book Company, Inc., New York, 1956), p. 146 ff.
4 J. E. Kilpatrick, J. Chern. Phys. 21, 1366 (1953); Ann. Rev.
Phys. Chern. 7, 70 (1956); Beckett, Green, and Woolley, ibid.
7,300 (1956).
6 H. W. Woolley, J. Research Natl. Bur. Standards 61, 469
(1958).
(4)
where We, WeX., Be, De, and a. are the usual spectroscopic
constants. The p.f. should be calculated from
' - "'Q2'
(i)" •. exp (- kT
Ei)'
Q2 ,lnt.~
,tnt.
and has been extensively used in this form for chemically inert atoms.l It is clear that a gas containing a
large fraction of molecules and treated by the use of
quantum partition functions that are good approximations near the most stable configurations approaches
more and more a classical assembly of atoms at higher
temperatures, and the virial coefficient formalism
becomes more and more appropriate since it avoids
some of the difficulties encountered in the use of partition functions near dissociation. It is this approach
that will form the basis for our treatment of the
thermodynamic properties of a gas composed of
chemically reacting atoms at high temperatures. In
this article we shall be concerned with a gas that contains only un-ionized atoms and "diatomic molecules,"
but at such temperatures that only a relatively small
fraction of "dimers" are involved so that the second
961
(Sb)
where Wi is the degeneracy and Ei energy to the minimum of the ith electronic state of the molecule. Equation (4) for E.}i) is an approximation, valid not too
near dissociation and, due to the (V+!)2 term, the
sum Eq. (Sa) will diverge if carried to infinity. Mayer
and Mayer6 have obtained expressions for thermodynamic functions from Eqs. (4) and (Sa) however, by
expanding some of the exponentials arising from Eq. (4)
and taking only the first two terms. At not too high
temperatures, the Boltzmann factor for the main part
of Ev,j overtakes the higher terms in this expansion
and the sum converges. At higher temperatures, however, errors from several sources are introduced and
are difficult to estimate. First, at high enough temperatures, the range of validity of Eq. (4) will no longer be
sufficient, and it must be emphasized that the range
important in the p.£. is not determined solely by the
magnitude of the Boltzmann factor for different energies but also by the density of states (the entropy
effect). Thus, even though the dissociation energy of
Na2 is kX8600oK, at 1600 0 K and 1 atmos the gas will
contain almost all monomers. Secondly, even in the
8 J. E. Mayer and M. G. Mayer, Statistical Mechanics (John
Wiley & Sons, Inc., New York, 1940), p. 160 if.
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962
O.
SINANOCLU
AND
range where Eq. (4) is still quite valid, the errors7
introduced in the approximate summation procedure
itself might become appreciable, especially if WeXe is
large, since each term in the sum Eq. (Sa) will have
some error due to the neglected terms of the exponential
series, and at higher energies there will be more and
more terms near the same energy. With strongly bound
atoms, i.e., when binding energies (U e) are greater
than several ev's, errors might become appreciable
above 3000-4000 oK ; on the other hand, with weakly
bound (U.<l ev, e.g., alkali vapors and Hg) atoms,
even at ordinary temperatures all the potential energy
curve [U(r) J contributes to Q2, and since energy levels
near dissociation generally are not known, it becomes necessary to revert to the virial coefficient treatment. Under the conditions of concern here, the use of
the classical second virial coefficient, Eq. (3), (quantum
corrections will be considered later) provides a good
approximation. Thus it becomes necessary to choose
suitable semiempirical potentials to represent the
interaction of atoms. On hopes to select a potential
which is applicable to a large number of systems and
adequate in the region of repulsive as well as attractive
forces. For chemically nonreacting systems (e.g.,
inert gases) and for repulsive electronic states (e.g.,
the 3~ state of Na 2) , two parameter functions that are
simply sums of an exponential or inverse power repulsion and inverse sixth-power London attraction are
quite adequate. Among these the Lennard-Jones (6-12)
potential has been most widely used to obtain the virial
coefficients and transport properties. t Tables of B* (T*)
have been extended by Epstein8 to the low reduced
temperatures [T*= (kT/U e) <0.3J necessary for application to atoms interacting much more strongly
than the inert gases. (He has also treated the LennardJones (6-9) potentia1. 9 ) However, with chemical interactions a two parameter potential is no longer satisfactory since the exchange attraction depends strongly
on the nature of the atoms involved. A very large
number of semiempirical potentials have been proposed by various investigators for such systems, i.e.
"diatomic molecules." An excellent critical review of
these has recently been given by Varshni.!O A good
potential must have a minimum number of parameters,
yet predict the remaining spectroscopic constants in
Eq. (4) with sufficient accuracy as well as being
adequately close to the dispersion forces at large
distances. The most widely known three-parameter
potential is that of Morse,
U*= U/U e= exp( -
2a~)
- 2 exp( -
a~),
(6)
where
a=re(k e/2U e )l;
-----
~=
(r/re)-1.
For a discussion of these errors, see also G. Baumann, Z.
Physik. Chern. (Frankfurt) 14, 113 (1958).
8 L. F. Epstein and G. M. Roe, J. Chern. Phys. 19, 1320 (1951).
9 L. F. Epstein, J. Chern. Phys. 20, 1665 (1952); L. F. Epstein
and C. J. Hibbert, ibid. 20, 752 (1952).
10 Y. P. Varshni, Revs. Modern Phys. 29, 664 (1957).
7
K.
S.
PITZER
This potential, although it has been used for the
calculation of thermodynamic properties of several
gases at high temperatures,u gives too much attraction
at long distances and predicts WeX., on the average,
within ±31.2% and ex e within ±33.1% for the twentyseven diatomic molecules studied.!O Hulburt and
Hirschfelder12 have tried to compensate for this excess
attraction by adding a term to the Morse function
with two additional parameters.
UH .H .= UMme+UeCX3 exp( -2x) (1+bx) ,
x=a~.
(7)
This potential has a total of five parameters and
makes use of all the available spectroscopic constants
given in Eq. (4); it is probably the most accurately
known potential over a wide range of r. The second
term in Eq. (7) introduces unnecessarily large "corrections" for X<O, but this defect is eliminated if we
replace x in e-2x by its absolute value [ x [. Equation (7)
is impractical for our use in the calculation of B (T)
because of the large number of its parameters, some
of which moreover, may not be available for some
systems. Of all the three-parameter potentials examined
by Varshni,t° the one proposed by Rydberg 13 in 1931
gives, on the average, the best prediction for both WeX.
(within ±23.1%) and exe (±28%). (Lippincott potential gives a better value for WeXe although not for ex e
at all; however, it gives U=O at r=O, and when corrected for this important defect, it loses the advantage
of its simple analytic form. The Rydberg function is
not infinite, at r= 0 either, but is sufficiently large.)
The Rydberg function, in general, will give considerably
less excess attraction at large r than the Morse function and has the simple analytical form
(8)
where
b' = r e(k e/ U e)!
and
~=r*-l=
(rlre)-1.
In Fig. 1, the Rydberg and Morse functions for the
3~u- excited electronic state of O2 are shown as well as a
curve (shown as "experimental") that Rydberg has
constructed by a graphical method from the vibrational
levels, which for this state are available up to quite
near dissociation. (The curves are from reference 13.)
The B (T) that we are about to give is necessary for the
thermodynamic properties of alkali vapors at ordinary
temperatures because of the weakness of interactions;
therefore, in Fig. 2 we examine and compare various
functions for sodium (1~). It is observed that the
11 C. W. Beckett and L. Haar, Proceedings oj the Joint Conference on Thermodynamic and Transport Properties of Fluids
(Institute of Mechanical Engineers, London, 1958), p. 27.
12 H. M. Hulburt and J. O. Hirschfelder, J. Chern. Phys. 9, 61
(1941) .
13 R. Rydberg, Z. Physik 73 376 (1931).
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EQUATION OF STATE, THERMODYNAMIC PROPERTIES OF GASES
Lennard-Jones (6-12) and (6-9) potentials give too
narrow a bowl. The Hulburt-Hirschfelder function may
be considered as the "true" potential. The attraction
for distances where overlap is negligible (i.e., ca r*> 2)
can be estimatedl4 using second-order perturbation
theory and is given in the form
(cI/r6) + (C2/,s) + (Ca/r10) + .. "
(9)
where Cl, C2, Ca, '" can be calculated from the atomic
energy levels and their oscillator strengths. Because of
their large polarizability, alkali atoms exhibit dispersion forces that are particularly large and even at r*= 2,
the multipole interaction terms, r- 8, r-10, contribute as
much as the r- 6 term.I4 A lower limit to the dispersion
attraction calculated from the CI, C2, C3 given by MargenauI4 for Na is also shown in Fig. 2. Both figures
indicate the considerable improvement resulting from
the use of Rydberg function over a wide range of r.
Hence in the following, this potential will be used in
Eq. (3) for the evaluation of BeT) and might be expected to be a good approximation except for ionicly
binding systems and for states that show some anomaly
such as a maximum in the potential curve.
SECOND VIRIAL COEFFICIENT
At high temperatures, usually more than one atomic
and diatomic electronic state need to be considered
when dealing with the binary interactions of an assembly of chemically reacting (i.e., spinwise unsaturated) atoms. For such systems B (T) must include
contributions from various potential energy curves.
An examination of Eqs. (2) and (5b) shows that the
complete B (T) is given by
B(T) = Li,,'Lexp( - E/kT)B(i)(T)
L~:::>Qj exp( - Ei/kT) J2 '
02
::l -0.4
"
lj(:l -06
~-~--
-0.8
Hulburf-Hirschfelde,
.•. ~.-.-. Oi spersion (Morgenou)
-1.0
0.5
1.5
1.0
2.5
2.0
r~=r/re
FIG. 2. Comparison of potential energy curves for the ground
state of Na..
excitation energy of one atom; E. is zero if the ith
diatomic potential dissociates into ground state atoms,
otherwise it is the energy difference between the dissociation products and the ground state atoms of the
ith diatomic electronic state and B(i) is the virial coefficient for the ith potential energy curve (taken to
become zero at its dissociation) and will be given by
Eq. (3) when quantum effects are negligible. If i is a
state for which the Rydberg potential, Eq. (6), is applicable, Bcl(i) (T) in reduced form is given by
°l°O
N
.
271" Bcl(')(T)
=
bl
!1-exp[O,;(1+x)e-x JI
-bit
. (x+b/)2dx,
(11)
where we have substituted
and
For each of the contributing electronic state potentials
there will be a different b', and a different reduced
temperature. Equation (11) can be exactly integrated
term by term after expanding the exponential and
denoting (x+b') by y; the result is
1000
2000
.:"3000
IE
b3B cl (T)
4000
:::>
1(0, b')
/
5000
3
02:-U
2
,/
6000
'I
-
ro
Experimental
= -
- - - Rydberg
---- Morse
7000
SOOOIr----'''''''-----------d
15
r
2.0
25
(A)
FIG. 1. Comparison of potential energy curves for the excited
a~ u- state of O. (reference 13).
14
04
(10)
where Wi and Qj are the electronic multiplicities of the
diatomic and atomic states, respectively. E j is the
2
963
H. Margenau, Revs. Modern Phys. 11, 1 (1939).
o'njoo° exp( -ny)(y-b'+1)ny dy,
~ n!
2
(12)
or
1(0 b')=- ~O'n~ (1-b')T(n-r+2)(n-r+!l
,
~
~
,~~
,
n-l
,==0
r.n
where 0' = 0 expb'.
However, this series is not desirable for the computation of I, because, for the 0 range of interest here,
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964
O. SINANOCLU AND K. S. PITZER
TABLE 1. Molecular constants& for the ground electronic
states of some diatomic molecules.
r, (AD)
We
(CITe l )
U,/k(OK)
---------
0.7417
2.672
3.078
3.923
3.2
2.667
1.207
Hz
Liz
Na2
K,
Hg2b
Iz
0,
b'
-------~--
4395.2
351.43
159.23
92.64
36.
214.6
1580.4
55 100
12 100
8600
6 020
755
18000
60 500
,-
2.04
3.28
3.71
4.26
8.66
7.02
4.51
All values are taken from reference 10 unless otherwise indicated.
b Douglas, Ball, and Ginnings, J. Research NatL Bur. Standards 46, 334
(14c) can be replaced by infinity, thus reducing Rk to
functions of () only. The functions Rk(O) obtained by
numerical integration are shown in Fig. 3 and can be
used to obtain the corrections to Eq. (15) at small O's.
We have already remarked that all the thermodynamic functions as corrections to those of the
monatomic gas could be obtained if, in addition to B,
its first and second temperature derivatives were
known. We differentiate Eq. (15) and note from Eq.
(14a) that
&
(1951).
and
02(d2Ak/d(2) = A k - 2 - Ak-l,
it is very slowly convergent and depends on b' in each
term so that it wonld be necessary to prepare tables of I
versus () and b'. Instead, 1(0, b') can be conveniently
evaluated in the following manner.
Let us break the integral in Eq. (11) into two parts,
1=10+ II =
+ f"{1-exP[()(1+x)e-
X
]
]
Hx+b')2dx.
1, I'"'-'I1+b"3/3)
-O(dI/dO) = T(dI/dT) "'T(dIl/dT)
=2A 2+(2b'+1)A 1+b'2A o,
(17)
-f)2(d21/d()2) = - P(d21/dT2)'::::.- T2(d 2h/dT2)
(13)
-1
Notice that 10 corresponds to the contribution of the
positive portion of U(r) and 11 of the negative portion,
and that forO> 1, 10 is usually a small fraction of h Now
II can be integrated exactly by substituting z for (x+ 1),
then expanding the exponential in Eq. (13) and we
obtain, after some manipulation,
(14a)
where
A k «() = E[(Oe)njnn+k].
(»
and
i~: {1 ~ exp[()(1 +x)e- I (X+b')2dx
X
obtaining (for
(16)
= b'2A_1 - (b'2- 2b' -1) Ao- (2b' -1) A 1 - 2A 2 ,
(18)
where again Ak'S are defined by Eq. (14b). The functions Ak(O) with k= -1,0, 1, 2, 3 were computed on an
IBM 701 computer for 0.05<0<10 and are given in
Table II. They might be useful also in other applications.
QUANTUM CORRECTION
When the energy of a vibrational quantum of the
diatomic molecule in the particular electronic state
considered is small compared to the thermal energy,
1.000,rl---'-~-"- , , - - - , -
(14b)
n=l
0.900
On the other hand, for 10 we write
10= (b"3/3)
b"2 Ro+ 2b" R 1 - R 2,
0.800
(14c)
0.700
where bl! = b'_l, and
i
on
Rk=
Zk
0.600 -
exp[ -()z exp(z+l) ]dz
(k=O, 1,2).
For () greater than about 1.0, which covers most of the
temperature range of interest as it is apparent from the
U e/ k's shown in Table I, the exponential part in 10 is
negligible and we get
CD
0500 -
-'"
0:: OAOO
0.300
0200
This provides a very good approximation to I at 0> ca
1.0, since not only the Rk terms in Eq. (14c) become
small with increasing 0, but also 10 itself becomes a
small fraction of 1. For instance, at ()=5 and b'=4,
II equals -3531 and 10'"'-'9. On the other hand, to obtain
10 for 1>0>0.1 the finite limits of the integrals in Eq.
R2
0.100
0.2
0.30.405
10
20
8- Ue
- kT
FIG. 3. The functions Rk (0) for the evaluation of 10.
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EQUATION OF STATE, THERMODYNAMIC PROPERTIES OF GASES
965
co
TABLE II. Functions Ak(/I) = 'Z,(/le)njnn+k for the evaluation of I, and its derivatives.
,,-,
/I
A-,
Ao
A,
A.
As
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.2
2.42.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
5.6
5.8
6.0
6.2
6.4
6.6
6.8
7.0
7.2
7.47.6
7.8
8.0
8.2
8.4
8.6
8.8
9.0
9.2
9.4
9.6
9.8
10.0
0.1454294
0.3110905
0.4988512
0.7107323
0.9489262
1.2157737
1.5138080
1.8457663
2.2145644
2.6233614
3.0755581
3.5747837
4.1249600
4.7302899
5.3953028
6.1248254
6.9240646
7.7985988
8.7544226
9.7979280
12.175992
14.993891
18.321669
22.239609
26.839591
32.226717
38. S21113
45.859932
S4.399717
64.318921
89.1374
122.3041
166.3919
224.7213
301.5696
402.4360
534.3777
706.4373
930.1832
1 220.3932
1 595.921
2080.788
2 705.S67
3 509.119
4540.793
5 863.191
7 555.657
9 718.646
12 479.257
15 998.152
20 478.25
26 175.67
33 413.45
42598.70
54 244.27
68 995.64
87 664.89
111 272.97
141 102.81
178 765.58
226 284.0
286 195.5
361 682.1
456 732.9
576 344.9
726 777.3
915 864.5
1 153 410.6
1 451.679.8
1 826 005.8
0.1406253
0.2910661
0.4519280
0.6238547
0.8075297
1.0036731
1. 2130494
1.4364687
1.6747851
1.9289047
2.1997876
2.4884453
2.7959528
3.1234435
3.4721212
3.8432533
4.2381846
4.6583360
5.1052126
5.5804003
6.622532
7.799371
9.127274
10.624526
12.311552
14.211168
16.348859
18.753076
21. 45S584
24.491840
31.72841
40.83745
52.28507
66.65079
84.6S469
107.19106
135.36992
170.56848
214.49466
269.26S41
337.5038
422.4589
528.1532
659.5655
822.8558
1 025.6435
1 277.3509
1 589.6233
1 976.8546
2 456.8284
3 051. 508
3 788.019
4 699.849
5 828.332
7 224.480
8951.228
11 086.218
13 725.200
16 986.248
21 014.936
25990.72
32 134.76
39 719.59
49 081.02
60 632.71
74 884.25
92 463.12
114 142.09
140 872.68
173826.14
0.1382542
0.2813178
0.4293886
0.5826738
0.7413899
0.9057626
1.0760279
1.2524322
1.4352327
1.6246982
1.8211096
2.0247594
2.2359550
2.4550151
2.6822751
2.9180836
3.1628058
3.4168234
3.6805351
3.9543564
4.534090
5.159749
5.835402
6.565494
7.354884
8.208885
9.133303
10.134492
11.219400
12.395624
15.056071
18.192229
21.895636
26.276051
31.465135
37.620921
44.933172
53.629894
63.985139
76.328449
91.05626
108.64570
129.67120
154.82451
184.93914
221.01958
264.27694
316.17137
378.46469
453.28256
543.1904
651.2856
781.3080
937.7753
1 126.1454
1 353.0128
1 626.3492
1955.7903
2 352.9877
2 832.0308
3409.966
4 107.411
4 949.316
5965.877
7 193.639
8 676.849
10 469.072
12635.178
15253.735
18 419.900
0.1370790
0.2765304
0.4184190
0.5628122
0.7097795
0.8593925
1.0117255
1.1668549
1.3248602
1.4858234
1.6498287
1.8169637
1. 9873194
2.1609886
2.3380685
2.5186591
2.7028639
2.8907901
3.0825489
3.2782541
3.6819835
4.1029784
4.5423110
5.0011293
5.4806639
5.9822363
6.5072637
7.0572675
7.6338828
8.2388638
9.541610
10.982364
12.580635
14.359025
16.343759
18.565307
2l.059094
23.866378
27.035254
30.621836
34.69168
39.32145
44.60088
50.63507
57.54734
65.48237
74.61016
85.13055
97.27870
111.33155
127.61541
146.51516
168.48491
194.06105
223.87739
258.68336
299.36567
346.97411
402.75227
468.17383
544.9870
635.2651
741.4701
866.5258
1 013.9060
1 187.7411
1 392.9423
1 635.3537
1 921.9324
2 260.9618
0.1364948
0.2741652
0.4130325
0.5531186
0.6944461
0.8370381
0.9809183
1.1261106
1.2726402
1.4205326
1. 5698139
1. 7205105
1.8726506
2.0262614
2.1813728
2.3380140
2.4962153
2.6560083
2.8174254
2.9804985
3.3117505
3.6500460
3.9956818
4.3489705
4.7102418
5.0798450
5.4581475
5.8455386
6.2424293
6.6492541
7.494573
8.385507
9.326555
10.322775
1l.379855
12.504211
13.703080
14.984650
16.358184
17.834192
19.424602
21.142980
23.004769
25.027578
27.231512
29.639545
32.277985
35.176964
38.371069
41.900019
45.80948
50.15206
54.98834
60.38827
66.43259
73.21468
80.84258
89.44145
99.15639
110.15578
122.63524
136.82211
152.98092
171. 41971
192.49742
216.63269
244.31403
276.11202
312.69370
354.83900
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966
O.
SINANOCLU AND
TABLE III. Calculations for Na vapor.
K.
S. PITZER
for Q2 in Eq. (2), then
Second virial coefficient
Q.orr=
ref.
12;
state: lit
5.0
RYd
B (I) in ce/mole L-J, 6-9
{L-J,6-12
32;
state: 8s
B(3)
0.195
ec/mole (L-J, 6-9)
B = (BRyd(1)+3B(3» /4 cc/mole
9
8
9
-492.3
jncal/mole
{Ryd
Anh
{Ryd
in cal/mole deg Anh
(S-SO)
(Cp-CpO) cal/deg mole, Ryd
-221.6
-193
-50.5
-41
-0.103 -0.0157
-0.093 -0.013
0.45
+"',
(22)
7rfh2re2cwe exp I U e l/kT
_
B ( T ) -Bcl +
12m!(kT)t
+ ...
.
(23)
Notice that the same expression is obtained if one
substitutes
Thermal properties at one atmos (neglecting 32; state)
(H-HO)
2
2 r. ki exp I U e I!kT
B( T) =Bct + (h /m) 12(27r)i(kT)!
or
0.140
+12.29 +37.99
-1656
where x= (hv/kT), and substituting in Eq. (2) we get
3.6
-6662.7 -2083.1
-3350 -1061
-2854
-904
;~-;~: =(1- ;~+ 5:~ .. -),
15
15
0.047
i.e., if
U(r)
= -
U.+tk.(r-r e )2+ ... ,
in Eq. (20). Equation (23) gives the first quantum
correction term as 89.3 cc/mole for Liz at 1720oK,
or roughly half a percent of B. Thus, for heavier elements and smaller e, the quantum correction ordinarily will be negligible.
APPLICATION
(19)
Eq. (3) might be expected to be a good approximation
to Eq. (2). This condition is satisfied for most systems
of interest here; nevertheless, it is possible to obtain an
estimate of the quantum effects on B (T). The usual
quantum correction on B (T) is ordinarily obtained
from a WKB approximation (or a solution of the
Bloch differential equation)! and the first two terms
are
where
Br(T)
_N-=..o-ico(exp[ - U(r)/kTJ) (au/ar)2 r2dr.
247rk 3 ']'3 0
(20)
Br(T) can be evaluated by substituting the Rydberg
potential in Eq. (20) and expanding the exponential;
however, the resulting series is inconvenient for computation. Instead, the first quantum correction can be
estimated by noticing that, at the lower temperatures
at which the quantum effect should be examined, the
greatest contribution to the integral in Eq. (20) comes
from the parabolic region of U(r). For instance, for Li
at 17200 K (0= 7.1), a graph of the integral in Eq. (20)
shows that there is a very tall and narrow peak at
r*=0.85. Since the quantum effect (only the diffraction effect is important and the symmetry effect need
not be considered) comes largely from the vibrational
motions, we write in the usual fashion
(21)
We shall now demonstrate the use of the equations
developed with an application to sodium vapor. As
we have already remarked, because of the relative weakness of their interactions, alkali vapors constitute a
case where the treatment is applicable at relatively low
temperatures and, for instance, would be useful in the
analysis of their vapor pressure data.
With alkali atoms it is necessary to consider the two
diatomic electronic states that arise, namely l~ and
3~. The potential curve for the 1~ state of Na already
has been discussed and the Rydberg potential will be
used for this. On the other hand, the a~ state is repulsive
and its potential energy curve arises from exchange
repulsion and only van der Waals attraction [Eq. (8) J
at distances greater than about r*= 2.
By using the constants given in Table I for the l~
state of Na, in connection with Eqs. (11), (15), (17),
and (18) and obtaining the functions Ak(O) from Table
II we have calculated B(1) (T) and its derivatives at
,
0
1
two temperatures, 1720 0 K and 2390 K. The resu ts are
shown in Table III. For comparison, the values of
B(!l(T) obtained for Lennard-Jones (12-6) and (9-6)
potentials from the tables in references 8 and 9
are also given; the differences are of the order of a
factor of two as would have been expected from a
comparison of the areas under the bowls of the potential energy curves in Fig. 2.
The contributions of the second virial coefficient
to the thermodynamic functions of Na vapor at one
atmos have been calculated from the relations l
H-IJO= (RT/V)[B- T(dB/dT) J,
S-so= -R{lnp+(T/V) (dB/dT)
(22)
+ ... },
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E QUA T ION 0 F
S TAT E, THE R MOD Y N A M I CPR 0 PER TIE S 0 F
and
Cp-Cpo= -R( (T2/V) (d 2B/dT2) + ... },
where HO, SO, and Cpo refer to one mole of ideal monatomic gas, using the Rydberg potential for the l~
state. These, and the values obtained by Benton and
Inatomi15 from the usual anharmonicity treatment 6 are
'
also compared in Table III.
To get the total B( T), the contribution of the 3~
s~ate must also be taken into account using Eq. (10).
Smce we do not have sufficient information on the potential energy curve of this state, we shall give only
an order of magnitude estimate.
The repulsive portion of the curve can be estimated
using the semiempirical method16 ; i.e., we write the
energy of the singlet state simply as Q+a, where Qis the
Coulombic and IX the exchange integral, then the energy of the repulsive state (3~) is Q-a, and we can
assume that p= Q/ (Q+a) is reasonably constant and
for Na is about 0.30. Thus, away from the dispersion
region, the repulsion is roughly -0.4U(1)(r) where
U(O(r) is the Rydberg curve for 1~. On the other hand
a~ larger distances (say beyond r= 6 A), the energy i~
given by Eq. (9). To get a rough estimate of B(3)(T),
we fit a Lennard-Jones (6-9) potential to this information since there is no exchange attraction for 3~.
15 A. Benton and T. H. Inatorni, J. Chern. Phys. 20, 1946
(1952); see also, Evans, Jacobson, Munson, and Wagman, J.
Research Natl. Bur. Standards 55, 83 (1955).
16 Glasstone, Laidler, and Eyring, The Theory of Rate Processes
(McGraw-Hill Book Company, Inc., New York 1941), p. 76.
GAS E S
967
Making Lennard-Jones (6-9) equal -0.4U(1)(r) at
y= 5.08 A and Eq. (8) at r= 6.94 A (see Fig. 2), we
obtain y.(3)=5.7 A and U.(3)=0.0289 ev. With these
parameters, B(3) (T) is obtained from the tables in
reference 9 (see Table III).
The total B (T) is given by
B( T)
= [B(1)(T) +3B(3) (T) J/4,
according to Eq. (10). Examination of results summarized in Table III shows that the contribution of the
triplet state to B (T) is about 0.54% at 1720 0 K and
about 5.8% at 2390°K. The thermodynamic functions
calculated from the Rydberg potential are somewhat
larger than those from the anharmonicity treatment as
would be expected. We should also remark that the
functions at pressures other than one atmos would be
very easily obtained from Eq. (22), whereas with the
chemical equilibrium treatment one has to :first evaluate
the equilibrium composition at each pressure before
obtaining the functions; we note that the B(T) treatment will give also the equation of state easily and
more correctly.
ACKNOWLEDGMENTS
We are indebted to Professor F. Harris and Professor
L. Brewer for various suggestions and discussions and
to Mr. J. Neuhaus and Mr. L. Davis of the University
Computer Center for computing the functions in
Table II. This research was carried out under the
auspices of the U. S. Atomic Energy Commission.
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