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20
Chapter-2
Equilibrium Composition and Thermodynamic
Properties of Hydrogen Plasma
It is well known that the thermodynamic and transport properties
depend directly on the
plasma composition, which further depends upon the inclusion of
electronically excited
states (EES) through the partition function. Thermodynamic
properties include mass
density, internal energy, enthalpy, specific heat and entropy.
The partition function plays
an important role in the determination of thermodynamic as well
as transport properties.
In this chapter, by evaluating the degree of ionization of
hydrogen thermal plasma, its
equilibrium composition and the thermodynamic properties have
been discussed both
for GS and ES hydrogen plasmas over wide range of temperature
and pressure. As the
thermodynamic properties depend upon the degree of ionization a
and the partition
function Hf . The partition function diverges due to statistical
weight 22ng n = , the
number of levels to be inserted in it are obtained by a simple
cutoff criterion. Thus,
the degree of ionization a varies with temperature and pressure.
The equilibrium
composition of ground and excited state plasma has thus been
obtained. The
thermodynamic properties e.g. specific heat at constant pressure
pc , the specific heat
at constant volume cv and the isentropic coefficient g vp cc /(=
) for both the ground
and the excited state plasmas has been calculated.
The method of computation of equilibrium composition and the
thermodynamic
properties has been discussed in Section 2.1. In Section 2.2,
Saha equation is used for
evaluating the degree of ionization and hence the equilibrium
composition for ground as
well as excited state hydrogen plasmas. Parameterization of
equilibrium composition for
hydrogen thermal plasma and truncation of partition function
using a cutoff criterion
have been discussed in Section 2.3. The equilibrium composition
of ground and
excited state plasmas has been obtained in Section 2.4 and 2.5.
Variation of degree of
ionizationa with temperature has been discussed in Section 2.6.
The expressions for the
thermodynamic properties for GS and ES hydrogen plasma has been
presented in Section
-
21
2.7. The internal contribution to specific heat has been
separately discussed in Section
2.8. The role of EES on the variation of these properties for
the two cases has been
described graphically in Section 2.9. Finally, the results and
discussions have been
presented in Section 2.10.
2.1 Method of computation
The method of computation starts with the determination of
electronic partition
function which through Saha equation gives the degree of
ionization
)/( Ne nn=a with en and Nn as the number densities of electrons
and nuclei
respectively. At a given temperature, the number of excited
states to be inserted
in the partition function depends upon pressure, thus the
partition function varies
with pressure. A strong decrease of the electronic partition
function with increase
of pressure has been observed. The expressions for the various
thermodynamic
properties have been written in terms of the degree of
ionization and the partition
function.
In order to estimate the effect of including the EES in the
atomic partition
function and its derivatives, we have selected specific heat at
constant pressure
)( prpfp ccc += where pfc and prc are the frozen and reactive
contributions to the
specific heat). The frozen part pfc is further the sum of the
translational and
internal contributions i.e. int2
5cRTcpf += . The ratios pfcc /int and pcc /int have been
worked out for hydrogen thermal plasma in the temperature range
10000–40000 K
and in the pressure range 1-102atm. At p = 10
2atm, the contribution from electronic
excitation is higher than the corresponding contribution from
the translational
energy. Following the same approach, the ratios of specific
heats at constant
pressure pc to the specific heat at constant volume vc i.e. g
(called the isentropic
coefficient) have been studied and the role of electronic
excitation has been
discussed. Electronic excitation appears in the frozen and
reactive part of these
properties with opposite signs respectively. Finally, detailed
discussion of the role
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22
of electronically excited states in affecting the various
thermodynamic properties of
hydrogen plasma has been presented.
2.2 Saha’s equation of ionization
Saha’s equation of ionization for hydrogen plasma is written
as
÷ø
öçè
æ -÷ø
öçè
æ=+
kT
I
fh
mkT
n
nnH
HH
He exp22 2
3
2
p (2.1)
The total number density n is given by eHH nnnn ++= + and the
total pressure
nkTp = . Equation (2.1) can be rewritten in terms of degree
of
ionization a = ÷÷ø
öççè
æ
+ +HH
e
nn
n as
÷ø
öçè
æ -÷ø
öçè
æ=- kT
I
fkT
h
mkTp H
H
exp22
)1(
23
22
2 pa
a
(2.2)
where Hf is the electronic partition function and IH is the
ionization energy of
atomic hydrogen. He nn , and +Hn are the number densities of
electrons, H- atoms and
protons respectively. Total number density n is given by
eHH nnnn ++= +
2.3 Parameterization of equilibrium composition and cutoff
criterion
The equilibrium composition for hydrogen thermal plasma as a
function of temperature
has been obtained by using Saha’s ionization equation, which is
written as
T
atm
eTnx
x /1076.152/372
4102806.3
21
´--´
=-
(2.3)
where x ÷ø
öçè
æ=n
ne is the concentration of ions or electrons and atmn is number
of
atmospheres. Firstly, the equilibrium composition for the ground
state plasma (e, H+, H)
is evaluated by using the above equation and then the atomic
hydrogen has been further
divided into excited states H (n) by Boltzmann law (Equation
(2.5)).
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23
As is well known that the partition function of atomic hydrogen
diverges due to
the degeneracy factor. A simple cutoff criterion based upon
confined- atom (CA) model
is adopted (Capitelli et al., 2003), i.e. by considering excited
states with classical Bohr
radius not exceeding the interparticle distance. The criterion
thus obtained is
3/12
max )/1( nnao ¢= (2.4)
where 0a is the Bohr radius and kTpn /=¢ is the total number
density of hydrogen
atoms. All the excited states with the principal quantum number
maxnn £ should be
considered in the partition function. The maximum number m axn
of excited states at
different pressures in temperature range 10000–50000 K are given
in Table 2.1.
Thus, the partition function Hf depends upon pressure and
temperature. Figure 2.1
displays its variation with temperature at different
pressures.
Figure 2.1. Partition function Hf vs. temperature. Curves (upper
to lower)
represent p=1, 10, 100 & 1000 atm respectively.
Temperature (K)
f H
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24
Table 2.1. The maximum number m axn of excited states to be
inserted in the
partition function at different pressures in temperature range
10000–50000 K.
Temperature
(K)
p=1 atm p=10 atm p=100 atm p=1000 atm
10000 14 9 6 4
12000 14 10 6 4
14000 15 10 7 4
16000 15 10 7 5
18000 16 10 7 5
20000 16 11 7 5
22000 16 11 7 5
24000 16 11 7 5
26000 17 11 7 5
28000 17 11 7 5
30000 17 11 8 5
32000 17 11 8 5
34000 17 12 8 5
36000 17 12 8 5
38000 18 12 8 5
40000 18 12 8 5
42000 18 12 8 5
44000 18 12 8 5
46000 18 12 8 5
48000 18 12 8 5
50000 18 12 8 5
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25
2.4 Equilibrium composition of ground state (GS) plasma
The concentration of electron eX , hydrogen ion +HX &
hydrogen atom HX in the
ground state plasma at p=1, 10& 100 atm respectively have
been evaluated at different
temperatures. It has been observed that += He XX at high
pressure (i.e. p=100 atm) is
less as compared to that at low pressure (p=1or 10 atm) (Singh
et al., 2008) because
ionization of hydrogen atoms take place at a high temperature
with increase in pressure.
2.5 Equilibrium composition of excited state (ES) plasma
Atomic hydrogen in the excited state plasma is divided into the
different possible excited
states depending upon pressure and temperature. We thus obtain
the excited state plasma
(e, H+, H(n)) with n=1, 2, 3 …, m axn where m axn =12, 12 and 7
have been used in the
calculation at p=1, 10& 100atm respectively.
The relative concentration of the ithatomic excited state is
obtained by
)exp()( kT
E
TZ
g
n
nii
T
i -= (2.5)
with å -= kTEi iegTZ /)(
where in , ig and iE are the number density, degeneracy and
energy of the thi atomic
excited state respectively. Tn and )(TZ are the total number
density and electronic
partition function of atomic hydrogen.
Table 2.2, 2.3 & 2.4 present the concentration of electron
eX , hydrogen ion +HX ,
hydrogen atom )1(HX in ground state for excited state plasma at
different
pressures. Concentration of various numbers of excited states
)(nHX vs. temperature at
different pressures are depicted in Figure 2.2. It has been
observed that with increase in
pressure, the concentration of the excited states increases as
well as the maxima of )(nHX
shifts towards higher temperature.
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26
Table 2.2. Concentrations of electron eX , hydrogen ion +HX ,
hydrogen atom )1(HX in the
ground state for excited state plasma at p=1 atm.
p=1 atm, n=12
Temperature (K)
+= He XX
)1(HX
10000 0.0213 0.9570
12000 0.0920 0.8140
14000 0.2310 0.5320
16000 0.3760 0.2380
18000 0.4550 0.0808
20000 0.4840 0.0255
22000 0.4930 0.0084
24000 0.4970 0.0029
26000 0.4990 0.0011
28000 0.4990 0.0011
30000 0.4990 0.0004
32000 0.5000 0.0002
34000 0.5000 0.0001
36000 0.5000 0.0000
38000 0.5000 0.0000
40000 0.5000 0.0000
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27
Table 2.3. Concentrations of electron eX , hydrogen ion +HX ,
hydrogen atom in
ground state )1(HX for excited state plasma at p=10 atm.
p =10 atm, n=12
Temperature (K)
+= He XX
)1(HX
10000 0.0069 0.9860
12000 0.0312 0.9360
14000 0.0903 0.8110
16000 0.1880 0.5980
18000 0.3010 0.3540
20000 0.3920 0.1680
22000 0.4460 0.0683
24000 0.4730 0.0264
26000 0.4860 0.0104
28000 0.4920 0.0043
30000 0.4950 0.0019
32000 0.4970 0.0009
34000 0.4980 0.0005
36000 0.4990 0.0002
38000 0.4990 0.0001
40000 0.4990 0.0001
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28
Table 2.4. Concentrations of electron eX , hydrogen ion +HX ,
hydrogen atom )1(HX
in ground state for excited state plasma at p=100 atm.
p=100 atm, n=7
Temperature (K)
+= He XX
)1(HX
10000 0.0022 0.9960
12000 0.0101 0.9790
14000 0.0306 0.9350
16000 0.0699 0.8490
18000 0.1300 0.7150
20000 0.2050 0.5460
22000 0.2830 0.3750
24000 0.3520 0.2330
26000 0.4040 0.1340
28000 0.4390 0.0740
30000 0.4610 0.0404
32000 0.4750 0.0223
34000 0.4830 0.0126
36000 0.4880 0.0074
38000 0.4920 0.0045
40000 0.4940 0.0028
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29
(a)
(b)
(c)
Figure 2.2. Concentration )(nHX versus temperature. Curves
(lower to upper) represent
)(nHX for (a) n=2-12 at p=1 atm (b) n=2-12 at p=10 atm and (c)
n=2-7 at p=100 atm.
Temperature (K)
)(nHX
0.000
0.002
0.004
0.006
0.008
0.010
0.012
10000 20000 30000 40000
)(nHX
0.0000
0.0005
0.0010
0.0015
0.0020
0.002
10000 20000 30000 40000 Temperature (K)
Temperature (K)
XH(n)
40000 30000 20000 10000
0.02
0.016
0.012
0.008
0.004
0
-
30
0.0
0.2
0.4
0.6
0.8
1.0
1.2
10000 20000 30000 40000
ES
GS
2.6 Variation of degree of ionization with temperature and
pressure
Using computer program based upon Equation (2.2), degree of
ionization a has
been evaluated at different pressures over a wide range of
temperature (i.e. from
10000-50000 K). Its variation with temperature has displayed in
Figure 2.3 for both GS
and ES plasmas at p=1& 100 atm. It is clear that with
increase of pressure, the degree of
ionization for ES plasma become less than that for GS
plasma.
(a)
(b)
Figure 2.3. Degree of ionization a vs. temperature for GS and ES
(upper and lower)
hydrogen plasma at (a) p=1 atm and (b) p=100 atm.
ES
Temperature (K)
GS
a
a
Temperature (K)
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31
2.7 Thermodynamic properties: Expressions
The expressions for thermodynamic properties such as enthalpy,
specific heat at constant
pressure and isentropic coefficient for GS and ES hydrogen
plasma has been presented.
The enthalpy of three component plasma (H, H+, e) has been
defined and its
dependence upon the electronic partition function has been
discussed in Section
2.7.1. Expressions for the specific heat at constant pressure pc
and the isentropic
coefficient vp cc /(=g ) have been given in Sections 2.7.2 &
2.7.3.
2.7.1 Enthalpy
Among the thermodynamic properties, the most important for
plasma modelling are the
enthalpy H, its derivative with respect to temperature T, and
the specific heat at constant
pressure pc . The peaks on these curves correspond to
dissociation mechanisms at low or
intermediate temperatures (around 4000 K for H2, and 7000 K for
N2 at atmospheric
pressure) and to ionization at high temperatures (mainly around
15000K for many species
having an ionization energy around 10–15 eV).
Starting with one mole of atomic hydrogen, then at a given
temperature H,
H+ and e have aa ,1- and a moles respectively , where a is the
degree of
ionization defined by
== Ne nn /a++ HH
e
nn
n , (2.6)
where HN nn , and +Hn are the number densities of nuclei, H
atoms and protons
respectively. Total enthalpy of this three component plasma (H ,
H+
, e) is
eH HHHH aaa ++-=+)1( (2.7)
where +HH HH , and eH are the molar enthalpies of H atoms,
protons and electrons
given by
HH
e
HH
ID
RTH
RTH
DERTH
++=
=
++=
+
22
5
2
5
22
5
-
32
where RT2
5 , D , IH and EH are the translational molar enthalpy of each
species,
dissociation energy of H2 , the ionization energy of H atoms and
the electronic
energy of atomic hydrogen respectively . Thus, 2
D and HI
D+
2 are the chemical
enthalpy of atomic hydrogen and protons. Thus, total enthalpy of
hydrogen plasma is
given by
HH ID
ERTH aaa ++-++=2
)1()1(2
5
The second term in the above equation is the internal enthalpy
with EH given by
÷ø
öçè
æ¶
¶=
T
fRTE HH
ln2 (2.8)
where Hf is the internal partition function defined by
)exp(å -=kT
gf nnHe
(2.9)
with
÷ø
öçè
æ -=2
11
nI Hne
and gn = 2n2
where n denotes the principal quantum number of atomic hydrogen
and gn is the
statistical weight.
2.7.2 Specific heat at constant pressure
When the derivative of total enthalpy H is taken with respect to
temperature T at
constant pressure , the dependence of the degree of ionization
with temperature
must be considered. Thus, we define two specific heats, the
first one, called frozen
specific heat is obtained by taking derivative of the total
enthalpy with respect to
temperature at constant a whereas in the second one, called
reactive specific heat
of the plasma, this constraint is eliminated.
Thus, the total specific heat of the hydrogen plasma is given
by
prpfp ccc += (2.10)
where the frozen specific heat pfc is
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33
VH
p
pf cRTT
Hc )1()1(
2
5aa -++=÷
ø
öçè
æ¶
¶=
(2.11)
where VHc is the internal specific heat of atomic hydrogen and
is given by
úû
ùêë
é
¶
¶+
¶¶
=÷ø
öçè
涶
=T
f
T
fR
T
Ec HH
V
HVH
ln
ln
ln
ln2
2
(2.12)
with Hf as the electronic partition function of atomic
hydrogen.
The reactive specific heat is given by
p
HH
pp
prT
EIRTT
Hc ÷
ø
öçè
涶
÷ø
öçè
æ -+=÷ø
öçè
涶
÷ø
öçè
涶
=aa
a a 25
,
22
2 2
5
2
)1(1÷ø
öçè
æ -+-
= HHpr EIRT
caa
(2.13)
where the degree of ionization α and its derivative (∂α/∂T)p
have been obtained
from the Saha’s equation.
2.7.3 Isentropic coefficient ( )g For the isentropic coefficient
( )vp cc /=g , we have
úúúú
û
ù
êêêê
ë
é
-+++-+-
-
-+++-+-=
VHHH
VHHH
cRERTIRT
cRERTIRT
)1(2
3)1()
2
3(
)2(
)1(1
)1(2
5)1()
2
5)(1(
1
2
2
2
2
aaaaa
aaaag
(2.14)
2.8 Internal specific heat
The internal specific heat of atomic hydrogen VHc depends upon
first and second
derivative of the partition function and can be written as
( ) 222 600,11 ÷ø
öçè
æ´-=t
EER
cVH
which is the product of two factors. The first factor is sharp
peaked curve at low
pressure due to large number of excited states whereas it is a
flattened curve at
high pressure due to smaller number of excited states. The
second factor is
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34
parabolically decreasing with temperature. The variation of
R
cVH at low and high
pressures is presented in Figure 2.4. The internal contribution
intc to the frozen part
pfc of the specific heat is again the product of two factors (i)
(1 -a
fraction of H atoms and (ii) R
cVH . The variation of R
cint with temperature is depicted
in Figure 2.5.
Figure 2.4. R
cVH vs. temperature. Curves (lower to upper) represent p=1, 10,
100
& 1000 atm respectively.
Figure 2.5. R
cint vs. temperature. Curves a, b, c & d represent p=1, 10
&100atm respectively.
Temperature (K)
R
cint
Temperature (K)
R
cVH
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35
2.9 Effect of electronically excited states on thermodynamic
properties
The role of electronically excited states (EES) in affecting
thermodynamic
properties of hydrogen thermal plasma has been examined over a
wide range of
temperature and pressure by taking GS and ES plasmas. The
enthalpy for GS and ES
plasma, total specific heat at constant pressure pc along with
its frozen and
reactive contributions have been worked out in detail for both
the cases. Their
pressure and temperature dependence has been depicted
graphically. The ratios pfcc /int
and pcc /int represent the contribution of electronic excitation
to the frozen and total
specific heats respectively. The isentropic coefficients g
exhibit strong dependence on
electronic excitation. Section 2.9.1 describes ratio of enthalpy
for GS and ES plasmas. In
Section 2.9.2 the frozen, reactive and total specific heat at
constant pressure for both the
cases has been described. In order to discuss the role of EES on
the specific heat at
constant pressure, internal contribution to the frozen and total
specific heat has been
obtained in Section 2.9.3.The total isentropic coefficient has
been obtained at different
pressures in Section 2.9.4, which shows strong minima
(especially at high pressures).
2.9.1 Enthalpy
The enthalpy for the hydrogen plasma has been evaluated. The
effect of electronically
excited states on the enthalpy has been graphically depicted by
the plot of ratio of
Figure 2.6. Ratio of enthalpy GS
ES
H
Hfor GS and ES hydrogen plasma at different
pressures.
p=1000atm
p=100atm
p=10atm
p=1atm
Temperature (K)
GS
ES
H
H
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36
enthalpy for GS and ES plasmas at different temperatures and
pressures and is displayed
in Figure 2.6.
2.9.2 Specific heat at constant pressure
In order to estimate the effect of electronically excited states
on the frozen pfc ,
reactive contributions prc and the total specific heat pc of
hydrogen thermal plasma,
the number of excited states to be included in the partition
function is first
determined using a simple cutoff criterion (based on confined
atom (CA) model).
Then a computer program has been developed to compute the degree
of
ionizationa and the various contributions to the specific heat
at different pressures
in the temperature range 10000 -50000 K. The results thus
obtained for pfc and prc are
presented in Figures 2.7 & 2.8 for p=1, 10& 100 atm. The
comparison of results of
pc for the excited state (ES) plasma with those of the ground
state (GS) plasma
have been made in Figure 2.9 which displays the variation of the
ratio )(
)(
GSc
ESc
p
p with
temperature for p=1, 10, 100 and 1000 atm respectively.
2.9.3 Internal contribution to specific heat
The role of electronic excitation on the frozen contribution of
specific heat intc
has been discussed and is given by:
VHcc )1(int a-=
Where the degree of ionization a and the internal specific heat
of atomic hydrogen
VHc depend upon electronic partition function. Comparison of in
tc with pfc and pc has
been attempted by plotting the ratios pfcc /int and pcc /int
with temperature at
different pressures in Figures 2.10& 2.11 respectively. It
may be mentioned here
that pfc is the sum of internal and translational contributions
i.e.
.)1(2
5intcRTc pf ++= a
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37
(a)
(b)
(c)
Figure 2.7. Frozen specific heat pfc vs. Temperature for ES and
GS hydrogen
plasmas at (a) p=1 atm (b) p=10 atm and (c) p=100 atm.
Temperature (K)
c p
f (J
/g/K
) c p
f (J
/g/K
)
GS
ES
Temperature (K)
c pf
(J/g
/K)
Temperature (K)
-
38
(a)
(b)
(c)
Figure 2.8. Reactive specific heat prc vs. temperature for ES
and GS hydrogen
plasmas at (a) p=1 atm (b) p=10 atm and (c) p=100 atm.
Temperature (K)
c pr
(J/g
/K) ES
Temperature (K)
c pr(
J/g/K
)
ES
Temperature (K)
c pr(
J/g/K
)
-
39
Figure 2.9. )(
)(
GSc
ESc
p
p vs. temperature. Curves a, b, c & d represent p= 1, 10,
100& 100
atm respectively.
Figure 2.10. pfcc /int vs. temperature. Curves a, b, c & d
represent p=1,10,
100&1000 atm respectively.
Temperature (K)
)(
)(
GSc
ESc
p
p
a
d
b
c
Temperature (K)
pfc
cint
-
40
Figure 2.11. pcc /int vs.temperature.Curves (lower to upper)
represent p=1, 10,
100& 1000 atm respectively.
2.9.4 Isentropic coefficient
Following the similar procedure, comparison of results for total
isentropic
coefficient g have been reported for both GS and ES plasmas in
Figures 2.12 at p=1
and 100atm.
pc
c int
Temperature (K)
-
41
(a)
(b)
Figure 2.12. Isentropic coefficient g vs. Temperature for ES and
GS hydrogen plasmas
at (a) p=1 atm and (b) p=100 atm.
g
Temperature
g
ES
Temperature (K)
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42
2.10 Results and Discussion
(i) From Figure 2.2, it has been observed that as pressure
increases, the population of
EES increases due to the fact that ionization of atoms occurs at
high temperature and
hence concentration of EES increases.
(ii) It has been observed from Figure 2.7, that for the ground
state (GS) plasma
Rcpf2
5)1( a+= and it behaves in a similar way as that of the degree
of ionization
a with temperature for all pressures whereas the peak observed
for the excited
state (ES) plasma is due to the addition of cint to the above
expression, thereby
indicating the role of EES. With increase of pressure,
ionization shifts towards
high temperature thereby, shifting the peak of cint for ES
plasma towards high
temperature (Figure 2.5).
(iii)The value of prc for ES plasma is lower than that of the GS
plasma in the
region where electronic excitation is dominant (Figure 2.8).
Ground state results
overestimate the reactive contribution by about 15% at p=100
atm. This is due to the
fact that the electronic energy EH of the atomic hydrogen
appears in the
expression for prc with negative sign [Equation (2.13)].
(iv) At low pressure p=1 atm, the internal contributions in pfc
and prc cancel each
other leading to a sort of compensation which is not observed at
high pressures
p=10-103atm.The deviation of results for GS and ES plasmas are
more
emphatically displayed in Figure 2.9 where the ratio )(
)(
GSc
ESc
p
p does not behave
monotonically, rather maxima and minima are observed at a given
pressure. The
differences strongly increase with increase of pressure. But for
1£p atm these
differences are negligible i.e. the ratio is practically
independent of temperature. This
fact mislead researchers in past for not considering
electronically excited states in
calculating thermodynamic properties of LTE plasmas.
(v) Regarding the estimation of internal contribution to
specific heat, Figure 2.10
displays maxima of pfc
cint at all pressures. At p=103 atm, this ratio is greater than
0.5
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43
i.e. contribution due to electronic excitation is higher than
the translational one. On the
other hand, the ratio pcc /int (Figure 2.11) behaves in a
similar way to that of
pfcc /int but the corresponding maxima decrease due to the
negative role played
by the electronic excitation in the reactive contribution prc of
the specific
heat. At p=103
atm, pcc /int =0.25 indicating thereby that the internal
contribution is
not negligible at high pressures.
(vi) In the absence of electronic excitation i.e. for the ground
state plasma fg
=5/3. The observed minima are due to the electronic excitation
and strongly depend
upon pressure ( 25.1=fg at p=10
3 atm). The contribution of electronic excitation tends
to disappear in the total isentropic coefficient g which include
translation, electronic
and reactive components (Figure 2.12 (a)). In fact, the computed
values for GS and
ES plasmas differ by not more than 7.5% due to some compensation
in different
terms of [Eq. (2.14)]. Thus, the contribution of electronic
excitation makes its
presence felt to total specific heat at high pressures but not
as emphatically as in the
case of frozen specific heat.
Thus, thermodynamic properties of a LTE hydrogen plasma depend
upon the
number of electronically excited states (EES) to be inserted in
the partition
function. This inclusion of EES increases the partition function
thereby affecting
the degree of ionization and the internal specific heat of high
temperature atomic
hydrogen plasmas especially at high pressures. Its strong
dependence on the frozen
specific heat of the plasma mixture has been observed but the
effect is negligible
on the total specific heat due to compensation between the
frozen and reactive
contributions to the total specific heat at low pressure. This
compensation fails at
pressures higher than one atm. As the equilibrium properties
such as enthalpy and
specific heat depend upon EES, therefore it is of interest to
see how these EES affect
various transport properties of thermal plasmas and the same
have been discussed in the
subsequent chapters.
