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UMYU Scientifica, Vol. 3 NO. 2, June 2024, Pp 146 – 158
146
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et al.,
/USci, 3(2): 146 – 158, June 2024
ORIGINAL RESEARCH ARTICLE
Exploring the Thermodynamic Characteristics of Isoelectronic Diatomic
Interstellar Molecular Species: Oxygen and Sulfur Containing Specie
Enock O. Oladimeji1,2 , Emmanuel E. Etim3* , Emmanuel C. Umeh1,4 , John P. Shinggu3,
Oluwatimilehin J. Oluwadare5; Oluwakemi M. Odeyemi6 and Humphrey S. Samuel3
1Theoretical Physics Group, Department of Physics, Federal University Lokoja, Lokoja, Nigeria.
2Institute of Physical Research and Technology, Peoples’ Friendship University of Russia, Moscow, Russia
3Department of Chemical Sciences, Federal University Wukari, Wukari, Nigeria.
4Department of Physics, Nigerian Defence Academy, Kaduna, Nigeria.
5Department of Physics, Federal University Oye-Ekiti, Oye-Ekiti, Nigeria.
6Department of Physics, Joseph Ayo Babalola University, Ikeji-Arakeji, Nigeria.
INTRODUCTION
A fascinating area of astrophysics that provides insight
into the intricate chemistry in the wide reaches of space is
the study of interstellar molecular species. These
extrasolar chemicals are essential for the development of
stars, planets, and even the beginnings of life. These
interstellar molecular species have several unique
thermodynamic characteristics, especially in the case of
isoelectronic diatomic molecules. Scientists may
investigate the complex interplay between energy,
temperature, and molecular behavior inside the interstellar
medium by learning about the thermodynamic
characteristics of these isoelectronic diatomic interstellar
molecules. These characteristics include entropy,
enthalpy, Free energy, heat capacity, internal energy,
etc.… (Cernicharo, Agúndez et al., 2021; Etim & Arunan,
2016, 2017; Shinggu et al., 2023.; Sil et al., 2018).
Isoelectronic diatomic interstellar molecular species,
regardless of the atomic elements of the molecules
involved, are molecules made up of two atoms with the
same amount of electrons. Diatomic interstellar
molecules are crucial in determining how the interstellar
medium is chemically structured because they have unique
electrical structures and may combine to generate a variety
of stable and reactive compounds of Oxygen (O) and
sulfur (S), bringing intriguing dynamics into play. These
molecules play pivotal roles in shaping the chemical
landscape of the cosmos, influencing the formation of
stars planets, and potentially the emergence of life itself.
Investigating the unique thermodynamic characteristics of
these molecules offers insights into their stability,
reactivity, and spectroscopic properties in the diverse
environments of interstellar space. Through the
application of thermodynamic principles, scientists aim to
unravel the complex interplay of energy, temperature, and
Correspondence: Dr. Emmanuel E. Etim. Department of Chemical Sciences, Federal University Wukari, Wukari, Nigeria.
emmaetim@gmail.com. Phone Number: +234 907 919 2231.
How to cite: Oladimeji, E. O., Etim, E. E., Umeh, E. C., Shinggu, J. P., Oluwadare, O. J., Odeyemi, O. M., & Samuel, H. S.
(2024). Exploring the Thermodynamic Characteristics of Isoelectronic Diatomic Interstellar Molecular Species: Oxygen and
Sulfur Containing Specie. UMYU Scientifica, 3(2), 146 – 158. https://doi.org/10.56919/usci.2432.016
ISSN: 2955 – 1145 (print); 2955 – 1153 (online)
https://doi.org/10.56919/usci.2432.016
A periodical of the Faculty of Natural and Applied Sciences, UMYU, Katsina
ABSTRACT
Interstellar molecular species, particularly isoelectronic diatomic molecules, exhibit distinct
thermodynamic traits, setting them apart from other molecular species. This study
investigates the thermodynamic properties of isoelectronic diatomic interstellar molecular
species containing oxygen and sulfur atoms, employing computational methods to analyze
entropy, free energy, heat capacity, and internal energy across a spectrum of interstellar
temperatures. Graphical representations highlight intriguing trends, revealing Oxygen and
sulfur-containing molecules' earlier responsiveness to temperature changes compared to
oxygen counterparts. Notably, molecule size emerges as a key determinant, with larger mass
molecules exhibiting higher entropy, free energy, and heat capacity. We showed the
isoelectronic effect of these Sulfur and Oxygen containing molecular species (OH, SN, CO,
CS, SiO, SiS, FeO, FeS, PO, PS O2, OS, ZnO, ZnS, TiO, TiS) on several interstellar molecules
at temperatures ranging from to (i.e., from the coldest place in the universe to
the mean temperature of the interstellar medium). These findings offer valuable insights into
the thermodynamic behavior of interstellar molecular species, paving the way for future
research on the role of oxygen and sulfur atoms in complex molecular systems.
ARTICLE HISTORY
Received March 10, 2024.
Accepted June 06, 2024.
Published June 24, 2024.
KEYWORDS
Interstellar medium (ISM);
Atoms-ISM; Isoelectronic
molecules; Laws of
thermodynamics, Astrochemistry
© The authors. This is an Open
Access article distributed under
the terms of the Creative
Commons Attribution 4.0 License
(http://creativecommons.org/
licenses/by/4.0)
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molecular behavior in isoelectronic diatomic interstellar
species, shedding light on their evolution and survival
(Agúndez et al., 2021; Cernicharo, Cabezas, Agúndez, et al.,
2021; Cernicharo, Cabezas, Endo, et al., 2021; Etim, 2023;
Etim, 2017; Etim et al., 2021; Etim & Arunan, 2016).
Oxygen and sulfur-based diatomic interstellar species
frequently develop through intricate chemical processes,
including ion-molecule interactions, photodissociation,
and radiative association mechanisms (Heays et al., 2017).
Theoretical simulations, laboratory studies, and
astronomical observations all help to reveal the
fundamental processes that led to their genesis and the
circumstances that made them possible. Scientists
investigate these diatomic species to understand the
complicated interstellar chemistry and learn more about
the development of molecular complexity (Samuel et al.,
2023). Diatomic interstellar species that include Oxygen
and sulfur also have significant astrophysical
consequences. For instance, sulfur monoxide (SO) and
sulfur dioxide (SO2) play important roles in the chemistry
of sulfur in interstellar clouds, influencing the creation of
other compounds containing sulfur. Diatomic species
that include Oxygen, such as the oxygen molecule (O2) and
carbon monoxide (CO), serve as important tracers of the
physical environments (Booth et al., 2023; Cabezas et al.,
2022; Samuel et al., 2023; Etim, Gorai, Das, & Arunan,
2018; Etim, Gorai, Das, Chakrabarti, et al., 2018; Mondal
et al., 2021; Etim and Arunan, 2017; Etim et al., 2017a;
Etim et al., 2017b).
Thermodynamics provides a framework for analysing and
understanding the changes in energy and matter, which
also offers insights into molecules' physical characteristics
and behavior. Scientists study these molecules' stability,
reactivity, and spectroscopic properties in the particular
conditions of interstellar space by applying
thermodynamic concepts to isoelectronic diatomic
interstellar species. Diatomic interstellar molecules
containing Oxygen and sulfur are fascinating research
targets because of their distinctive thermodynamic
features, such as entropy, enthalpy, free energy, internal
energy, and heat capacity. These qualities control the
stability and reactivity of these molecules in the interstellar
medium through factors including bond energies,
enthalpy changes, and equilibrium constants.
Investigating these thermodynamic properties contributes
to our knowledge of how these species develop, evolve,
and survive in various interstellar environments, from
dense molecular clouds to diffuse interstellar medium
(Samuel et al., 2023).
This work aims to derive the partition functions and
establish suitable thermodynamic properties for
isoelectronic diatomic interstellar molecular species
containing Oxygen and Sulfur. The parameters obtained
in this study will find applications in modeling a wide
range of astrophysical environments, such as the
interstellar medium, protoplanetary disks, star-forming
regions, etc. The outline of our paper is as follows: In
Section 2, we present the methodology of our work, while
in Section 3, we discuss the derived results, and we retire
with concluding remarks in Section 4.
THEORY AND CALCULATION
For all the presented quantum chemical computations, the
GAUSSIAN 09 software suite has been a very useful tool
(Frisch et al., 1999). This program's computation of the
partition function is succinctly described as explored by
(McQuarrie & Simon, 1999) and in the white paper titled
"Thermochemistry in GAUSSIAN" where they provided a
detailed description of the equations utilized in the
program's calculation of the various partition functions
(Ochterski, 2000) Following this trend, we also applied
the GAUSSIAN 09 software suite to derive the partition
functions (i.e., total , translational , electronic ,
rotational and vibrational ) of isoelectric interstellar
molecules made of Sulfur and Oxygen atoms as shown in
Table 1 (OH, SN, CO, CS, SiO, SiS, FeO, FeS, PO, PS O2,
OS, ZnO, ZnS, TiO, TiS). The partition functions were
computed through the optimization and frequency
calculations of the Oxygen and sulfur-containing
interstellar molecules using the popular compound model
G4, the fourth-order Gaussian theory. Since the particles
are assumed to not interact, the equations in this study
apply to ideal gases for all the systems in consideration.
The degree of the system's non-ideality under examination
determines the error induced by this assumption.
However, it is necessary to state that since the systems
under discussion are all isomers with several similar
features, this inaccuracy can be negligible (Etim & Arunan,
2017). For this work, we shall remind our readers of the
textbook definition of partition functions used in deriving
our data. We begin with the translational partition
function, which is defined as:
Where = translational partition function; = mass of
the molecule; = Boltzmann constant; =Temperature;
=Plank’s constant; =pressure. The equation for
determining the electronic partition function is given
as:
As outlined in the paper, is the energy of the level,
and is the degeneracy of the energy level. The first
(which is greater than ) and higher excited states are
assumed to be inaccessible at any temperature, thereby
reducing to:
This equation is simply the electronic spin multiplicity of
the molecule under consideration. Equations (4) and (5)
are used in computing the rotational partition function for
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linear molecules and nonlinear polyatomic molecules
respectively.
The parameter in Eqs. (4) and (5) is called the symmetry
number, which is the number of indistinguishable
orientations of the molecule. The quantity ,
is the characteristic temperature for rotation (in the ,
or plane).
For all the calculations reported here, only equilibrium
structures were considered with no imaginary frequency.
Hence, the vibrational partition function is computed
considering real modes only. The zero-point energy is
included in computing the vibrational partition function;
therefore, the first vibrational level is chosen to be the zero
of the energy and the partition function for each
vibrational level is given as
while the overall vibrational partition function is given as
where is the index of vibrational modes and is the
characteristic temperature for vibration . (McQuarrie &
Simon, 1999)
Table 1:
Molecular partition function(q) of vibrational temperature (k) of isoelectronic interstellar molecular species
containing Oxygen and Sulfur.
Molecules
Partition Function (KCal/Mol)
Vibrational
Temprature (K)
HO
0
0.889
0.592
5.49
6.971
5524.89
HS
0
0.889
0.592
4.002
5.483
4027.81
CO
0
0.889
0.592
3.041
4.522
4153.35
CS
0
0.889
0.592
1.901
3.382
1906.76
SiO
0
0.889
0.592
1.703
3.185
1703.03
SiS
0
0.889
0.592
1.135
2.617
1083.89
FeO
0
0.889
0.592
1.38
2.861
1359.77
FeS
0
0.889
0.592
0.995
2.477
911.99
PO
0
0.889
0.592
3.622
5.103
3644.91
PS
0
0.889
0.592
1.919
3.401
1925.78
O2
0
0.889
0.592
1.814
3.295
1817.44
OS
0
0.889
0.592
1.409
2.89
1391.76
TiO
0
0.889
0.592
1.698
3.179
1697.31
TiS
0
0.889
0.592
1.150
2.631
1100.71
OH+
0
0.889
0.592
4.644
6.125
4674.16
SH+
0
0.889
0.592
3.857
5.338
3881.60
NO+
0
0.889
0.592
3.098
4.576
3117.43
NS+
0
0.889
0.592
1.803
3.285
1806.61
Following the description of the Partition functions ,
(i.e., translational, electronic, rotational, and vibrational
motion) of these isoelectronic interstellar molecules, the
application of statistical mechanics to derive the
thermodynamic properties of these interstellar species is
apparent since all these properties may be defined in terms
of the partition function (Franz, 2014; Garanin, 2017;
Gosachinskij & Morozova, 1996; Ochterski & Ph, 2000;
Pathria & Paul, 2011; Rudoy & Oladimeji, 2017) The
equations used in the calculation of these thermodynamic
properties such as entropy , internal energy , free
energy , and heat capacity resulting from (total ,
translational , electronic , rotational and
vibrational ) partition functions as stated below are
described in detail in ‘Thermochemistry in Gaussian”
(Ochterski, 2000) which are equivalent to standard texts
on thermodynamics (Adkins, 1979; Bipin K. Agarwal &
Eisner, 1989; Franz, 2014; Gallavotti, 1999; Garanin,
2017; Matus et al., 2019; Pathria & Beale, 2011; Pathria &
Paul, 2011; Penrose, 1979; Sears & Salinger, 1982; Seddon
& Gale, 2002; Shell, 2015; Smirnov, 2006; Stowe, 2007)
These terms are grouped according to the translational,
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electronic , rotational and vibrational motion is
derived by simply using the relations:
(8)
(9)
(10)
respectively (University, 2008; Vaz, 2004) Where
=number of moles, =Boltzmann constant, =gas
constant, =volume and =Temperature.
In performing the computation of these properties, we
observe the interstellar isomers at different temperatures
from (the temperature of a Black hole) up to
(the mean temperature of interstellar gas)
(Chiang et al., 2020; Goldsmith et al., 1969) at constant
pressure since the temperature plays a huge factor in
observing thermodynamic properties and the thermal
pressure of this region is at equilibrium. In section 3, we
evaluate the numerical values of the thermodynamic
properties of OH, SN, CO, CS, SiO, SiS, FeO, FeS, PO,
PS O2, OS, TiO, TiS, ZnO, ZnS, NS, and NO and present
the results with relevant discussion.
RESULTS AND DISCUSSION
Following the method of statistical thermodynamics
introduced in Section 2 to calculate the thermodynamics
properties of isoelectric molecules made of Sulfur and
Oxygen atoms located in the interstellar medium, we
further analyzed and discussed our results by computing
these properties, i.e., the entropy , Free Energy , Heat
capacity , and Internal energy of interstellar molecules
of interest against several values of interstellar temperature
.
We begin our analysis with entropy’s relation to
temperature , as illustrated in Figure 1. It is observed
that all the molecules initially remain relatively
unresponsive to the increase in temperature till
where ’s entropy level began to rise as other
molecules followed at a later temperature increase. This
pattern was observed among all the interstellar molecules
but at different temperatures and entropic values. The
molecule appears to have the best response to
temperature increase while is the least. For every
observed molecule, the molecules that contain the Sulfur
atom react earlier to temperature increase than their
counterparts made up of Oxygen atoms (Heay et al., 2017).
which has the smallest mass, has the least entropy,
while has the most mass conversely has the highest
entropy. This shows that the higher the mass, the higher
the molecule's entropy. The continuous increase in
entropy as the temperature increases indicates its
agreement with the second law of thermodynamics, often
referred to as the generalized second law of
thermodynamics (GSLT), as this increase is from the
enhanced thermal energy of the molecules, which leads to
increased disorderliness in the form of random motion
(Nammas, 2018; Tu et al., 2019).
As shown in Figure 2, the free energy of all the molecules
reduces linearly with increased temperature. This is most
evident in oxygen molecules and least in which at
, has reduced to and
respectively. At (the approximate
temperature of our sun’s surface), the free energy has
reduced to and
respectively. The plot also depicts that OH, which has the
smallest mass, has the least free energy, while FeS has the
most mass and, hence, the highest free energy (Shinggu et
al., 2023b; Alahira et al., 2024). This divergence of the Free
energy continues with an increase in temperature,
especially at the (the mean temperature of interstellar gas).
To observe the Free Energy of molecules in this region,
the region is assumed to be isolated and at constant
pressure , volume . In general, it appears that molecules
made up of Oxygen tend to lose their initial Free Energy
faster than their Sulfur-contained counterparts as the
temperature increases. This means the lower the mass, the
faster the free energy is lost.
The reaction of the molecule’s Internal energy in
relation to Temperature as illustrated in Figure 3,
appears to be the most interesting result in this work. The
molecule appears to react early with an increase in
temperature, such that at , its internal energy
is while other molecules remain unreactive
till . However, the rate of the Internal energy
remained steady from . This pattern appears to
reverse with other molecules, especially which is the
least reactive molecule at lower temperatures, even till
. However, from , we observe a
spontaneous increase in the value of Internal energy
from the least reactive molecules, while the most reactive
molecules at lower temperatures maintained a steady
increase. This led to total reversal at higher temperatures.
Following the premise that the interstellar region is
observed as a closed system through which energy can be
transferred, the volume of matter that passes through it is
very insignificant in relation to the vast size of the
interstellar region. As shown in Figure 4, at absolute zero,
the heat capacity of all the observed molecules remains
non-zero, as expected for a non-adiabatic system.
However, the rate of increase in the molecule’s heat
capacity in relation to temperature differs. The and
remain the most reactive molecules, while the least
reactive are the and , this differences between the
most reactive and least reactive molecules continues to
widen as their value diverges. This high reactivity of
is due to its large mass compared to the other molecules,
meaning the increase in mass leads to an increase in the
heat capacity of a molecule. Figure 4 is a graphical
representation that indicates that among the substances
plotted, , characterized by the smallest mass, exhibits
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the lowest heat capacity. Conversely, , with the
greatest mass among the molecules considered,
demonstrates the highest capacity. It is necessary to note
that the heat capacity for all the sulfide-containing
molecules responds better to an increase in temperature
than their -oxide counterpart.
Figure 1: Plot of Entropy (S) against Temperature (T).
Figure 2: Free Energy (F) plot against Temperature (T).
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CONCLUSION
In this work, we observed the thermodynamic properties
of isoelectronic diatomic interstellar molecular species,
Oxygen, and Sulfur containing species by using
GUASSIAN 09 to derive the partition functions (total ,
translational , electronic , rotational and
vibrational ), the thermodynamic properties, e.g.,
entropy , internal energy , free energy , and heat
capacity in relation to the temperature from (the
temperature of a Black hole) up to (the mean
temperature of interstellar gas) was further computed and
discussed. The thermodynamic analysis revealed
intriguing trends among isoelectric molecules in the
interstellar medium. Entropy (S) increased with
temperature, indicating compliance with the second law of
thermodynamics. Molecules with Sulfur atoms exhibited
earlier reactivity to temperature changes, potentially linked
to higher mass and entropy. Free Energy (E) decreased
linearly with temperature, with Oxygen-containing
molecules showing a faster decline, highlighting the
impact of mass on energy reduction rates. Internal energy
(U) displayed dynamic behavior, with FeS reacting early
and steadily increasing, while other molecules showed
delayed reactions and reversal at higher temperatures,
suggesting complex molecular interactions. Heat capacity
(C) varied significantly among molecules, with heavier
molecules like FeS and SiS exhibiting higher reactivity,
emphasizing mass's influence on capacity, particularly
evident with increasing temperature. The increase in mass
of the molecules leads to a slower loss in free energy The
data computed will be useful in future thermodynamic
studies on the effect of Oxygen and Sulfur atoms in
interstellar molecular species made up of more than 2
atoms.
CONFLICT OF INTEREST STATEMENT
The authors declare that there is no conflict of interest.
FUNDING DECLARATION
The authors have no relevant financial or non-financial
interests to disclose.
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APPENDIX 1
Table 2:
Mean thermodynamic properties of interstellar molecules with temperature
Temp
Internal Energy
Free Energy
Entropy
Heat Capacity
0
0
0
21.7861495
10
207.861495
-1.33559E-22
23.76350195
21.7861495
20
415.7229902
-2.67118E-22
23.76350195
21.7861495
30
466418.0708
-4.00677E-22
23.76350195
21.7861495
40
6.47089E+11
-5.34236E-22
23.76350195
21.7861495
50
3.1378E+15
-6.67795E-22
23.76350195
21.7861495
60
8.98946E+17
-8.01354E-22
23.76350195
21.7861495
70
5.11459E+19
-9.34913E-22
23.76350196
21.7861495
80
1.05954E+21
-1.06847E-21
23.76350206
21.7861495
90
1.11922E+22
-1.20203E-21
23.76350302
21.78614951
100
7.37819E+22
-1.33559E-21
23.76350831
21.78614954
200
3.5785E+26
-2.67118E-21
23.77977112
21.78635577
300
6.07693E+27
-4.00677E-21
23.9573471
21.78965226
400
2.52769E+28
-5.34236E-21
24.39779454
21.80071921
500
6.02325E+28
-6.67795E-21
25.02716906
21.82086771
600
1.08935E+29
-8.01354E-21
25.74668996
21.84893997
700
1.6845E+29
-9.34913E-21
26.49144065
21.88324478
800
2.36208E+29
-1.06847E-20
27.22654967
21.92230041
900
3.1023E+29
-1.20203E-20
27.93523665
21.96496729
1000
3.89056E+29
-1.33559E-20
28.61048774
22.01040285
2000
1.29809E+30
-2.67118E-20
33.68748559
22.53437127
3000
2.28035E+30
-4.00677E-20
36.92362169
23.10055007
4000
3.28229E+30
-5.34236E-20
39.2676277
23.67807472
5000
4.29226E+30
-6.67795E-20
41.10065483
24.26022164
6000
5.30626E+30
-8.01354E-20
42.6044212
24.84469726
7000
6.32258E+30
-9.34913E-20
43.87877117
25.43050876
8000
7.34035E+30
-1.06847E-19
44.98425117
26.01715708
9000
8.3591E+30
-1.20203E-19
45.96028719
26.60436411
10000
9.37852E+30
-1.33559E-19
46.83396566
27.19196261
20000
1.95877E+31
-2.67118E-19
52.58963151
33.07657208
30000
2.98051E+31
-4.00677E-19
55.95946906
38.96589095
40000
4.00244E+31
-5.34236E-19
58.35090499
44.85638806
50000
5.02447E+31
-6.67795E-19
60.20599854
50.74735655
60000
6.04653E+31
-8.01354E-19
61.72178181
56.63856075
70000
7.06861E+31
-9.34913E-19
63.0033879
62.52989965
80000
8.09071E+31
-1.06847E-18
64.11358156
68.42132273
90000
9.11282E+31
-1.20203E-18
65.09285114
74.31280194
100000
1.01349E+32
-1.33559E-18
65.9688435
80.20432043
200000
2.03563E+32
-2.67118E-18
71.73191302
139.1203697
300000
3.05777E+32
-4.00677E-18
75.1031225
198.0368905
400000
4.07991E+32
-5.34236E-18
77.49503868
256.9535292
500000
5.10205E+32
-6.67795E-18
79.35035452
315.870215
600000
6.12419E+32
-8.01354E-18
80.86625855
374.7869243
700000
7.14634E+32
-9.34913E-18
82.14793745
433.7036472
800000
8.16848E+32
-1.06847E-17
83.25817837
492.6203784
900000
9.19062E+32
-1.20203E-17
84.23748035
551.5371153
1000000
1.02128E+33
-1.33559E-17
85.11349589
610.4538561
2000000
2.04342E+33
-2.67118E-17
90.87663951
1199.621351
