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Physics:Spin isomers of hydrogen - HandWiki
Physics:Spin isomers of hydrogen
From HandWiki
Molecular hydrogen occurs in two isomeric forms, one
with its two proton nuclear spins aligned parallel
(orthohydrogen), the other with its two proton
spins aligned antiparallel (parahydrogen).[1]
These two forms are often referred to as spin
isomers.[2]
Parahydrogen is in a lower energy state than is
orthohydrogen. At room temperature and
thermal equilibrium, thermal excitation causes
hydrogen to consist of approximately 75%
orthohydrogen and 25% parahydrogen. When
hydrogen is liquified at low temperature, there is
a slow spontaneous transition to a
predominantly para ratio, with the released
energy having implications for storage.
Essentially pure parahydrogen form can be
obtained at very low temperatures, but it is not
possible to obtain a sample containing more
than 75% orthohydrogen by heating.
Short description: Spin states of hydrogen
Spin isomers of molecular hydrogen
A mixture or 50:50 mixture of ortho- and parahydrogen can be made in the laboratory by passing it over
an iron(III) oxide catalyst at liquid nitrogen temperature (77 K)[3] or by storing hydrogen at 77 K for 2–3
hours in the presence of activated charcoal.[4] In the absence of a catalyst, gas phase parahydrogen takes
days to relax to normal hydrogen at room temperature while it takes hours to do so in organic
solvents.[4]
Contents
1
2
3
4
5
6
7
8
9
Nuclear spin states of H2
Allowed rotational energy levels
Thermal properties
History
Use in NMR and MRI
Deuterium
Other substances with spin isomers
References
Further reading
Nuclear spin states of H2
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Each hydrogen molecule (H2) consists of two hydrogen atoms linked by a covalent bond. If we neglect
the small proportion of deuterium and tritium which may be present, each hydrogen atom consists of
one proton and one electron. Each proton has an associated magnetic moment, which is associated with
the proton's spin of 1​ ⁄2. In the H2 molecule, the spins of the two hydrogen nuclei (protons) couple to
form a triplet state known as orthohydrogen, and a singlet state known as parahydrogen.
The triplet orthohydrogen state has total nuclear spin I = 1 so that the component along a defined axis
can have the three values MI = 1, 0, or −1. The corresponding nuclear spin wavefunctions are |↑↑⟩,
(|↑↓⟩ + |↓↑⟩) and |↓↓⟩. This uses standard bra–ket notation; the symbol ↑ represents the spin-up
1
√2
wavefunction and the symbol ↓ the spin-down wavefunction for a nucleus, so ↑↓ means that the first
nucleus is up and the second down. Each orthohydrogen energy level then has a (nuclear) spin
degeneracy of three, meaning that it corresponds to three states of the same energy (in the absence of a
magnetic field).[1] The singlet parahydrogen state has nuclear spin quantum numbers I = 0 and MI = 0,
with wavefunction
(|↑↓⟩ − |↓↑⟩). Since there is only one possibility, each parahydrogen level has a spin
1
√2
degeneracy of one and is said to be non-degenerate.[1]
Allowed rotational energy levels
Since protons have spin 1​ ⁄2, they are fermions and the permutational antisymmetry of the total H2
wavefunction imposes restrictions on the possible rotational states of the two forms of H2.[1]
Orthohydrogen, with symmetric nuclear spin functions, can only have rotational wavefunctions that are
antisymmetric with respect to permutation of the two protons, corresponding to odd values of the
rotational quantum number J; conversely, parahydrogen with an antisymmetric nuclear spin function, can
only have rotational wavefunctions that are symmetric with respect to permutation of the two protons,
corresponding to even J.[1]
The para form whose lowest level is J = 0 is more stable by 1.455 kJ/mol[5][6] than the ortho form whose
lowest level is J = 1. The ratio between numbers of ortho and para molecules is about 3:1 at standard
temperature where many rotational energy levels are populated, favoring the ortho form as a result of
thermal energy. However, at low temperatures only the J = 0 level is appreciably populated, so that the
para form dominates at low temperatures (approximately 99.8% at 20 K).[7] The heat of vaporization is
only 0.904 kJ/mol. As a result, ortho liquid hydrogen equilibrating to the para form releases enough
energy to cause significant loss by boiling.[5]
Thermal properties
Applying the rigid rotor approximation, the energies and degeneracies of the rotational states are given
by:[8]
J (J + 1)ℏ
EJ =
2
;
2I
g J = 2J + 1
.
The rotational partition function is conventionally written as:
∞
Z rot = ∑ g J e
−E J /k B T
.
J =0
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However, as long as the two spin isomers are not in equilibrium, it is more useful to write separate
partition functions for each:
Z para =
∑ (2J + 1)e
−J (J +1)ℏ
2
/2I k B T
even J
Z ortho = 3 ∑ (2J + 1)e
−J (J +1)ℏ
2
/2I k B T
odd J
The factor of 3 in the partition function for orthohydrogen accounts for the spin degeneracy associated
with the +1 spin state; when equilibrium between the spin isomers is possible, then a general partition
function incorporating this degeneracy difference can be written as:
∞
Z equil = ∑ (2 − (−1)
J
)(2J + 1)e
−J (J +1)ℏ
2
/2I k B T
J =0
The molar rotational energies and heat capacities are derived for any of these cases from:
U rot = RT
C v,
rot
=
2
(
∂ ln Z rot
)
∂T
∂U rot
∂T
Plots shown here are molar rotational energies and heat capacities for ortho- and parahydrogen, and the
"normal" ortho:para ratio (3:1) and equilibrium mixtures:
Because of the antisymmetry-imposed restriction on possible
rotational states, orthohydrogen has residual rotational energy
at low temperature wherein nearly all the molecules are in the
J = 1 state (molecules in the symmetric spin-triplet state cannot
fall into the lowest, symmetric rotational state) and possesses
nuclear-spin entropy due to the triplet state's threefold
degeneracy. The residual energy is significant because the
rotational energy levels are relatively widely spaced in H2; the
gap between the first two levels when expressed in temperature
units is twice the characteristic rotational temperature for H2:
E J =1 − E J =0
kB
ℏ
= 2θ rot =
2
≈ 174.98 K
kB I
.
Molar rotational energy ER/R in kelvins, or
This is the T = 0 intercept seen in the molar energy of
equivalently mean molecular rotational
orthohydrogen. Since "normal" room-temperature hydrogen is a
energy εrot/kB in kelvins
3:1 ortho:para mixture, its molar residual rotational energy at
low temperature is (3/4) × 2Rθrot ≈ 1091 J/mol, which is
somewhat larger than the enthalpy of vaporization of normal
hydrogen, 904 J/mol at the boiling point, Tb ≈ 20.369 K.[9] Notably, the boiling points of parahydrogen
and normal (3:1) hydrogen are nearly equal; for parahydrogen ∆Hvap ≈ 898 J/mol at Tb ≈ 20.277 K, and it
follows that nearly all the residual rotational energy of orthohydrogen is retained in the liquid state.
However, orthohydrogen is thermodynamically unstable at low temperatures and spontaneously
converts into parahydrogen.[10] This process lacks any natural de-excitation radiation mode, so it is slow
in the absence of a catalyst which can facilitate interconversion of the singlet and triplet spin states.[10]
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At room temperature, hydrogen contains 75% orthohydrogen, a
proportion which the liquefaction process preserves if carried
out in the absence of a catalyst like ferric oxide, activated
carbon, platinized asbestos, rare earth metals, uranium
compounds, chromic oxide, or some nickel compounds to
accelerate the conversion of the liquid hydrogen into
parahydrogen. Alternatively, additional refrigeration equipment
can be used to slowly absorb the heat that the orthohydrogen
fraction will (more slowly) release as it spontaneously converts
into parahydrogen. If orthohydrogen is not removed from
rapidly liquified hydrogen, without a catalyst, the heat released
during its decay can boil off as much as 50% of the original
liquid.
History
Molar heat capacities; only rotational and
spin contribution is shown. Total value is
1.5R higher due to translational degrees
of freedom (rotational degrees were
included in the rigid rotor approximation
itself).
The unusual heat capacity of hydrogen was discovered in 1912
by Arnold Eucken.[11] The two forms of molecular hydrogen
were first proposed by Werner Heisenberg and Friedrich Hund
in 1927. Taking into account this theoretical framework, pure
parahydrogen was first synthesized by Paul Harteck and Karl
Friedrich Bonhoeffer in 1929 at the Kaiser Wilhelm Institute for
Physical Chemistry and Electrochemistry.[12] When Heisenberg was awarded the 1932 Nobel prize in
physics for the creation of quantum mechanics, this discovery of the "allotropic forms of hydrogen" was
singled out as its most noteworthy application.[13] Further work on the properties and chemical reactivity
of parahydrogen was carried out in the following decade by Elly Agallidis and Georg-Maria Schwab.[14]
Modern isolation of pure parahydrogen has since been achieved using rapid in-vacuum deposition of
millimeters thick solid parahydrogen (p–H2) samples which are notable for their excellent optical
qualities.[15]
Use in NMR and MRI
When an excess of parahydrogen is used during hydrogenation reactions (instead of the normal mixture
of orthohydrogen to parahydrogen of 3:1), the resultant product exhibits hyperpolarized signals in
proton NMR spectra, an effect termed PHIP (Parahydrogen Induced Polarisation) or, equivalently,
PASADENA (Parahydrogen And Synthesis Allow Dramatically Enhanced Nuclear Alignment; named for
first recognition of the effect by Bowers and Weitekamp of Caltech),[16] a phenomenon that has been
used to study the mechanism of hydrogenation reactions.[17][18]
Signal amplification by reversible exchange (SABRE) is a technique to hyperpolarize samples without
chemically modifying them. Compared to orthohydrogen or organic molecules, a much greater fraction
of the hydrogen nuclei in parahydrogen align with an applied magnetic field. In SABRE, a metal center
reversibly binds to both the test molecule and a parahydrogen molecule facilitating the target molecule
to pick up the polarization of the parahydrogen.[19] This technique can be improved and utilized for a
wide range of organic molecules by using an intermediate "relay" molecule like ammonia. The ammonia
efficiently binds to the metal center and picks up the polarization from the parahydrogen. The ammonia
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then transfers it other molecules that don't bind as well to the metal catalyst.[20] This enhanced NMR
signal allows the rapid analysis of very small amounts of material and has great potential for applications
in magnetic resonance imaging.
Deuterium
Diatomic deuterium (D2) has nuclear spin isomers like diatomic hydrogen, but with different populations
of the two forms because the deuterium nucleus (deuteron) is a boson with nuclear spin equal to one.[21]
There are six possible nuclear spin wave functions which are ortho or symmetric to exchange of the two
nuclei, and three which are para or antisymmetric.[21] Ortho states correspond to even rotational levels
with symmetric rotational functions so that the total wavefunction is symmetric as required for the
exchange of two bosons, and para states correspond to odd rotational levels.[21] The ground state (J = 0)
populated at low temperature is ortho, and at standard temperature the ortho:para ratio is 2:1.[21]
Other substances with spin isomers
Other molecules and functional groups containing two hydrogen atoms, such as water[22] and methylene
(CH2),[23] also have ortho- and para- forms (e.g. orthowater and parawater), but this is of little
significance for their thermal properties.[23] Their ortho:para ratios differ from that of dihydrogen. The
ortho and para forms of water have recently been isolated.[24][25]
Molecular oxygen (O2) also exists in three lower-energy triplet states and one singlet state, as groundstate paramagnetic triplet oxygen and energized highly reactive diamagnetic singlet oxygen. These
states arise from the spins of their unpaired electrons, not their protons or nuclei.
References
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2. Matthews, M.J.; Petitpas, G.; Aceves, S.M. (2011). "A study of spin isomer conversion kinetics in supercritical
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3. Matsumoto, Mitsuru; Espenson, James H. (2005). "Kinetics of the Interconversion of Parahydrogen and
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4. Aroulanda, Christie; Starovoytova, Larisa; Canet, Daniel (2007). "Longitudinal Nuclear Spin Relaxation ofOrthoandPara-Hydrogen Dissolved in Organic Solvents". The Journal of Physical Chemistry A 111 (42): 10615–10624.
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"This source says 0.34 kcal/mol (= 1.4 kJ/mol)".
7. Rock, Peter A., Chemical thermodynamics; principles and applications (Macmillan 1969) Table p. 478 shows
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der Wissenschaften (Berlin). Sitzungsberichte (1912): S. 141–151
12. Michael Polanyi and His Generation: Origins of the Social Construction of Science (https://books.google.ca/bo
oks?id=7CLTNAzcynMC&pg=PA119&lpg=PA119&dq=Heisenberg+ortho+para+hydrogen&source=bl&ots=4
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vC_gQ6AEITjAH#v=onepage&q=Heisenberg%20ortho%20para%20hydrogen&f=false) Mary Jo Nye,
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13. Werner Heisenberg – Facts (https://www.nobelprize.org/nobel_prizes/physics/laureates/1932/heisenberg-fact
s.html) Nobelprize.org
14. "Elly Schwab-Agallidis" (http://jupiter.chem.uoa.gr/thanost/schwab/) (in el). University of Athens, Department
of Chemistry. http://jupiter.chem.uoa.gr/thanost/schwab/.
15. Rapid Vapor Deposition of Millimeters Thick Optically Transparent Solid Parahydrogen Samples for Matrix
Isolation Spectroscopy - Storming Media (https://web.archive.org/web/20120831182954/http://www.storming
media.us/72/7208/A720893.html)
16. Bowers, C. R.; Weitekamp, D. P. (1986). "Transformation of symmetrization order to nuclear-spin magnetization
by chemical reaction and nuclear magnetic resonance" (https://authors.library.caltech.edu/10855/1/BOWprl86.
pdf). Physical Review Letters 57 (21): 2645–2648. doi:10.1103/physrevlett.57.2645 (https://dx.doi.org/10.1103%
2Fphysrevlett.57.2645). PMID 10033824 (http://www.ncbi.nlm.nih.gov/pubmed/10033824).
Bibcode: 1986PhRvL..57.2645B (http://adsabs.harvard.edu/abs/1986PhRvL..57.2645B).
https://authors.library.caltech.edu/10855/1/BOWprl86.pdf.
17. Duckett, S. B.; Mewis (2013). Improving NMR and MRI Sensitivity With Parahydrogen. Topics in Current
Chemistry. 338. 75–103. doi:10.1007/128_2012_388 (https://dx.doi.org/10.1007%2F128_2012_388). ISBN 9783-642-39727-1.
18. Adams, R. W.; Aguilar, J. A.; Atkinson, K. D.; Cowley, M. J.; Elliott, P. I.; Duckett, S. B.; Green, G. G.; Khazal, I. G. et
al. (2009). "Reversible interactions with para-hydrogen enhance NMR sensitivity by polarization transfer" (http
s://www.pure.ed.ac.uk/ws/files/11451966/Reversible_Interactions_with_para_Hydrogen_Enhance_NMR_Sensitiv
ity_by_Polarization_Transfer.pdf). Science 323 (5922): 1708–11. doi:10.1126/science.1168877 (https://dx.doi.or
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19. Eshuis, Nan; Aspers, Ruud L.E.G.; van Weerdenburg, Bram J.A.; Feiters, Martin C.; Rutjes, Floris P.J.T.; Wijmenga,
Sybren S.; Tessari, Marco (2016). "Determination of long-range scalar 1 H– 1 H coupling constants responsible
for polarization transfer in SABRE". Journal of Magnetic Resonance 265: 59–66. doi:10.1016/j.jmr.2016.01.012 (h
ttps://dx.doi.org/10.1016%2Fj.jmr.2016.01.012). ISSN 1090-7807 (http://www.worldcat.org/issn/1090-7807).
PMID 26859865 (http://www.ncbi.nlm.nih.gov/pubmed/26859865). Bibcode: 2016JMagR.265...59E (http://adsa
bs.harvard.edu/abs/2016JMagR.265...59E).
20. Iali, Wissam; Rayner, Peter J.; Duckett, Simon B. (2018). "Using para hydrogen to hyperpolarize amines, amides,
carboxylic acids, alcohols, phosphates, and carbonates" (http://www.pubmedcentral.nih.gov/articlerender.fcgi?
tool=pmcentrez&artid=5756661). Science Advances 4 (1): eaao6250. doi:10.1126/sciadv.aao6250 (https://dx.d
oi.org/10.1126%2Fsciadv.aao6250). ISSN 2375-2548 (http://www.worldcat.org/issn/2375-2548).
PMID 29326984 (http://www.ncbi.nlm.nih.gov/pubmed/29326984). Bibcode: 2018SciA....4O6250I (http://adsab
s.harvard.edu/abs/2018SciA....4O6250I).
21. Hollas, J. Michael (1996). Modern Spectroscopy (3rd ed.). John Wiley and Sons. p. 115. ISBN 0-471-96523-5.
22. Tikhonov, Vladimir I.; Volkov, Alexander A. (28 June 2002). "Separation of water into its ortho and para
isomers" (https://science.sciencemag.org/content/296/5577/2363). Science 296 (5577): 2363.
doi:10.1126/science.1069513 (https://dx.doi.org/10.1126%2Fscience.1069513). PMID 12089435 (http://www.nc
bi.nlm.nih.gov/pubmed/12089435). https://science.sciencemag.org/content/296/5577/2363. Retrieved 16 July
2021.
23. Shinitzky, Meir; Elitzur, Avshalom C. (2006). "Ortho–para spin isomers of the protons in the methylene group".
Chirality 18 (9): 754–756. doi:10.1002/chir.20319 (https://dx.doi.org/10.1002%2Fchir.20319). PMID 16856167
(http://www.ncbi.nlm.nih.gov/pubmed/16856167).
24. "Two different forms of water isolated for first time" (https://www.bbc.com/news/science-environment-443056
63). 30 May 2018. https://www.bbc.com/news/science-environment-44305663. "If the nuclear spins of the two
hydrogen atoms in water are oriented in the same direction, it is called ortho-water. If they are arranged in
different directions, it is known as para-water."
25. Kilaj, Ardita; Gao, Hong; Rosch, Daniel; Rivero, Uxia; Kupper, Jochen; Willitsch, Stefan (29 May 2018).
"Observation of different reactivities of para and ortho-water towards trapped diazenylium ions" (https://www.
nature.com/articles/s41467-018-04483-3.pdf). Nature Communications 9 (Article 2096): 2096.
doi:10.1038/s41467-018-04483-3 (https://dx.doi.org/10.1038%2Fs41467-018-04483-3). PMID 29844308 (htt
p://www.ncbi.nlm.nih.gov/pubmed/29844308). PMC 5974139 (http://www.pubmedcentral.gov/articlerender.fc
gi?tool=pmcentrez&artid=5974139). https://www.nature.com/articles/s41467-018-04483-3.pdf. Retrieved 16
July 2021. "As an example, we investigate the proton-transfer reaction of water with ionic diazenylium
(N2H+)".
Further reading
Aline Léon, Ed. 2008, Hydrogen Technology: Mobile and Portable Applications, pp. 93–101, New York,
NY:Springer Science & Business, ISBN:3-540-69925-2, see [1] (https://books.google.com/books?isb
n=3540699252), accessed 10 May 2015.
Tikhonov V. I., Volkov A. A. (2002). "Separation of water into its ortho and para isomers" (https://se
manticscholar.org/paper/f8a3fc7fa7b2e4ae17d2181bf87dc669870b93bd). Science 296 (5577):
2363. doi:10.1126/science.1069513 (https://dx.doi.org/10.1126%2Fscience.1069513).
PMID 12089435 (http://www.ncbi.nlm.nih.gov/pubmed/12089435).
https://semanticscholar.org/paper/f8a3fc7fa7b2e4ae17d2181bf87dc669870b93bd.
{{cite book
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| title = Rapid Vapor Deposition of Millimeters Thick Optically Transparent Solid Parahydrogen Samples
for Matrix Isolation Spectroscopy | author = Mario E. Fajardo; Simon Tam | publisher = Edwards AFB
(Propulsion Directorate West):USAF Research Lab
Bowers, C. R.; Weitekamp, D. P. (1986). "Transformation of symmetrization order to nuclear-spin
magnetization by chemical reaction and nuclear magnetic resonance" (https://authors.library.caltec
h.edu/10855/1/BOWprl86.pdf). Phys. Rev. Lett. 57 (21): 2645–2648. doi:10.1103/physrevlett.57.2645
(https://dx.doi.org/10.1103%2Fphysrevlett.57.2645). PMID 10033824 (http://www.ncbi.nlm.nih.gov/
pubmed/10033824). Bibcode: 1986PhRvL..57.2645B (http://adsabs.harvard.edu/abs/1986PhRvL..57.
2645B). https://authors.library.caltech.edu/10855/1/BOWprl86.pdf.
{{cite book
| title = Orthohydrogen, parahydrogen and heavy hydrogen | author = A. Farkas | series = The
Cambridge series of physical chemistry | publisher = Cambridge, U.K.:Cambridge University Press (https://
www.wikipedia.org/wiki/Cambridge%20University%20Press) | year = 1935
Bonhoeffer KF, Harteck P (1929). "Para- and ortho hydrogen". Zeitschrift für Physikalische Chemie B
4 (1–2): 113–141.
Oxford Instruments, Date Unknown, "Boosting the Sensitivity of NMR Spectroscopy using
Parahydrogen"
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