Treatment of Thermal Non-Equilibrium Dissociation Rates: Application to $\rm H_2$

Aaron M. Brandis; Alex T. Carroll; Brett A. Cruden; Guillaume Blanquart; Jacob Wolmer

REVIEW 4 major objections 5 minor 134 references

Treatment of Thermal Non-Equilibrium Dissociation Rates: Application to $\rm H_2$

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper claims that non-equilibrium dissociation of a diatomic gas can be reduced to a one-temperature source term with two fitted rates.

desk verdict The QSS source-term equivalence is a real and useful result; the pre-QSS correction is a fitted correlation that needs independent validation before it carries the load. read the letter →

arxiv 2501.01626 v1 pith:ZSMHEY3Q submitted 2025-01-03 physics.chem-ph

Alex T. Carroll , Jacob Wolmer , Guillaume Blanquart , Aaron M. Brandis , Brett A. Cruden This is my paper · ORCID

classification physics.chem-ph

keywords non-equilibriumdissociationquasi-steady-statemasterequationspre-QSScorrectionhydrogenone-temperaturemodelrateconstantfitsthermochemicalnonequilibrium

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the full state-resolved chemistry of a dissociating diatomic can be collapsed, under the quasi-steady-state assumption, into a chemical source term that depends only on the translational temperature $T_{\rm t}$ and the fraction of dissociation $\phi_{\rm A}$. It derives the exact QSS source term from the master equations, then adds a simple pre-QSS correction factor $\eta(T_{\rm t})$. For $\rm H_2$ dissociation with the third bodies $\rm H_2$, $\rm H$, and $\rm He$, the resulting two-parameter expression reproduces the number-density and rovibrational-energy profiles of detailed master-equation simulations. If correct, this gives computational fluid dynamics a cheap way to include non-equilibrium dissociation without solving additional internal-energy equations.

What carries the argument

The central object is the quasi-steady-state decomposition of the rovibrational distribution, $$\vec{\psi}_{\rm A_2} = \vec{\psi}_{\rm A_2,nr}\left(1 - \frac{\phi_{\rm A}^2}{\chi}\right) + \frac{\phi_{\rm A}^2}{\chi}\,\vec{1},$$ which splits the QSS solution into a non-recombining part and a recombining part, each a function of $T_{\rm t}$ alone. Feeding this into the aggregate rate expression makes $k_{\rm d}$ a convex combination of $k_{\rm d,nr}$ and $k_{\rm d,th}$, and substituting into the source term cancels $k_{\rm d,th}$ exactly, producing Eq. (41). For the pre-QSS region the machinery is the slowest eigenmode of the relaxation matrix: $\eta(T_{\rm t}) \equiv k_{\rm d,nr}/(2(\lambda_1-\lambda_0))$ converts the time-dependent relaxation into a function of $\phi_{\rm A}$, removing the dependence on third-body number density.

What would settle it

Set up a 0-D isothermal, isochoric master-equation simulation of $\rm H_2$ dissociation in a $\rm H_2$/\rm H$/\rm He$ mixture at a number density outside the fitted range, say $n_{\rm M}=10^{17}\ \mathrm{cm^{-3}}$ at $T_{\rm t}=12{,}000\ \mathrm{K}$, and compare the predicted $n_{\rm H}(t)$ and rovibrational energy history against Eq. (78) using the paper's $\eta$ and $k_{\rm d,nr}$ fits; systematic disagreement beyond the factor-of-two uncertainty would falsify the transferability of $\eta$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the QSS chemical source term for dissociation, $$\frac{dn_{\rm A}}{dt} = 2n_{\rm A_2} n_{\rm M} k_{\rm d,nr} - 2n_{\rm A}^2 n_{\rm M} \frac{k_{\rm d,nr}}{K_{\rm eq}},$$ depends only on the non-recombining dissociation rate constant $k_{\rm d,nr}(T_{\rm t})$; the thermal-limit rate constant drops out exactly through the progress variable $\phi_{\rm A}^2/\chi$. The pre-QSS extension replaces $k_{\rm d,nr}$ by $$k_{\rm d,pre\text{-}QSS} = k_{\rm d,nr}\left[1 - (1-\varepsilon)\exp\left(-\sqrt{-\ln(1-\phi_{\rm A})/\eta}\right)\right],$$ so the complete source term is a function of $T_{\rm t}$ and $\phi_{\rm A}$ alone, with inputs $k_{\rm d,nr}(T_{\rm t})$ and $\eta(T_{\rm t})$. The authors show that this reproduction holds for the majority of the tested master-equation cases for $\rm H_2$ with the third bodies $\rm H_2$, $\rm H$, and $\rm He$, and they use the same framework to reinterpret reported shock-tube rates as $k_{\rm d,nr}$ data.

Load-bearing premise

The load-bearing premise is that the pre-QSS correction factor $\eta(T_{\rm t})$, fitted to a small set of 0-D isothermal and isochoric master-equation cases for three third bodies, is a function of translational temperature alone and transfers to other number densities, mixtures, and flow conditions.

Editorial extensions

If this is right

A one-temperature bulk-species model can capture both QSS and pre-QSS non-equilibrium dissociation for $\rm H_2$ in the tested regimes, without separate vibrational or rotational temperatures.
The same $k_{\rm d,nr}$ fit must be used for both the production and consumption terms in the QSS source term; the apparent recombination constant is $k_{\rm d,nr}/K_{\rm eq}$, not the physical $k_{\rm r}$.
The transition from the non-recombining QSS limit to thermal equilibrium is captured implicitly by Eq. (41), so no separate fit of $k_{\rm d,th}$ is needed in the source term.
The new fits of $k_{\rm d,nr}/K_{\rm eq}$ for $\rm H_2$, $\rm H$, and noble-gas third bodies are claimed valid from 200 to 20,000 K with uncertainty under a factor of two.
The fraction of dissociation that happens in the pre-QSS region is a function of temperature alone once $\eta(T_{\rm t})$ is known, independent of number density.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A genuinely independent test would run master-equation simulations at number densities and in $\rm H_2$/\rm H$/\rm He$ mixtures outside the cases used to fit $\eta$, since the paper's $\eta$ is fitted to the same 0-D isothermal, isochoric reactor data used for validation.
The exactness of Eq. (41) is structural, not $\rm H_2$-specific, so the same QSS source-term identity should transfer to other diatomics such as $\rm N_2$ or $\rm O_2$; only the fits of $k_{\rm d,nr}$ and $\eta$ would need to be re-established.
If the linear mixture rule is applied, the per-third-body $\eta(T_{\rm t})$ fits could be summed over species, but the paper notes the mixture rovibrational distribution need not match any single-bath distribution, so this is an extrapolation rather than a proven result.
For recombination-dominated flows the model may lose accuracy, because those flows overpopulate excited states while the QSS assumption here is tied to the underpopulated, dissociation-dominated distribution; the authors flag this as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth a careful read. The paper does two genuinely useful things. First, it makes explicit that under the QSS assumption the dissociation/recombination source term collapses to Eq. (41), a one-temperature expression in kd,nr and Keq alone. That is a clean analytic result, and the master-equation comparisons at 4,000 and 6,000 K back it up. Second, the H2 dissociation rate compilation and the new kd,nr/Keq fits are a practical contribution; the paper makes a decent case that the old Leibowitz rates are badly off at low temperature and that a factor-of-two uncertainty bound is more honest than the order-of-magnitude or 0.25x-3x bounds used in prior entry studies. The discussion of the three inconsistent Leibowitz model variants is a nice detail.

The soft spot is the pre-QSS correction, just as the stress-test says. In Section III A, eta(Tt) is obtained by least-squares fitting Eq. (75) to the very same master-equation kd(phi_H) cases that are then shown as the validating comparisons in Figures 3, 6, and 7. All of those cases share one initial rovibrational condition (Tr,0 = Tv,0 = 1000 K) and nearly fixed number densities; only Tt varies. So the claim that kd,pre-QSS depends only on Tt and phi_A, and that a one-temperature bulk model captures both QSS and pre-QSS behavior, is not independently established beyond that fitted test set. The derivation itself also leans on several uncontrolled approximations: single-slowest-mode dominance, psi_inf = psi_nr, and alpha = phi_A. None of this makes the QSS result wrong, but it means the pre-QSS expression is presently a calibrated correlation, not a validated predictive model.

Two smaller gripes: no code or data are shipped, and the factor-of-two uncertainty on the rate fits is based on visual scatter rather than formal UQ. Both are fixable.

Who is this for? Anyone doing CFD for hydrogen-helium planetary entry, and anyone who wants a compact source term for QSS dissociation. The paper deserves a serious referee: the QSS equivalence and the rate review are solid enough that the manuscript should not be desk-rejected, but the referee should push for an out-of-sample test of eta (different densities, initial temperatures, or mixtures) and for release of fitting scripts and extracted data.

Referee Report

4 major / 5 minor

Summary. The paper derives, from the rovibrational master equations, rate expressions for diatomic dissociation in thermal non-equilibrium, and applies them to H2. The central QSS result is Eq. (41), which states that when the QSS assumption holds the chemical source term can be written as dnA/dt = 2 nA2 nM kd,nr - 2 nA^2 nM kd,nr/Keq, with kd,nr a function of Tt alone. The paper then adds a pre-QSS correction leading to Eq. (75), making kd a function of Tt and the dissociation fraction phi_A only. The model is compared with master-equation simulations of 0-D isothermal/isochoric H2 dissociation with H2, H, and He as third bodies, and an extensive literature review yields Arrhenius fits for kd,nr/Keq from 200 to 20,000 K with claimed uncertainty better than a factor of two.

Significance. If the QSS source-term equivalence of Eq. (41) is accepted, it is a significant and elegant result: a one-temperature bulk-species model captures the full QSS dissociation/recombination source term without fitting a recombination rate separately. The paper's algebraic derivation of Eq. (41) from the QSS distribution is internally consistent and is verified against master-equation number densities at 4,000 and 6,000 K, where pre-QSS effects are negligible. The literature review of H2 dissociation rate constants, with new fits and a factor-of-two uncertainty estimate, is also a useful contribution for ice-giant entry modeling. The pre-QSS correction, however, is the load-bearing novelty for high-temperature flows, and its validation is currently tied to the fitting procedure and to a narrow set of test conditions, so the broader claim that kd,pre-QSS depends only on Tt and phi_A is not yet independently established.

major comments (4)

[III A, Eq. (75), Figs. 3, 6, 7] The pre-QSS correction is fitted and validated on the same data. Section III A states that the eta values are computed by a least-squares fit to the master equation results, and the resulting corrected kd curves are then shown to reproduce those same master equation curves in Fig. 3 and, after integration, in Figs. 6 and 7. The agreement in the pre-QSS region is therefore partly circular. The only genuinely independent checks are the 4,000 and 6,000 K number-density and energy profiles, where the pre-QSS contribution is small. To support the central claim that Eq. (75) is a predictive one-temperature source term, the authors should validate eta(Tt) on master-equation cases not used in the fit, e.g., different initial Tr/Tv values, different number densities, or post-shock (non-isothermal) conditions.
[II D 2, Eqs. (66), (73), (74), (75)] Several uncontrolled approximations enter the pre-QSS derivation: the slow-mode dominance of Eq. (66), the identification of psi_infinity with psi_nr, the first-order Taylor expansion used to derive Eq. (73) while the exponential form is retained in Eq. (74), and the alpha approx phi_A substitution. These approximations are stated but not quantified, and no sensitivity analysis is provided. Since Eq. (75) is the basis for the claimed extension beyond QSS, the manuscript should either bound the error of these approximations against full master-equation solutions or clearly limit the claimed validity range to the conditions that are actually tested.
[III A / III C, Figs. 1-7] All fitted and validated master-equation cases share nearly identical initial conditions: Tr,0 = Tv,0 = 1000 K, nH2,0 = 1e18 cm^-3, and nH,0 = nHe,0 = 5e17 cm^-3, with only Tt varied. The derivation of Eq. (75) removes the nM dependence present in Eq. (70), but this removal is not empirically verified over a range of densities or mixture compositions. Consequently, the transferability of eta(Tt) to other number densities, initial internal temperatures, and H2/H/He mixtures is the key unproven premise for CFD application, and the linear-mixture rule of Appendix A does not cure this because no mixed-bath master-equation data are presented.
[Section III C / Conclusion (Eq. (78))] The claim that a one-temperature bulk-species model captures both QSS and pre-QSS dissociation in practical flows also relies on the source-term form of Eq. (78), in which the recombination term is written as kd,pre-QSS/Keq. The paper correctly notes in Section II C 3 that this term is not the physical kr, but the pre-QSS version of this equivalence has not been independently verified: the integrated number-density profiles in Fig. 6 at high Tt use the same fitted eta and the same isothermal, isochoric, inert-bath assumptions as the fit. A test where recombination dominates, or where the bath is not inert, would be needed before the source term is asserted to be general for entry-flow conditions.

minor comments (5)

[Table II] The table heading reads 'Experimental Sources for High Temperature Rate Constants' but the table lists discharge-flow tube studies at Tt <= 350 K; the heading should be 'Low Temperature'.
[Fig. 4] The horizontal axis is labeled only with numeric tick values (0.05, 0.1, 0.15, 0.2); the axis label, presumably 10^4/Tt or similar, is missing.
[III A, Fig. 3 caption] The caption says 'dashed blue and dash-dotted red lines' but the text refers to 'dash-dotted red and dashed blue lines'; the order should be made consistent so the reader knows which line is QSS and which is pre-QSS.
[Appendix C] In the paragraph describing the three Leibowitz rate sets, 'Leibowitz and Kuo128' should presumably be 'Leibowitz and Kuo129'; the current reference numbering is inconsistent with the bibliography.
[Eq. (10) and Eq. (15)] The symbol QA2 is used both for the total internal partition function and, in context, for the bulk quantity; near Eq. (10) the distinction between QA2 and QA2(J,nu) should be stated explicitly to avoid confusion.

Circularity Check

3 steps flagged · score 6.0 of 10

Pre-QSS correction η(Tt) is least-squares fitted to the same master-equation cases used for validation; the QSS source-term derivation itself is self-contained.

fitted input called prediction [Section III A, Eq. (75) and Fig. 3]
"The η values used in the pre-QSS expressions are computed for each third-body and temperature case using a least squares fit to the master equation results. ... Additionally, with the simple pre-QSS correction of Eq. (75), the kd values are also able to reproduce the results of the master equation calculations in the pre-QSS regions."

The pre-QSS rate expression Eq. (75) is presented as reproducing the master-equation kd(φA) curves, but its only non-derived input η(Tt) is obtained by least-squares fitting to those very same curves. The agreement is therefore enforced by the fit rather than established independently; reproducing the fitted data does not validate the functional form outside the fitted points.
fitted input called prediction [Section III C, Fig. 6]
"For the pre-QSS expression, ε is set to 10−3 and the η(Tt) fits obtained in the previous section are used. ... with the pre-QSS correction, the predicted profiles are in much better agreement with the master equation results."

The number-density profiles in Fig. 6 are integrated using η(Tt) fits obtained from the same 0-D isothermal/isochoric master-equation cases (initial Tr,0 = Tv,0 = 1000 K; nH2,0 = 1e18 cm^-3; nH,0 = nHe,0 = 5e17 cm^-3) against which they are compared. The pre-QSS correction is thus not a fresh prediction for these cases; the agreement is a consistency check of the integration, not evidence of transferability to other densities, mixtures, or flow histories.

1 more flagged steps

fitted input called prediction [Section III B, Arrhenius fits of η(Tt)]
"The parameters for the Arrhenius fits are determined by fitting η(Tt) as a function of temperature (in K) across each of the master equation cases discussed in the previous section. ... using these Arrhenius fits in Eq. (77) instead of the individual master equation-extracted η values produced effectively identical predictions."

The Arrhenius fits are refits of η values extracted from the same master-equation cases, and the 'effectively identical predictions' are comparisons against those same cases. This step does not add independent evidence for the generality of η(Tt); it only compresses the already-fitted values into a smooth temperature function.

full rationale

The QSS derivation is genuinely self-contained: Eq. (41) is an exact algebraic consequence of the QSS distribution and the source-term equivalence, with no fitted parameters entering the derivation, and it is verified against master-equation results only after kd,nr is taken from the literature or from the master-equation plateaus. The circularity is localized to the pre-QSS correction. The paper explicitly fits η(Tt) by least squares to the master-equation kd(φA) curves and then validates Eq. (75) and Eq. (78) on those same isothermal/isochoric cases, so the pre-QSS agreement is partly an in-sample fit. The claimed transferability of η(Tt) to other number densities, mixtures, and flow conditions is asserted but not independently tested. The kd,nr fits in Section IV are a separate literature review and are not circular. Overall, the central QSS claim remains independent, but the pre-QSS 'prediction' reduces in part to its own fitting data, giving a partial circularity score of 6.

Assumptions & free parameters 9 free parameters · 9 assumptions · 0 invented entities

No new physical entities are proposed. The model's added degrees of freedom are three eta(Tt) Arrhenius fits and three kd,nr/Keq power-law fits, plus two numerical constants (delta and epsilon) and one digitization reconciliation factor. Most auxiliary assumptions are standard QSS and master equation modeling choices; the pre-QSS eigenmode and Taylor-expansion steps are ad hoc and are the main source of model risk.

free parameters (9)

eta_H2_fit = 6.489e1 T^-0.50 exp(-2.221e4/T)
Pre-QSS correction for M=H2, least-squares fit to master equation extracted eta values in Section III B.
eta_H_fit = 1.897e3 T^-0.66 exp(-4.236e4/T)
Pre-QSS correction for M=H, fitted to master equation results.
eta_He_fit = 1.668e6 T^-1.41 exp(-4.035e4/T)
Pre-QSS correction for M=He, fitted to master equation results.
kd_nr_over_Keq_H2_fit = 1.779e17 T^-0.69 cm6/mol2/s
Chosen to lie in the middle of the scatter of experimental and computational data in Section IV D.
kd_nr_over_Keq_H_fit = 5.461e17 T^-0.61 cm6/mol2/s
Constrained mainly by computational data for M=H because reliable low-temperature experimental data are lacking.
kd_nr_over_Keq_noble_gas_fit = 8.168e17 T^-0.95 cm6/mol2/s
Single fit for He, Ne, Ar, Xe, and Kr, assuming differences between noble gas third bodies are negligible.
delta = 0.25
Arbitrary cutoff between pre-QSS, non-recombining QSS, and recombining regions.
epsilon = 1e-3
Floor value required for numerical integration so kd is not zero at phi_A=0; stated to be insensitive below this threshold.
Keq_mod_factor_H2 = 2
Ad hoc factor needed to reproduce equilibrium number densities from digitized Kim and Boyd H2 plots, footnote 70.

assumptions (9)

domain assumption State-specific rate constants depend only on translational temperature Tt because the third body's internal state is fixed.
Stated in Section II A; key to the kd,nr(Tt) functional dependence.
standard math Micro-reversibility relates forward and backward state-specific rates via equilibrium populations.
Eqs. (8)-(9), foundational for the QSS formulation and Eq. (18).
domain assumption The ground rovibrational state does not satisfy QSS and its equation is dropped to resolve the over-constrained system.
Section II C 1, standard Park-style QSS reduction.
domain assumption In pre-QSS, recombination is negligible, the reactor is isothermal and isochoric, and the third body density is constant.
Section II D 1, restricts validity to early dissociation in a dilute bath.
ad hoc to paper Only the slowest eigenmode of the relaxation matrix survives, and the steady pre-QSS distribution equals the non-recombining QSS distribution psi_nr.
Eqs. (66)-(68); the paper notes M and M-tilde are not exactly equal, so psi_inf and psi_nr are not formally equivalent.
ad hoc to paper A first-order Taylor expansion of exp(-a t) can be used to derive t(alpha) while the final expression retains the exponential.
Transition from Eq. (72) to Eq. (74); validity requires n_M(lambda1-lambda0)t small, which is not established for later times.
domain assumption The linear mixture rule applies for multicomponent mixtures, with the internal distribution a branching-ratio weighted sum of single-bath distributions.
Appendix A, Eq. (A3); the paper calls the validity of the linear mixture rule an open question.
domain assumption Low-temperature discharge-flow data are interpreted as third-order recombination with no internal-distribution dependence.
Section IV B and Appendix B; justified by a Lindemann reduced-pressure estimate.
domain assumption Noble gas third bodies He, Ne, Ar, Xe, and Kr have identical dissociation rate constants.
Section IV D; based on scatter being larger than reported differences between noble gases.

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Cite this review

Pith. "Pith review of Treatment of Thermal Non-Equilibrium Dissociation Rates: Application to $\rm H_2$." pith.science (2026). https://pith.science/paper/ZSMHEY3Q

@misc{pith2026250101626,
  author       = {Pith},
  title        = {Pith review of: Treatment of Thermal Non-Equilibrium Dissociation Rates: Application to $\rm H_2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSMHEY3Q}},
  note         = {Machine review of arXiv:2501.01626}
}

abstract

This work presents a detailed description of the thermochemical non-equilibrium dissociation of diatomic molecules, and applies this theory to the case of $\rm H_2$ dissociation. The master equations are used to derive corresponding aggregate rate constant expressions that hold for any degree of thermochemical non-equilibrium. These general expressions are analyzed in three key limits/ regimes: the thermal equilibrium limit, the quasi-steady-state (QSS) regime, and the pre-QSS regime. Under several simplifying assumptions, an analytical source term expression that holds in all of these regimes, and is only a function of the translational temperature, $T_{\rm t}$, and the fraction of dissociation, $\phi_{\rm A}$, is proposed. This expression has two input parameters: the QSS dissociation rate constant in the absence of recombination, $k_{\rm d,nr}(T_{\rm t})$, and a pre-QSS correction factor, $\eta(T_{\rm t})$. The value of $\eta(T_{\rm t})$ is evaluated by comparing the predictions of the proposed expression against existing master equation simulations of a 0-D isothermal and isochoric reactor for the case of $\rm H_2$ dissociation with the third-bodies $\rm H_2$, $\rm H$, and $\rm He$. Despite its simple functional form, the proposed expression is able to reproduce the master equation results for the majority of the tested conditions. The best fit of $k_{\rm d,nr}(T_{\rm t})$ is then evaluated by conducting a detailed literature review. Data from a wide range of experimental and computational studies are considered for the third-bodies $\rm H_2$, $\rm H$, and inert gases, and fits that are valid from 200 to 20,000 K are proposed. From this review, the uncertainty of the proposed fits are estimated to be less than a factor of two.

Figures

Figures reproduced from arXiv: 2501.01626 by the authors.

**Figure 2.** FIG. 2: Rotational and vibrational temperature profiles for the master equation calculations of [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

**Figure 3.** FIG. 3: Aggregate dissociation rate constants for the master equation calculations of Kim and [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4: Extracted [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]

**Figure 5.** Figure 5: FIG. 5: Fraction of dissociation that occurs in the pre-QSS (blue lines), non-recombining QSS [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

**Figure 6.** Figure 6: FIG. 6: Number density profiles for the master equation calculations of Kim and Boyd [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

**Figure 7.** Figure 7: FIG. 7: Average rovibrational energy profiles for the master equation calculations of Kim and [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

**Figure 8.** Figure 8: FIG. 8: Contours of the estimated fraction of dissociation, [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]

**Figure 9.** Figure 9: FIG. 9: Review of rate constant data for M = H [PITH_FULL_IMAGE:figures/full_fig_p037_9.png]

**Figure 10.** Figure 10: FIG. 10: Rate constant data for M = H [PITH_FULL_IMAGE:figures/full_fig_p039_10.png]

**Figure 11.** Figure 11: FIG. 11: Rate constant fits of Leibowitz [PITH_FULL_IMAGE:figures/full_fig_p046_11.png]

Discussion (0). Continue with ORCID to comment.

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Therefore, these values have all been obtained via digitizations

note Number densities, rovibrational temperatures, and rate constants for these studies were only reported as plots. Therefore, these values have all been obtained via digitizations. Stop
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Additionally, for the case of M=H, k_ d values were extracted from the number density profiles assuming n_ M was fixed and equal to n_ H,0

note For the case of M= H_2 , a modified equilibrium constant, K_ eq,mod =2K_ eq , was required to reproduce the equilibrium number densities plotted by Kim and Boyd Kim2012 . Additionally, for the case of M=H, k_ d values were extracted from the number density profiles assumi...
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author author N. Singh \ and\ author T. Schwartzentruber ,\ title title Nonequilibrium internal energy distributions during dissociation , \ 10.1073/pnas.1713840115 journal journal Proceedings of the National Academy of Sciences \ volume 115 ,\ pages 47–52 ( year 2017 ) NoStop

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

2010

[2] [2]

merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

2010

[3] [3]

merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

2010

[4] [4]

merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

2010

[5] [5]

merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

2010

[6] [6]

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doi:10.1086/160617

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Hollenbach \ and\ author C

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doi:10.1086/167595

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author author C. A. \ Chang \ and\ author P. G. \ Martin ,\ title title Partially dissociative jump shocks in molecular hydrogen , \ 10.1086/170420 journal journal The Astrophysical Journal \ volume 378 ,\ pages 202 ( year 1991 ) NoStop

work page doi:10.1086/170420