Enthalpy of the N⁺ + H₂ → NH⁺ + H Reaction—Experimental Study of the Reverse Process

This work aims to resolve the longstanding question of the enthalpy of the nearly thermoneutral reaction

$$\begin{eqnarray}&&{{\rm{N}}}^{+}+{{\rm{H}}}_{2}\mathop{\to }\limits^{{k}_{f}}{\mathrm{NH}}^{+}+{\rm{H}},\end{eqnarray} \tag{ 1 }$$

which initiates the formation of nitrogen hydrides in interstellar clouds (Le Gal et al. 2014; Gerin et al. 2016). We will evaluate the reaction enthalpy at 0 K, which is equal to the energy difference between the product and reactant ground states, ΔE⁰. This quantity is an essential parameter in calculations of state-specific NH⁺ formation rate coefficients in astrochemical models (Grozdanov et al. 2016; Gómez-Carrasco et al. 2022). At temperatures relevant to the chemistry of molecular clouds (Millar 2015), the temperature dependence of the rate coefficient, k_f , of this forward reaction can be represented by the Arrhenius dependence, ${k}_{f}={k}^{0}\exp (-{E}_{{\rm{A}}}/({k}_{{\rm{B}}}T))$, with activation energy E_A ≈ 19 meV (Marquette et al. 1985; Plašil et al. 2022). The presence of activation energy indicates that the reaction cross section has a certain threshold energy E_T (Menzinger & Wolfgang 1969). However, neither the activation energies nor the threshold energies of cross sections can be directly identified with the reaction enthalpy, as the data can also be explained by the presence of energy barriers. To elucidate the issue, the variation of the activation energy with the internal excitation and isotopic composition of reactants has been studied in a number of experiments (Marquette et al. 1985, 1988; Gerlich 1993; Zymak et al. 2013; Plašil et al. 2022) and some have also deduced the threshold energy (Ervin & Armentrout 1987; Marquette et al. 1988; Gerlich 1993; Tosi et al. 1994). A notable result was obtained by Tosi et al. (1994), who observed that the change of the energy threshold due to deuteration cannot be explained by the different zero-point energies of the deuterated species, suggesting that the energy threshold is not purely due to reaction enthalpy and an energy barrier is involved. On the other hand, several studies (Adams & Smith 1985; Marquette et al. 1988; Plašil et al. 2022), have observed isotope effects consistent with the hypothesis of no barrier. The recent study by Plašil et al. (2022) shows that the available data are compatible with the hypothesis of no reaction barrier when only low-temperature thermal reaction rate coefficients with known error estimates are taken into account. The isotopic variation of activation energies in accordance with the hypothesis of the absence of energy barriers is a strong indication in favor of this hypothesis.

The reaction enthalpy can, in principle, be calculated as the difference between the H₂ and NH⁺ dissociation energies. However, despite the recent theoretical and experimental works (Tarroni et al. 1997; Hübers et al. 2009; Shi et al. 2009; Lecointre et al. 2010; Beloy et al. 2011; Zhang et al. 2017; Yang et al. 2019; Ghosh et al. 2022; Gómez-Carrasco et al. 2022), the dissociation energy of NH⁺ has not been determined with sufficient precision yet and the most recent theoretical reaction studies still approximate the energy difference with the experimental activation energies (Gómez-Carrasco et al. 2022). We have therefore chosen an experimental approach to study the reaction endothermicity by measuring the rate coefficient k_r of the reverse reaction

$$\begin{eqnarray}&&{\mathrm{NH}}^{+}+{\rm{H}}\mathop{\to }\limits^{{k}_{r}}{{\rm{N}}}^{+}+{{\rm{H}}}_{2}.\end{eqnarray} \tag{ 2 }$$

Knowing the forward and reverse thermal rate coefficients and accounting for all the relevant states of the NH₂ ⁺ system (see Figure 1), we can calculate ΔE⁰ using the detailed balance principle.

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The experiments were carried out in the Atomic Beam with 22-pole Trap instrument (AB-22PT; Gerlich 1992, 2012; Borodi et al. 2009; Plasil et al. 2011; Roučka et al. 2015). The NH⁺ ions were produced by electron bombardment from two different precursor gas mixtures in the storage ion source (Gerlich 1992)—either N₂: H₂ or NH₃:He mixtures in pressure ratios of 1: 2 or 1: 10, respectively (with identical resulting rate coefficients within the error margin). After mass selection, the ions were injected into a linear 22-pole radiofrequency trap, where they were cooled by collisions with helium buffer gas. To freeze out the N₂ or NH₃ leaking from the ion source and to avoid parasitic reactions with these species, the trap was kept at temperatures T_22PT < 30 K. Based on our previous studies of endothermic reactions (Mulin et al. 2015; Roučka et al. 2018; Plašil et al. 2022) and photodetachment spectroscopy (Plašil et al. 2023), we assume that neither the translational nor the internal temperatures of NH⁺ exceed the trap temperature by more than 10 K under the present operating conditions. We therefore denote both temperatures as ${T}_{{\mathrm{NH}}^{+}}$, which is given as ${T}_{{\mathrm{NH}}^{+}}={T}_{22\mathrm{PT}}+(5\pm 5)\,{\rm{K}}$. The H atoms were produced in a radiofrequency discharge in pure H₂ and they were passing through a cold nozzle (accommodator) with temperature T_acc in the range from 7 to 150 K. The H atom temperature T_H is close to T_acc (Borodi et al. 2009) and we assume T_H = T_acc. The atoms exit the accommodator in an effusive beam, which passes through the trap and can be blocked using a mechanical shutter.

The effective translational temperature T_t of the collisions between H atoms with mass M_H and NH⁺ ions with mass ${M}_{{\mathrm{NH}}^{+}}$ is given as (Light et al. 1969; Gerlich & Horning 1992; Paul et al. 1995) ${T}_{{\rm{t}}}=({T}_{{\rm{H}}}{M}_{{\mathrm{NH}}^{+}}+{T}_{{\mathrm{NH}}^{+}}{M}_{{\rm{H}}})/({M}_{{\mathrm{NH}}^{+}}+{M}_{{\rm{H}}})$. The most relevant parameter for determining the rate coefficients of endothermic reactions is the total available collisional energy—the sum of the translational and internal energies ${E}_{\mathrm{coll}}({T}_{{\mathrm{NH}}^{+}},{T}_{{\rm{H}}})={E}_{\mathrm{int}}({T}_{{\mathrm{NH}}^{+}})+\tfrac{3}{2}{k}_{{\rm{B}}}{T}_{{\rm{t}}}$. We therefore define the corresponding collisional temperature T_coll through the following implicit formula, ${E}_{\mathrm{int}}({T}_{\mathrm{coll}})+\tfrac{3}{2}{k}_{{\rm{B}}}{T}_{\mathrm{coll}}={E}_{\mathrm{coll}}$, as the temperature of hypothetical thermal reactants with total collisional energy E_coll.

The effective number density of H atoms, n_H, was determined by chemical probing with CO₂ ⁺ using the reaction CO₂ ⁺ + H → HOC⁺ + O (Borodi et al. 2009). The n_H calibration and reaction measurements were always carried out at similar values of the trap temperature, effective trapping potential, and accommodator temperature in order to maintain the same overlap of the H beam with the ion cloud and the level of dissociation (Roučka et al. 2015).

The typical experimental data are shown in Figure 2. Every measurement is performed with the beam shutter open (H on) and closed (H off) to resolve the reactions with H atoms from other background reactions. Since our quadrupole mass filter cannot resolve ions with the same number of nucleons, there is also a minor amount of ¹⁵N⁺ isotope mixed in with the studied ¹⁴NH⁺ ions. The dominant loss process for the ¹⁴NH⁺ ions is the fast hydrogen abstraction reaction with the background H₂ gas (Rednyk et al. 2019):

$$\begin{eqnarray}&&{\mathrm{NH}}^{+}+{{\rm{H}}}_{2}\to {\mathrm{NH}}_{2}^{+}+{\rm{H}},\end{eqnarray} \tag{ 3 }$$

whereas the ¹⁵N⁺ are destroyed by the much slower reaction (1). This is manifested by the bi-exponential decay of the number of ions with m/z = 15. These different loss rates allow us to resolve ¹⁴NH⁺ from ¹⁵N⁺ in the data analysis. The loss of ¹⁵N⁺ on the timescale of our experiment is practically negligible and we assume that its reaction rate coefficient with H₂ is equal to that of ¹⁴N⁺ (Zymak et al. 2013).

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By least-squares fitting (Newville et al. 2014, 2020) of the measured data with a kinetic model that includes reactions (1), (2), and (3), we obtain the rates of N⁺ production from NH⁺ with the beam on and off (r_on and r_off, respectively), where r_off is typically close to zero (see Figure 2). The production rates of N⁺ due to the interaction of NH⁺ with H atoms are calculated as r = r_on − r_off. The rate coefficients of the reverse reaction are finally obtained by dividing the rate with the H atom number density calibrated at equivalent conditions using chemical probing.

The measurements were performed at four H atom temperatures—7, 50, 75, and 100 K—and ${T}_{{\mathrm{NH}}^{+}}$ varied from 15 to 30 K. Since the translational temperature T_t is predominantly determined by T_H, we group the results by the accommodator temperature and calculate a weighted average of the measured reaction rate coefficients in each group. The averaged reverse reaction rate coefficients are plotted as a function of T_t and T_coll in the upper panel of Figure 3. We also plot the data as a function of ${T}_{{\mathrm{NH}}^{+}}$ (measurements were done mostly at ion temperatures ${T}_{{\mathrm{NH}}^{+}}\approx 17\ {\rm{K}}$ and 27 K). The data point at the lowest T_t and T_coll (T_H = 7 K) has large statistical uncertainty due to the low number density of H atoms in these conditions. The reaction rate coefficients have no significant dependence on temperature. Hence, the reverse reaction has no significant activation energy.

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To obtain quantitative information about the reaction enthalpy, we employ the principle of detailed balance. The full derivation of this principle from quantum mechanics can be found, e.g., in Henriksen & Hansen (2008), Light et al. (1969), and Sakurai & Napolitano (2010). For a general bimolecular reaction ${\rm{A}}+{\rm{B}}\underset{{k}_{r}}{\overset{{k}_{f}}{\rightleftharpoons }}{\rm{C}}+{\rm{D}}$, the forward and reverse reaction rate coefficients are related by the formula

$$\begin{eqnarray}&&\displaystyle \frac{{k}_{f}}{{k}_{r}}={\left(\displaystyle \frac{{\mu }_{\mathrm{CD}}}{{\mu }_{\mathrm{AB}}}\right)}^{3/2}\displaystyle \frac{{q}_{{\rm{C}}}{q}_{{\rm{D}}}}{{q}_{{\rm{A}}}{q}_{{\rm{B}}}}\exp \left(-\displaystyle \frac{{\rm{\Delta }}{E}^{0}}{{k}_{{\rm{B}}}T}\right),\end{eqnarray} \tag{ 4 }$$

where q_X denotes the internal partition sum of species X and μ_XY is the reduced mass of species X and Y (Light et al. 1969).

For H atoms, we take into account the nuclear spin contribution ${q}_{{\rm{H}}}^{\mathrm{NS}}=2I+1\,=\,2$ (I = 1/2 for ¹H) and the electron spin multiplicity ${q}_{{\rm{H}}}^{{\rm{S}}}=2S+1\,=\,2$ of the ²S ground state, which leads to ${q}_{{\rm{H}}}={q}_{{\rm{H}}}^{\mathrm{NS}}{q}_{{\rm{H}}}^{{\rm{S}}}=4$. For N⁺ ions, ${q}_{{{\rm{N}}}^{+}}={q}_{{{\rm{N}}}^{+}}^{\mathrm{NS}}{q}_{{{\rm{N}}}^{+}}^{\mathrm{FS}}$, where ${q}_{{{\rm{N}}}^{+}}^{\mathrm{NS}}=2I+1\,=\,3$ (I = 1 for ¹⁴N; Fuller 1976) and we sum over the ${}^{3}{{\rm{P}}}_{{j}_{a}}$ fine-structure (FS) states ${q}_{{{\rm{N}}}^{+}}^{\mathrm{FS}}={\sum }_{{j}_{a}=0}^{2}(2{j}_{a}+1)\exp (-{E}_{{j}_{a}}/{k}_{{\rm{B}}}T)$, with FS energies ${E}_{{j}_{a}}$ (Kramida et al. 2020). The partition sum of H₂ is calculated over the five lowest rotational states of the vibrational ground state, accounting for the different nuclear spin multiplicities g_J of odd and even (ortho and para) rotational states, ${q}_{{{\rm{H}}}_{2}}={\sum }_{J}{g}_{J}(2J+1)\exp \left(-{E}_{J}/{k}_{{\rm{B}}}T\right)$, where g_J = 1 and 3 for even and odd J, respectively. The H₂ rotational energies E_J are calculated from the molecular constants in Huber & Herzberg (1979). In the case of NH⁺, we consider the vibrational ground states of the two lowest electronic states, ²Π and ⁴Σ⁻, where we account for the rotational states with J ≤ 19/2 according to the effective Hamiltonian of Kawaguchi & Amano (1988). We checked that the neglect of the higher energy levels of NH⁺ and H₂ does not influence the results of our analysis. The nuclear spin partition function of NH⁺ is the product of the hydrogen and nitrogen nuclear spin partition functions, ${q}_{{\mathrm{NH}}^{+}}^{\mathrm{NS}}\,={q}_{{\rm{H}}}^{\mathrm{NS}}{q}_{{{\rm{N}}}^{+}}^{\mathrm{NS}}=6$.

When the partition sums are known, we can express ΔE⁰ from the measured forward and reverse reaction rate coefficients by applying Equation (4) to reaction (1) as

$$\begin{eqnarray}&&{\rm{\Delta }}{E}^{0}={k}_{{\rm{B}}}T\mathrm{ln}\left(\displaystyle \frac{{k}_{r}}{{k}_{f}}{\left(\displaystyle \frac{{\mu }_{{\mathrm{NH}}^{+},{\rm{H}}}}{{\mu }_{{{\rm{N}}}^{+}},{{\rm{H}}}_{2}}\right)}^{3/2}\displaystyle \frac{{q}_{{\mathrm{NH}}^{+}}{q}_{{\rm{H}}}}{{q}_{{{\rm{N}}}^{+}}{q}_{{{\rm{H}}}_{2}}}\right).\end{eqnarray} \tag{ 5 }$$

To evaluate k_f , we first calculate the rate coefficients of reaction (1) with ortho- and para-H₂ from the experimental data with normal and para-enriched H₂ (Zymak et al. 2013), and we take k_f as a weighted average of the state-specific reaction rate coefficients over the thermal population of H₂ nuclear spin states. The systematic uncertainties are not provided by Zymak et al. (2013) and we account for the typical 20% uncertainty of pressure determination and temperature uncertainty defined as T_coll = T_22PT + (5 ± 5) K. The thermal rate coefficient of the forward reaction can be obtained down to approximately 20 K (see Figure 3). At lower temperatures, the rate coefficient for reaction with para-H₂ is below the noise level in the measurements with para-enriched H₂. Hence, the reaction enthalpy cannot be extracted from the present data obtained at 7 K accommodator temperature.

When calculating ΔE⁰ from Equation (5), we assumed that the temperature in the present experiment is equal to either T_t or T_coll. This corresponds to two extreme cases: if we take T = T_t, we assume that the internal energy of NH⁺ is not relevant for promoting the (possibly endothermic) reverse reaction. Conversely, assuming T = T_coll means that the internal energy is equivalent to translational energy in promoting the reverse reaction. For completeness, we also consider the case $T={T}_{{\mathrm{NH}}^{+}}$, i.e., that only the internal energy is relevant.

As expected, the detailed balance analysis leads to positive ΔE⁰, i.e., the reverse reaction is exothermic. The obtained data shown in the lower panel of Figure 3 as a function of T_t, T_coll, or ${T}_{{\mathrm{NH}}^{+}}$ do not depend on temperature, in accordance with the fact that ΔE⁰ is a constant. Averaging the data obtained at different accommodator temperatures leads to ${\rm{\Delta }}{E}^{0}({T}_{{\rm{t}}})\,={18.3}_{-4.5}^{+3.1}\ \mathrm{meV}$, ${\rm{\Delta }}{E}^{0}({T}_{\mathrm{coll}})={18.4}_{-2.8}^{+1.8}\ \mathrm{meV}$, and ${\rm{\Delta }}{E}^{0}({T}_{{\mathrm{NH}}^{+}})\,={18.1}_{-1.3}^{+1.0}\ \mathrm{meV}$ when assuming T = T_t, T = T_coll, and $T={T}_{{\mathrm{NH}}^{+}}$, respectively. The averages are obtained as arithmetic means of the data and the stated error estimates represent the mean upper/lower systematic error bounds due to the systematic uncertainties (40% of n_H and ${T}_{{\mathrm{NH}}^{+}}\pm 5\,{\rm{K}}$). Regardless of our assumptions on temperature, ΔE⁰ is contained between 13.8 and 21.4 meV, i.e., our experimental value of ΔE⁰ can be represented as ΔE⁰ = (17.6 ± 3.8) meV or, after rounding, as

$$\begin{eqnarray}&&{\rm{\Delta }}{E}^{0}=(18\pm 4)\ \mathrm{meV}.\end{eqnarray} \tag{ 6 }$$

The vibrationless energy change ΔE_e (Grozdanov et al. 2016; Plašil et al. 2022) can be calculated by correcting for the zero-point vibrational energies (ZPEs) of reactants and products. Subtracting the ZPE of NH⁺ (186.9 meV Colin 1989) and adding the ZPE of H₂ (270.2 meV Huber & Herzberg 1979) leads to

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{e}=(101\pm 4)\ \mathrm{meV}.\end{eqnarray} \tag{ 7 }$$

The resulting vibrationless energy change is consistent with ΔE_e = (103 ± 3) meV obtained by Plašil et al. (2022) in the ion trap study of N⁺ + HD/D₂ reactions under the assumption of no reaction barriers.

The measured value of the reaction enthalpy ΔE⁰ can be utilized to calculate the dissociation energy of NH⁺ in the ²Π ground state toward the N⁺ + H asymptote as

$$\begin{eqnarray}&&\begin{array}{c}{D}_{0}({{\rm{N}}}^{+}{\textstyle \text{--}}{\rm{H}},{}^{2}{\rm{\Pi }})\\ ={D}_{0}({\rm{H}}{\textstyle \text{--}}{\rm{H}},{}^{1}{{\rm{\Sigma }}}_{g})-{\rm{\Delta }}{E}^{0}=(4460\pm 4){\rm{m}}{\rm{e}}{\rm{V}},\end{array}\end{eqnarray} \tag{ 8 }$$

where the H₂ dissociation energy is D₀(H − H, ¹Σ_g ) = 4478.1 meV (Liu et al. 2009). This value is in agreement with the theoretically calculated dissociation energy D₀(N⁺–H, ²Π) = 4462 meV (Tarroni et al. 1997). The dissociation energy toward the N + H⁺ asymptote can be calculated from the known ionization potentials (IPs) of H and N (Kramida et al. 2020), resulting in D₀(N–H⁺, ²Π) = D₀(N⁺–H, ²Π) + IP(H) − IP(N) = (3525 ± 4) meV, in agreement with the experimental values (3531 ± 3) meV of Adams & Smith (1985) and (3524 ± 3) meV of Marquette et al. (1988), which were obtained from measurements of k_f with para-enriched and normal H₂, HD, and D₂ with the assumption of no energy barrier. Since the process NH⁺ (²Π) → N (⁴S) + H+(¹S) involves an electronic transition, some works report instead the adiabatic energy of dissociation from the NH⁺ (⁴Σ⁻) state (Tarroni et al. 1997; Amero & Vázquez 2005). This can be calculated as D₀(N–H⁺, ⁴Σ⁻) = D₀(N–H⁺, ²Π) − ΔE_ΣΠ = (3481 ± 4) meV, where ΔE_ΣΠ = 44.0 meV is the energy difference between the lowest levels of the ⁴Σ⁻ and ²Π electronic states (Kawaguchi & Amano 1988). The theoretically determined value D₀(N–H⁺, ⁴Σ⁻) = 3496 meV (Tarroni et al. 1997) is in satisfactory agreement with our data, considering that the theory neglects the spin–orbit interaction. More information concerning NH⁺ dissociation energies can be found in Amero & Vázquez (2005) and Tarroni et al. (1997).

We have used the principle of detailed balance to derive the enthalpy of reaction (1) from the measurements of forward and reverse reaction rate coefficients. This method is accurate if the reactants both in the forward and reverse reaction measurements are in thermal equilibrium. In our study, we had to deal with a few deviations from thermal equilibrium:

(1) The nuclear spin states of H₂ in the k_f measurement were not in equilibrium. However, measurements with two different nuclear spin state populations (Zymak et al. 2013) allowed us to derive the thermal reaction rate coefficients.

(2) The FS-level population of N⁺ in the k_f measurement was not known. However, our experimental results (Zymak et al. 2013; Plašil et al. 2022) are compatible with the assumption that the FS of N⁺ ions is relaxed efficiently in collisions with He, and this conclusion is also supported by a recent theoretical prediction of relatively fast FS relaxation in collisions of N⁺ with H₂ and He (Gueguen & Lique 2023).

(3) The internal and translational temperatures, ${T}_{{\mathrm{NH}}^{+}}$ and T_t, are not in equilibrium, but we show in Figure 3 that the obtained reaction enthalpy is independent of the assumptions concerning the role of the translational and internal temperature of the reactants, owing to the fact that the reverse reaction rate coefficients are nearly constant throughout the studied temperature range.

The present value of ΔE⁰ is compared to the previously measured activation energies, E_A, and energy thresholds, E_T, of the forward reaction in Figure 4. If there are no barriers on the reaction paths, E_T should be equal to ΔE⁰ and E_A should converge to ΔE⁰ at low temperatures (k_B T ≪ E_A; Menzinger & Wolfgang 1969). The good agreement of the previously determined E_A and E_T with the measured value of ΔE⁰ confirms that the assumption of no barriers is valid (the agreement with the data of Marquette et al. 1985; Gerlich 1993 with unknown uncertainties cannot be statistically evaluated). The height of the hypothetical energy barrier can be calculated as E_b = E_A − ΔE⁰ and using the most recent value of E_A = (19.3 ± 2.7) meV for H₂ in thermodynamic equilibrium (TDE; Plašil et al. 2022) leads to E_b = (2 ± 5) meV. Therefore, the height of the possible barrier is, within the accuracy of our measurement, equal to zero. This result confirms that the interpretation of the activation energy as reaction endothermicity in the previous theoretical (Grozdanov & McCarroll 2015; Grozdanov et al. 2016; Yang et al. 2019; Gómez-Carrasco et al. 2022) and astrochemical (Dislaire et al. 2012; Roueff et al. 2015) studies was justified, providing solid ground for further research of the N⁺ + H₂ collision system.

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We are grateful to the late Prof. Dieter Gerlich for helpful discussions during the conception of the present work. We thank TU Chemnitz and the DFG for lending us the 22-pole instrument. This work was partly supported by GACR 21-28560S.

Software: lmfit (Newville et al. 2014, 2020).