Quantifying the nitrogen isotope effects during photochemical equilibrium between NO and NO₂: implications for δ¹⁵N in tropospheric reactive nitrogen

Nitrogen isotope fractionations between nitrogen oxides (NO and NO₂) play a significant role in determining the nitrogen isotopic compositions (δ¹⁵N) of atmospheric reactive nitrogen. Both the equilibrium isotopic exchange between NO and NO₂ molecules and the isotope effects occurring during the NO_x photochemical cycle are important, but both are not well constrained. The nighttime and daytime isotopic fractionations between NO and NO₂ in an atmospheric simulation chamber at atmospherically relevant NO_x levels were measured. Then, the impact of NO_x level and NO₂ photolysis rate on the combined isotopic fractionation (equilibrium isotopic exchange and photochemical cycle) between NO and NO₂ was calculated. It was found that the isotope effects occurring during the NO_x photochemical cycle can be described using a single fractionation factor, designated the Leighton cycle isotope effect (LCIE). The results showed that at room temperature, the fractionation factor of nitrogen isotopic exchange is 1.0289±0.0019, and the fractionation factor of LCIE (when O₃ solely controls the oxidation from NO to NO₂) is 0.990±0.005. The measured LCIE factor showed good agreement with previous field measurements, suggesting that it could be applied in an ambient environment, although future work is needed to assess the isotopic fractionation factors of $\mathrm{NO}+{\mathrm{RO}}_{\mathrm{2}}/{\mathrm{HO}}_{\mathrm{2}}\to {\mathrm{NO}}_{\mathrm{2}}$. The results were used to model the NO–NO₂ isotopic fractionations under several NO_x conditions. The model suggested that isotopic exchange was the dominant factor when NO_x>20 nmol mol⁻¹, while LCIE was more important at low NO_x concentrations (<1 nmol mol⁻¹) and high rates of NO₂ photolysis. These findings provided a useful tool to quantify the isotopic fractionations between tropospheric NO and NO₂, which can be applied in future field observations and atmospheric chemistry models.

1 Introduction

The nitrogen isotopic composition (δ¹⁵N) of reactive nitrogen compounds in the atmosphere is an important tool in understanding the sources and chemistry of atmospheric NO_x (NO+NO₂). It has been suggested that the δ¹⁵N value of atmospheric nitrate (HNO₃, nitrate aerosols and nitrate ions in precipitation and snow) imprints the δ¹⁵N value of NO_x sources (Elliott et al., 2009; Kendall et al., 2007); thus many studies have used the δ¹⁵N values of atmospheric nitrate to investigate NO_x sources (Chang et al., 2018; Felix et al., 2012; Felix and Elliott, 2014; Gobel et al., 2013; Hastings et al., 2004, 2009; Morin et al., 2009; Park et al., 2018; Walters et al., 2015, 2018). However, there remain questions about how isotopic fractionations that may occur during photochemical cycling of NO_x could alter the δ¹⁵N values as it partitions into NO_y (NO_y = atmospheric nitrate, NO₃, N₂O₅, HONO, etc.; Chang et al., 2018; Freyer, 1991; Hastings et al., 2004; Jarvis et al., 2008; Michalski et al., 2005; Morin et al., 2009; Zong et al., 2017). Similarly, other complex reactive nitrogen chemistry, such as nitrate photolysis and redeposition in ice and snow (Frey et al., 2009), may impact the δ¹⁵N of NO_y and atmospheric nitrate. The fractionation between NO and NO₂ via isotope exchange has been suggested to be the dominant factor in determining the δ¹⁵N of NO₂ and ultimately atmospheric nitrate (Freyer, 1991; Freyer et al., 1993; Savarino et al., 2013; Walters et al., 2016). However, isotopic fractionations occur in most, if not all, NO_x and NO_y reactions, while most of these are still unknown or, if calculated (Walters and Michalski, 2015), unverified by experiments. Since the atmospheric chemistry of NO_y varies significantly in different environments (e.g., polluted vs. pristine, night vs. day), the isotopic fractionations associated with NO_y chemistry are also likely to vary in different environments. These unknowns could potentially bias conclusions about NO_x source apportionment reached when using nitrogen isotopes. Therefore, understanding the isotopic fractionations between NO and NO₂ during photochemical cycling could improve our understanding of the relative role of sources versus chemistry for controlling the δ¹⁵N variations in atmospheric NO₂ and nitrate.

In general, there are three types of isotopic fractionation effects associated with NO_x chemistry (Fig. 1a). The first type is the equilibrium isotopic effect (EIE), i.e., isotope exchange between two compounds without forming new molecules (Urey, 1947; Bigeleisen and Mayer, 1947), which for nitrogen isotopes in the NO_x system is the ${}^{\mathrm{15}}\mathrm{NO}{+}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\leftrightarrow {}^{\mathrm{14}}\mathrm{NO}{+}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}$ exchange reaction (Begun and Melton, 1956; Walters et al., 2016). The second type is the kinetic isotopic effect (KIE) associated with difference in isotopologue rate coefficients during unidirectional reactions (Bigeleisen and Wolfsberg, 1957). In the NO_x system this KIE would manifest in the oxidation of NO into NO₂ by ${\mathrm{O}}_{\mathrm{3}}/{\mathrm{HO}}_{\mathrm{2}}/{\mathrm{RO}}_{\mathrm{2}}$. The third type is the photochemical isotope fractionation effect (PHIFE; Miller and Yung, 2000), which for NO_x is the isotopic fractionation associated with NO₂ photolysis. All three fractionations could impact the δ¹⁵N value of NO₂ and consequently atmospheric nitrate, but the relative importance of each may vary.

Figure 1 (a) A sketch of the isotopic fractionation processes between NO and NO₂; both fractionation factors are determined in this work. (b) Results from five dark experiments (red circles) yielded a line with slope of 28.1 ‰ and an α(NO₂–NO) value of 1.0289, while the results from five UV irradiation experiments (blue squares) showed a smaller slope. (c) Results from five UV irradiation experiments (blue squares) and a previous field study (purple triangle), comparing to the dark experiments (red circle). The three lines represent different (α₂−α₁) values: the (α₂−α₁) $=-\mathrm{10}$ ‰ line showed the lowest RMSE to our experimental data as well as the previous field observations. The error bars in panels (b) and (c) represented the combined uncertainties of NO_x concentration measurements and isotopic analysis.

The limited number of studies on the EIE in the NO_x cycle have significant uncertainties. Discrepancies in the EIE for ${}^{\mathrm{15}}\mathrm{NO}{+}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\leftrightarrow {}^{\mathrm{14}}\mathrm{NO}{+}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}$ have been noted in several studies. Theoretical calculations predicted isotope fractionation factors (α) ranging from 1.035 to 1.042 at room temperature (Begun and Fletcher, 1960; Monse et al., 1969; Walters and Michalski, 2015) due to the different approximations used to calculate harmonic frequencies in each study. Likewise, two separate experiments measured different room temperature fractionation factors of 1.028±0.002 (Begun and Melton, 1956) and 1.0356±0.0015 (Walters et al., 2016). A concern in both experiments is that they were conducted in small chambers with high NO_x concentrations (hundreds of micromoles per mole), significantly higher than typical ambient atmospheric NO_x levels (usually less than 0.1 µmol mol⁻¹). Whether the isotopic fractionation factors determined by these experiments are applicable in the ambient environment is uncertain because of possible wall effects and formation of higher oxides, notably N₂O₄ and N₂O₃ at these high NO_x concentrations.

Even less research has examined the KIE and PHIFE occurring during NO_x cycling. The KIE of NO+O₃ has been theoretically calculated (Walters and Michalski, 2016) but has not been experimentally verified. The NO₂ PHIFE has not been experimentally determined or theoretically calculated. As a result, field observation studies often overlook the effects of PHIFE and KIE. Freyer et al. (1993) measured NO_x concentrations and the δ¹⁵N values of NO₂ over a 1-year period at Julich, Germany, and inferred a combined NO_x isotope fractionation factor (EIE+KIE+PHIFE) of 1.018±0.001. Freyer et al. (1993) suggested that the NO_x photochemical cycle (KIE and PHIFE) tends to diminish the equilibrium isotopic fractionation (EIE) between NO and NO₂. Even if this approach were valid, applying this single fractionation factor elsewhere, where NO_x and O₃ concentrations and actinic fluxes are different, would be tenuous given that these factors may influence the relative importance of EIE, KIE and PHIFE (Hastings et al., 2004; Walters et al., 2016). Therefore, to quantify the overall isotopic fractionations between NO and NO₂ at various tropospheric conditions, it is crucial to know (1) isotopic fractionation factors of EIE, KIE and PHIFE individually and (2) the relative importance of each factor under various conditions.

In this work, we aim to quantify the nitrogen isotope fractionation factors between NO and NO₂ at photochemical equilibrium. First, we measure the N isotope fractionations between NO and NO₂ in an atmospheric simulation chamber at atmospherically relevant NO_x levels. Then, we provide mathematical solutions to assess the impact of NO_x level and NO₂ photolysis rate (j(NO₂)) on the relative importance of EIE, KIE and PHIFE. Subsequently we use the solutions and chamber measurements to calculate the isotopic fractionation factors of EIE, KIE and PHIFE. Lastly, using the calculated fractionation factors and the equations, we model the NO–NO₂ isotopic fractionations at several sites to illustrate the behavior of δ¹⁵N values of NO_x in the ambient environment.

2 Methods

The experiments were conducted using a 10 m³ atmospheric simulation chamber at the National Center for Atmospheric Research (see descriptions in Appendix A and Zhang et al., 2018). A set of mass flow controllers was used to inject NO and O₃ into the chamber. NO was injected at 1 L min⁻¹ from an in-house NO∕N₂ cylinder (133.16 µmol mol⁻¹ NO in ultrapure N₂), and O₃ was generated by flowing zero air through a flow tube equipped with a UV Pen-Ray lamp (UVP LLC., CA) into the chamber at 5 L min⁻¹. NO and NO₂ concentrations were monitored in real time by chemiluminescence with a detection limit of 0.5 nmol mol⁻¹ (model CLD 88Y, Eco Physics, MI) as were O₃ concentrations using a UV absorption spectroscopy with a detection limit of 0.5 nmol mol⁻¹ (model 49, Thermo Scientific, CO). In each experiment, the actual amounts of NO and O₃ injected were calculated using measured NO_x and O₃ concentrations after a steady state was reached (usually within 1 h). The wall loss rate of NO₂ was tested by monitoring O₃ (29 nmol mol⁻¹) and NO_x (62 nmol mol⁻¹) over a 4 h period. After the NO and NO₂ concentrations reached a steady state, no decrease in NO₂ concentrations was observed, showing that chamber wall loss was negligible.

Three experiments were conducted to measure the δ¹⁵N value of the tank NO (i.e., the δ¹⁵N value of total NO_x). In each of these experiments, a certain amount of O₃ was first injected into the chamber, then approximately the same amount of NO was injected into the chamber to ensure 100 % of the NO_x was in the form of NO₂ with little O₃ (<15 nmol mol⁻¹) remaining in the chamber such that the O₃+NO₂ reaction was negligible. The NO₂ in the chamber was then collected and its δ¹⁵N value measured, which equates to the δ¹⁵N value of the tank NO.

Two sets of experiments were conducted to separately investigate the EIE, KIE and PHIFE. The first set of experiments was conducted in the dark. In each of these dark experiments, a range of NO and O₃ ([O₃] < [NO]) was injected into the chamber to produce NO–NO₂ mixtures with [NO]∕[NO₂] ratios ranging from 0.43 to 1.17. The N isotopes of these mixtures were used to investigate the EIE between NO and NO₂. The second set of experiments was conducted under irradiation of UV lights (300–500 nm; see Appendix A for irradiation spectrum). Under such conditions, NO, NO₂ and O₃ reached a photochemically steady state, which combined the isotopic effects of EIE, KIE and PHIFE.

In all experiments, the concentrations of NO, NO₂ and O₃ were allowed to reach a steady state, and the product NO₂ was collected from the chamber using a honeycomb denuder tube. After the NO, NO₂ and O₃ concentrations reached a steady state, well-mixed chamber air was drawn out through a 40 cm long Norprene thermoplastic tubing at 10 L min⁻¹ and passed through a honeycomb denuder system (Chemcomb 3500, Thermo Scientific). Based on flow rate, the NO₂ residence time in the tubing was less than 0.5 s; thus in the light-on experiments where NO and O₃ coexisted, the NO₂ produced inside the transfer tube through NO+O₃ reactions should be <0.03 nmol mol⁻¹ (using the upper limit of NO and O₃ concentrations in our experiments). The honeycomb denuder system consisted of two honeycomb denuder tubes connected in series. Each honeycomb denuder tube is a glass cylinder 38 mm long and 47 mm in diameter and consists of 212 hexagonal tubes with inner diameters of 2 mm. Before collecting samples, each denuder tube was coated with a solution of 10 % KOH and 25 % guaiacol in methanol and then dried by flowing N₂ gas through the denuder tube for 15 s (Williams and Grosjean, 1990; Walters et al., 2016). The NO₂ reacted with the guaiacol coating and was converted into ${\mathrm{NO}}_{\mathrm{2}}^{-}$ that was retained on the denuder tube wall (Williams and Grosjean, 1990). NO was inert to the denuder tube coating: a control experiment sampled pure NO using the denuder tubes, which did not show any measurable ${\mathrm{NO}}_{\mathrm{2}}^{-}$. The NO₂ collection efficiency of a single honeycomb denuder tube was tested in another control experiment: air containing 66 nmol mol⁻¹ of NO₂ was drawn out of the chamber through a denuder tube, and the NO₂ concentration at the exit of the tube holder was measured and found to be below the detection limit (<1 nmol mol⁻¹), suggesting that the collection efficiency was nearly 100 % when [NO₂] <66 nmol mol⁻¹. Furthermore, when the denuder system consisted of two denuder tubes in series, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the second denuder was below the detection limit, indicating trivial NO₂ breakthrough. Each NO₂ collection lasted for 0.5–3 h in order to collect enough ${\mathrm{NO}}_{\mathrm{2}}^{-}$ for isotopic analysis (at least 300 nmol). After collection, the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ was leached from each denuder tube by rinsing thoroughly with 10 mL deionized water into a clean polypropylene container and stored frozen until isotopic analysis. Isotopic analysis was conducted at the Purdue Stable Isotope Laboratory. For each sample, approximately 50 nmol of the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ extract was mixed with 2 M sodium azide solution in an acetic acid buffer in an airtight glass vial, then shaken overnight to completely reduce all the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ to N₂O_(g) (Casciotti and McIlvin, 2007; McIlvin and Altabet, 2005). The product N₂O was directed into a Thermo Scientific GasBench equipped with a cryotrap, then the δ¹⁵N of the N₂O was measured using a Delta-V isotope ratio mass spectrometer (IRMS). Six coated denuder tubes that did not get exposed to NO₂ were also analyzed using the same chemical procedure, which did not show any measurable signal on the IRMS, suggesting that the blank from both the sampling process and the chemical conversion process was negligible. The overall analytical uncertainty for δ¹⁵N analysis was 0.5 ‰ (1σ) based on replicate analysis of in-house ${\mathrm{NO}}_{\mathrm{2}}^{-}$ standards.

3 Results and discussions

3.1 Equilibrium isotopic fractionation between NO and NO₂

The equilibrium isotope fractionation factor, α(NO₂–NO), is the ¹⁵N enrichment in NO₂ relative to NO and is expressed as the ratio of rate constants k₂∕k₁ of two reactions:

$$\begin{array}{}\text{(R1)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}{+}^{\mathrm{14}}\mathrm{NO}\to {}^{\mathrm{15}}\mathrm{NO}{+}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}},\\ & \text{rate constant}=k_{\mathrm{1}}\end{array}\text{(R2)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{15}}\mathrm{NO}{+}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\to {}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}{+}^{\mathrm{14}}\mathrm{NO},\\ & \text{rate constant}=k_{\mathrm{2}}=k_{\mathrm{1}}\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right),\end{array}\end{array}$$

where k₁ is the rate constant of the isotopic exchange, which was previously determined to be $\mathrm{8.14}\times {\mathrm{10}}^{-\mathrm{14}}$ cm³ s⁻¹ (Sharma et al., 1970). The reaction time required for NO–NO₂ to reach isotopic equilibrium was estimated using the exchange rate constants in a simple kinetics box model (BOXMOX; Knote et al., 2015). The model predicts that at typical NO_x concentrations used during the chamber experiments (7.7–62.4 nmol mol⁻¹), isotopic equilibrium would be reached within 15 min (see Appendix B). Since the sample collection usually started 1 h after NO_x was well mixed in the chamber, there was sufficient time to reach full isotope equilibrium. The isotope equilibrium fractionation factor (α(NO₂–NO)) is then calculated to be

$$\begin{array}{}\text{(1)}& \mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)={\displaystyle \frac{\left[{}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}\right]\left[{}^{\mathrm{14}}\mathrm{NO}\right]}{\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\right]\left[{}^{\mathrm{15}}\mathrm{NO}\right]}}={\displaystyle \frac{R\left({\mathrm{NO}}_{\mathrm{2}}\right)}{R\left(\mathrm{NO}\right)}},\end{array}$$

where R(NO,NO₂) are the ¹⁵N∕¹⁴N ratios of NO and NO₂. By definition, the δ¹⁵N(NO) is $\left(R\right(\mathrm{NO})/R(\mathrm{reference})-\mathrm{1})\times \mathrm{1000}$ ‰, and δ¹⁵N(NO₂) is $\left(R\right({\mathrm{NO}}_{\mathrm{2}})/R(\mathrm{reference})-\mathrm{1})\times \mathrm{1000}$ ‰, but hereafter, the δ¹⁵N values of NO, NO₂ and NO_x are referred to as δ(NO), δ(NO₂) and δ(NO_x), respectively. Equation (1) leads to

$$\begin{array}{}\text{(2)}& \mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }\left(\mathrm{NO}\right)=\left(\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}\right)(\mathrm{1}+\mathit{\delta }(\mathrm{NO}\left)\right).\end{array}$$

Using Eq. (2) and applying NO_x isotopic mass balance ($\mathit{\delta }\left({\mathrm{NO}}_x\right)=f\left({\mathrm{NO}}_{\mathrm{2}}\right)\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)+(\mathrm{1}-f({\mathrm{NO}}_{\mathrm{2}}\left)\right)\mathit{\delta }\left(\mathrm{NO}\right)$, $f\left({\mathrm{NO}}_{\mathrm{2}}\right)=\left[{\mathrm{NO}}_{\mathrm{2}}\right]/\left(\right[\mathrm{NO}]+[{\mathrm{NO}}_{\mathrm{2}}\left]\right)$) yields

$$\begin{array}{}\text{(3)}& {\displaystyle \frac{\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }\left({\mathrm{NO}}_x\right)}{\mathrm{1}+\left({\mathrm{NO}}_{\mathrm{2}}\right)}}={\displaystyle \frac{\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}}{\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}\left(\mathrm{1}-f\left({\mathrm{NO}}_{\mathrm{2}}\right)\right).\end{array}$$

Here, δ(NO_x) equals the δ¹⁵N value of the cylinder NO, and f(NO₂) is the molar fraction of NO₂ with respect to total NO_x. Three experiments (Table 1) that measured δ(NO_x) showed consistent δ(NO_x) values of $-\mathrm{58.7}\pm \mathrm{0.8}$ ‰ (n=3), indicating that δ(NO_x) remained unchanged throughout the experiments (as expected for isotope mass balance). Thus, the δ(NO_x) can be treated as a constant in Eq. (3), and the linear regression of $(\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }({\mathrm{NO}}_x\left)\right)/(\mathrm{1}+\mathit{\delta }({\mathrm{NO}}_{\mathrm{2}}\left)\right)$ versus 1−f(NO₂) should have an intercept of 0 and a slope of $\left(\mathit{\alpha }\right({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO})-\mathrm{1})/\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)$.

Table 1 Experimental conditions; concentrations of NO, NO₂ and O₃ at a steady state; and measured δ(NO₂) values.

The plot of $(\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }({\mathrm{NO}}_{\mathrm{x}}\left)\right)/(\mathrm{1}+\mathit{\delta }({\mathrm{NO}}_{\mathrm{2}}\left)\right)$ as a function of 1−f(NO₂) values from five experiments yields an α(NO₂–NO) value of 1.0289±0.0019 at room temperature (Fig. 1b and Table 1). This fractionation factor is comparable to previously measured values but with some differences. Our result agrees well with the α(NO₂–NO) value of 1.028±0.002 obtained by Begun and Melton (1956) at room temperature. However, Walters et al. (2016) determined the α(NO₂–NO) values of NO–NO₂ exchange in a 1 L reaction vessel, which showed a slightly higher α(NO₂–NO) value of 1.035. This discrepancy might originate from rapid heterogeneous reactions on the wall of the reaction vessel at high NO_x concentrations and the small chamber size used by Walters et al. (2016). They used a reaction vessel made of Pyrex, which is known to absorb water (Do Remus et al., 1983; Takei et al., 1997) and can react with NO₂, forming HONO, HNO₃ and other N compounds. Additionally, previous studies have suggested that Pyrex walls enhance the formation rate of N₂O₄ by over an order of magnitude (Barney and Finlayson-Pitts, 2000; Saliba et al., 2001), which at isotopic equilibrium is enriched in ¹⁵N compared to NO and NO₂ (Walters and Michalski, 2015). Therefore, their measured α(NO₂–NO) might be slightly higher than the actual α(NO₂–NO) value. In this work, the 10 m³ chamber has a much smaller surface-to-volume ratio relative to Walters et al. (2016), which minimizes wall effects, and the walls were made of Teflon, which minimizes NO₂ surface reactivity, as evidenced by the NO₂ wall loss control experiment. Furthermore, the low NO_x mixing ratios in our experiments minimized N₂O₄ and N₂O₃ formation. At NO and NO₂ concentrations of 50 nmol mol⁻¹, the steady-state concentrations of N₂O₄ and N₂O₃ were calculated to be 0.014 and 0.001 pmol mol⁻¹, respectively (Atkinson et al., 2004). Therefore, we suggest that our measured α(NO₂–NO) value (1.0289±0.0019) may better reflect the room temperature (298 K) NO–NO₂ EIE in the ambient environment.

Unfortunately, the chamber temperature could not be controlled, so we were not able to investigate the temperature dependence of the EIE. Hence, we speculate that the α(NO₂–NO) follows a similar temperature dependence pattern calculated in Walters et al. (2016). Walters et al. (2016) suggested that the α(NO₂–NO) value would be 0.0047 higher at 273 K and 0.002 lower at 310 K relative to room temperature (298 K). Using this pattern and our experimentally determined data, we suggest that the α(NO₂–NO) values at 273, 298 and 310 K are 1.0336±0.0019, 1.0289±0.0019 and 1.0269±0.0019, respectively. This 0.0067 variation at least partially contributes to the daily and seasonal variations in δ¹⁵N values of NO₂ and nitrate in some areas (e.g., polar regions with strong seasonal temperature variation). Thus, future investigations should be conducted to verify the EIE temperature dependence.

3.2 Kinetic isotopic fractionation of Leighton cycle

The photochemical reactions of NO_x will compete with the isotope exchange fractionations between NO and NO₂. The NO–NO₂ photochemical cycle in the chamber was controlled by the Leighton cycle: NO₂ photolysis and the NO+O₃ reaction. This is because there were no volatile organic compounds (VOCs) in the chamber, so no RO₂ was produced, which excludes the NO+RO₂ reaction. Likewise, the low water vapor content (relative humidity <10 %) and the minor flux of photons (<310 nm) results in minimal OH production and hence little HO₂ formation, and subsequently a trivial amount of NO₂ would be formed by NO+HO₂. Applying these limiting assumptions, the EIE between NO and NO₂ (Reactions R1–R2) is only competing with the KIE (Reactions R3–R4) and the PHIFE in Reactions (R5)–(R6):

$$\begin{array}{}\text{(R3)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\to {}^{\mathrm{14}}\mathrm{NO}+\mathrm{O},\\ & \text{rate constant}=j\left({\mathrm{NO}}_{\mathrm{2}}\right)\end{array}\text{(R4)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}\to {}^{\mathrm{15}}\mathrm{NO}+\mathrm{O},\\ & \text{rate constant}=j\left({\mathrm{NO}}_{\mathrm{2}}\right){\mathit{\alpha }}_{\mathrm{1}}\end{array}\text{(R5)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{14}}\mathrm{NO}+{\mathrm{O}}_{\mathrm{3}}\to {}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}+{\mathrm{O}}_{\mathrm{2}},\\ & \text{rate constant}=k_{\mathrm{5}}\end{array}\text{(R6)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{15}}\mathrm{NO}+{\mathrm{O}}_{\mathrm{3}}\to {}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}+{\mathrm{O}}_{\mathrm{2}},\\ & \text{rate constant}=k_{\mathrm{5}}{\mathit{\alpha }}_{\mathrm{2}},\end{array}\end{array}$$

in which j(NO₂) is the NO₂ photolysis rate ($\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$ s⁻¹ in these experiments), k₅ is the rate constant for the NO+O₃ reaction ($\mathrm{1.73}\times {\mathrm{10}}^{-\mathrm{14}}$ cm³ s⁻¹; Atkinson et al., 2004), and α_1,2 are isotopic fractionation factors for the two reactions. Previous studies (Freyer et al., 1993; Walters et al., 2016) have attempted to assess the competition between EIE (Reactions R1–R2), KIE and PHIFE (Reactions R3–R6), but none of them quantified the relative importance of the two processes, nor were α₁ or α₂ values experimentally determined. Here we provide the mathematical solution of EIE, KIE and PHIFE to illustrate how Reactions (R1)–(R6) affect the isotopic fractionations between NO and NO₂.

First, the NO₂ lifetime with respect to isotopic exchange with NO (τ_exchange) and photolysis (τ_photo) was determined:

$$\begin{array}{}\text{(4)}& {\displaystyle }{\mathit{\tau }}_{\mathrm{exchange}}={\displaystyle \frac{\mathrm{1}}{k_{\mathrm{1}}\left[\mathrm{NO}\right]}}\text{(5)}& {\displaystyle }{\mathit{\tau }}_{\mathrm{photo}}={\displaystyle \frac{\mathrm{1}}{j\left({\mathrm{NO}}_{\mathrm{2}}\right)}}.\end{array}$$

We then define an A factor:

$$\begin{array}{}\text{(6)}& A=\left\{\begin{array}{ll}\frac{{\mathit{\tau }}_{\mathrm{exchange}}}{{\mathit{\tau }}_{\mathrm{photo}}}& \text{when }j\left({\mathrm{NO}}_{\mathrm{2}}\right)\ne \mathrm{0}\\ \mathrm{0}& \text{when }j\left({\mathrm{NO}}_{\mathrm{2}}\right)=\mathrm{0}.\end{array}\right.\end{array}$$

Using Reactions (R1)–(R6) and Eqs. (1)–(6), we solved steady-state δ(NO₂) and δ(NO) values (see calculations in Appendix C). Our calculations show that the δ(NO₂)–δ(NO) and δ(NO₂)–δ(NO_x) values at a steady state can be expressed as functions of α₁, α₂, α(NO₂–NO) and A:

$$\begin{array}{}\text{(7)}& {\displaystyle }\begin{array}{rl}\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }\left(\mathrm{NO}\right)& ={\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}\right)A+\left(\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}\right)}{{\mathit{\alpha }}_{\mathrm{2}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}\\ & \left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\\ & \approx {\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}\right)A+\left(\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}\right)}{A+\mathrm{1}}}\\ & \left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\end{array}\text{(8)}& {\displaystyle }\begin{array}{rl}\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }\left({\mathrm{NO}}_x\right)& ={\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}\right)A+\left(\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}\right)}{{\mathit{\alpha }}_{\mathrm{2}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}\\ & \left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\left(\mathrm{1}-f\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\\ & \approx {\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}\right)A+\left(\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}\right)}{A+\mathrm{1}}}\\ & \left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\left(\mathrm{1}-f\left({\mathrm{NO}}_{\mathrm{2}}\right)\right).\end{array}\end{array}$$

Equation (7) shows the isotopic fractionation between NO and NO₂; δ(NO₂)–δ(NO) is mainly determined by A; the EIE factor (α(NO₂–NO)−1) and the (α₂−α₁) factor assuming 1+δ(NO₂) is close to 1. This (α₂−α₁) represents a combination of KIE and PHIFE, suggesting that they act together as one factor; therefore, we name the (α₂−α₁) factor the Leighton cycle isotopic effect (LCIE). Using measured δ(NO₂) and δ(NO_x) values, A values (Table 1), and the previously determined α(NO₂–NO) value, we plot $\frac{\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }\left({\mathrm{NO}}_x\right)}{\left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\left(\mathrm{1}-f\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)}$ $\left(\text{equals }\frac{\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-\mathit{\delta }\left(\mathrm{NO}\right)}{\left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)}\right)$ against the A value and use Eqs. (7) and (8) to estimate the (α₂−α₁) value (Fig. 1c). The plot shows that the best fit for the LCIE factor is ($-\mathrm{10}\pm \mathrm{5}$) ‰ (root mean square error, RMSE, was lowest when ${\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}=-\mathrm{10}$ ‰). The uncertainties in the LCIE factor are relatively higher than that of the EIE factor, mainly because of the accumulated analytical uncertainties at low NO_x and O₃ concentrations and low A values (0.10–0.28) due to the relatively low j(NO₂) value ($\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$ s⁻¹) under the chamber irradiation conditions.

This LCIE factor determined in our experiments is in good agreement with theoretical calculations. Walters and Michalski (2016) previously used an ab initio approach to determine an α₂ value of 0.9933 at room temperature, 0.9943 at 237 K and 0.9929 at 310 K. The total variation in α₂ values from 273 to 310 K is only 1.4 ‰, significantly smaller than our experimental uncertainty (±5 ‰). The α₁ value was calculated using a zero-point energy (ZPE) shift model (Miller and Yung, 2000) to calculate the isotopic fractionation of NO₂ by photolysis. Briefly, this model assumes both isotopologues have the same quantum yield function, and the PHIFE was only caused by the differences in the ¹⁵NO₂ and ¹⁴NO₂ absorption cross section as a function of wavelength; thus α₁ values do not vary by temperature. The ¹⁵NO₂ absorption cross section was calculated by shifting the ¹⁴NO₂ absorption cross section by the ¹⁵NO₂ zero-point energy (Michalski et al., 2004). When the ZPE shift model was used with the irradiation spectrum of the chamber lights, the resulting α₁ value was 1.0023. Therefore, the theoretically predicted α₂−α₁ value should be −0.0090, i.e., $-\mathrm{9.0}\pm \mathrm{0.7}$ ‰ when the temperature ranges from 273 to 310 K. This result shows excellent agreement with our experimentally determined room temperature α₂−α₁ value of $-\mathrm{10}\pm \mathrm{5}$ ‰ .

This model was then used to evaluate the variations in α₁ under different lighting conditions. The tropospheric ultraviolet and visible (TUV) radiation model (TUV5.3.2; Madronich and Flocke, 1999) was used to calculate the solar wavelength spectrum at three different conditions: early morning or late afternoon (solar zenith angle =85^∘), midmorning or afternoon (solar zenith angle =45^∘), and 12:00 local time (LT; solar zenith angle =0^∘). These spectra were used in the ZPE shift model to calculate the α₁ values, which are 1.0025, 1.0028 and 1.0029 at solar zenith angles of 85, 45 and 0^∘, respectively. These values, along with the predicted α₁ value in the chamber, showed a total span of 0.6 ‰ (1.0026±0.0003), which is again significantly smaller than our measured uncertainty. Therefore, we suggest that our experimentally determined LCIE factor ($-\mathrm{10}\pm \mathrm{5}$ ‰) can be used in most tropospheric solar irradiation spectra.

The equations can also be applied in tropospheric environments to calculate the combined isotopic fractionations of EIE and LCIE for NO and NO₂. First, the NO₂ sink reactions (mainly NO₂+OH in the daytime) are at least 2–3 orders of magnitude slower than the Leighton cycle and the NO–NO₂ isotope exchange reactions (Walters et al., 2016); therefore their effects on the δ(NO₂) should be minor. Second, although the conversion of NO into NO₂ in the ambient environment is also controlled by NO+RO₂ and HO₂ in addition to NO+O₃ (e.g., King et al., 2001), Eq. (7) still showed good agreement with field observations in previous studies. Freyer et al. (1993) determined the annual average daytime δ(NO₂)–δ(NO) at Julich, Germany, along with average daytime NO concentration (9 nmol mol⁻¹, similar to our experimental conditions) to be $+\mathrm{18.03}\pm \mathrm{0.98}$ ‰. Using Eq. (7), assuming the daytime average j(NO₂) value throughout the year was $(\mathrm{5.0}\pm \mathrm{1.0})\times {\mathrm{10}}^{-\mathrm{3}}$, and a calculated A value from measured NO_x concentration ranged from 0.22 to 0.33, the average NO–NO₂ fractionation factor was calculated to be $+\mathrm{19.8}\pm \mathrm{1.4}$ ‰ (Fig. 1c), in excellent agreement with the measurements in the present study. This agreement suggests that the $\mathrm{NO}+{\mathrm{RO}}_{\mathrm{2}}/{\mathrm{HO}}_{\mathrm{2}}$ reactions might have similar fractionation factors as NO+O₃. Therefore, we suggest that Eqs. (7) and (8) can be used to estimate the isotopic fractionations between NO and NO₂ in the troposphere.

3.3 Calculating nitrogen isotopic fractionations of NO–NO₂

First, Eq. (7) was used to calculate the Δ(NO₂–NO)=δ(NO₂)–δ(NO) at a wide range of NO_x concentrations and f(NO₂) and j(NO₂) values (Fig. 2a–d), assuming $\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\approx \mathrm{1}$. j(NO₂) values of 0 s⁻¹ (Fig. 2a), $\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$ s⁻¹ (Fig. 2b), $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{3}}$ s⁻¹ (Fig. 2c) and $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{2}}$ s⁻¹ (Fig. 2d) were selected to represent nighttime, dawn (as well as the laboratory conditions of our experiments), daytime average and 12:00 LT, respectively. Each panel represented a fixed j(NO₂) value, and the Δ(NO₂–NO) values were calculated as a function of the A value, which was derived from NO_x concentration and f(NO₂). The A values have a large span, from 0 to 500, depending on the j(NO₂) value and the NO concentration. When A=0 (j(NO₂)=0) and f(NO₂)<1 (meaning NO and NO₂ coexist and [O₃] = 0), Eqs. (7) and (8) become Eqs. (2) and (3), showing that the EIE was the sole factor; the Δ(NO₂–NO) values were solely controlled by EIE, which has a constant value of +28.9 ‰ at 298 K (Fig. 2a). When j(NO₂)>0, the calculated Δ(NO₂–NO) values showed a wide range from −10.0 ‰ (controlled by LCIE factor: ${\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}=-\mathrm{10}$ ‰) to +28.9 ‰ (controlled by EIE factor: $\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}=+\mathrm{28.9}$ ‰). Figure 2b–d display the transition from an LCIE-dominated regime to an EIE-dominated regime. The LCIE-dominated regime is characterized by low [NO_x] (<50 pmol mol⁻¹), representing remote ocean areas and polar regions (Beine et al., 2002; Custard et al., 2015). At this range the A value can be greater than 200; thus Eq. (7) can be simplified as $\mathrm{\Delta }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)={\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}$, suggesting that the LCIE almost exclusively controls the NO–NO₂ isotopic fractionation. The Δ(NO₂–NO) values of these regions are predicted to be <0 ‰ during most times of the day and $<-\mathrm{5}$ ‰ at 12:00 LT. On the other hand, the EIE-dominated regime was characterized by high [NO_x] (>20 nmol mol⁻¹) and low f(NO₂) (<0.6), representative of regions with intensive NO emissions, e.g., near roadside or stack plumes (Clapp and Jenkin, 2001; Kimbrough et al., 2017). In this case, the τ_exchange are relatively short (10–50 s) compared to the τ_photo (approximately 100 s at 12:00 LT and 1000 s at dawn); therefore the A values are small (0.01–0.5). The EIE factor in this regime thus is much more important than the LCIE factor, resulting in high Δ(NO₂–NO) values (>20 ‰). Between the two regimes, both EIE and LCIE are competitive, and therefore it is necessary to use Eq. (7) to quantify the Δ(NO₂–NO) values.

Figure 2 Calculating isotopic fractionation values between NO–NO₂ (Δ(NO₂–NO); a–d) and NO_x–NO₂ (Δ(NO₂–NO_x); e–h) at various j(NO₂), NO_x level and f(NO₂) using Eqs. (7) and (8). Each panel represents a fixed j(NO₂) value (on the upper-right side of each panel), and the fractionation values are shown by color. Lines are contours with the same fractionation values at an interval of 5 ‰; the contour line representing 0 ‰ was marked in each panel except for panels (a) and (e).

Figure 2 also implies that changes in the j(NO₂) value can cause the diurnal variations in Δ(NO₂–NO) values. Changing j(NO₂) would affect the value of A and consequently the NO–NO₂ isotopic fractionations in two ways: (1) changes in the j(NO₂) value would change the photolysis intensity and therefore the τ_photo value; (2) in addition, changes in the j(NO₂) value would also alter the steady-state NO concentration, therefore changing the τ_exchange (Fig. 2c). The combined effect of these two factors on the A value varies along with the atmospheric conditions and thus needs to be carefully calculated using NO_x concentration data and atmospheric chemistry models.

We then calculated the differences in δ¹⁵N values between NO₂ and total NO_x, e.g., Δ(NO₂–NO_x)=δ(NO₂)–δ(NO_x) in Fig. 2e–h. Since Δ(NO₂–NO_x) are connected through the observed δ¹⁵N of NO₂ (or nitrate) to the δ¹⁵N of NO_x sources, this term might be useful in field studies (e.g., Chang et al., 2018; Zong et al., 2017). The calculated Δ(NO₂–NO_x) values (Fig. 2e–h) also showed an LCIE-dominated regime at low [NO_x] and an EIE-dominated regime at high [NO_x]. The Δ(NO₂–NO_x) values were dampened by the 1−f(NO₂) factor comparing to Δ(NO₂–NO), as shown in Eqs. (3) and (8): Δ(NO₂–NO_x)=Δ(NO₂–NO) (1−f(NO₂)). At high f(NO₂) values (>0.8), the differences between δ(NO₂) and δ(NO_x) were less than 5 ‰; thus the measured δ(NO₂) values were similar to δ(NO_x), although the isotopic fractionation between NO and NO₂ could be noteworthy. Some ambient environments with significant NO emissions or high NO₂ photolysis rates usually have f(NO₂) values between 0.4 and 0.8 (Mazzeo et al., 2005; Vicars et al., 2013). In this scenario, the Δ(NO₂–NO_x) values in Fig. 2f–h showed wide ranges of −4.8 ‰ to +15.6 ‰, −6.0 ‰ to +15.0 ‰ and −6.3 ‰ to +14.2 ‰ at $j\left({\mathrm{NO}}_{\mathrm{2}}\right)=\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$, $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{3}}$ and $\mathrm{1}\times {\mathrm{10}}^{-\mathrm{2}}$ s⁻¹, respectively. These significant differences again highlighted the importance of both LCIE and EIE (Eqs. 7 and 8) in calculating the Δ(NO₂–NO_x). In the following discussion, we assume that (1) the α₁ value remains constant (see discussion above), (2) the $\mathrm{NO}+{\mathrm{RO}}_{\mathrm{2}}/{\mathrm{HO}}_{\mathrm{2}}$ reactions have the same fractionation factors (α₂) as NO+O₃, and (3) both EIE and LCIE do not display significant temperature dependence. We then use Eqs. (7) and (8) and this laboratory-determined LCIE factor (−10 ‰) to calculate the nitrogen isotopic fractionation between NO and NO₂ at various tropospheric atmospheric conditions.

4 Implications

The daily variations in Δ(NO₂–NO_x) values at two roadside NO_x monitoring sites were predicted to demonstrate the effects of NO_x concentrations to the NO–NO₂ isotopic fractionations. Hourly NO and NO₂ concentrations were acquired from a roadside site at Anaheim, CA (https://www.arb.ca.gov, last access: 9 August 2019), and an urban site at Evansville, IN (http://idem.tx.sutron.com, last access: 9 August 2019), on 25 July 2018. The hourly j(NO₂) values output from the TUV model (Madronich and Flocke, 1999) at these locations were used to calculate the daily variations in Δ(NO₂–NO_x) values (Fig. 3a, b) by applying Eq. (8) and assuming $\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\approx \mathrm{1}$. Hourly NO_x concentrations were 12–51 nmol mol⁻¹ at Anaheim and 9–38 nmol mol⁻¹ at Evansville, and the f(NO₂) values at both sites did not show significant daily variations (0.45±0.07 at Anaheim and 0.65±0.08 at Evansville), likely because the NO_x concentrations were controlled by the high NO emissions from the road (Gao, 2007). The calculated Δ(NO₂–NO_x) values using Eq. (8) showed significant diurnal variations. During the nighttime, the isotopic fractionations were solely controlled by the EIE; the predicted Δ(NO₂–NO_x) values were $+\mathrm{14.5}\pm \mathrm{2.0}$ ‰ and $+\mathrm{8.7}\pm \mathrm{2.1}$ ‰ at Anaheim and Evansville, respectively. During the daytime, the existence of LCIE lowered the predicted Δ(NO₂–NO_x) values to $+\mathrm{9.8}\pm \mathrm{1.7}$ ‰ at Anaheim and $+\mathrm{3.1}\pm \mathrm{1.5}$ ‰ at Evansville, while the f(NO₂) values at both sites remained similar. The lowest Δ(NO₂–NO_x) values for both sites (+7.0 ‰ and +1.7 ‰) occurred around 12:00 LT, when the NO_x photolysis was the most intense. In contrast, if one neglects the LCIE factor in the daytime, the Δ(NO₂–NO_x) values would be $+\mathrm{12.9}\pm \mathrm{1.5}$ ‰ and $+\mathrm{10.0}\pm \mathrm{1.6}$ ‰, respectively, an overestimation of 3.1 ‰ and 6.9 ‰. These discrepancies suggested that the LCIE played an important role in the NO–NO₂ isotopic fractionations, and neglecting it could bias the NO_x source apportionment using δ¹⁵N of NO₂ or nitrate.

Figure 3 NO_x concentrations and calculated Δ(NO₂–NO_x) values at four sites. Stacked bars show the NO and NO₂ concentrations extracted from monitoring sites (a–c) or calculated using a 0-D box model (d); the red lines are Δ(NO₂–NO_x) values at each site. Note that the NO_x concentration (left y axis in panel d) is different from the rest.

The role of LCIE was more important in less polluted sites. The Δ(NO₂–NO_x) values were calculated for a suburban site near San Diego, CA, USA, again using the hourly NO_x concentrations (https://www.arb.ca.gov; Fig. 3c) and j(NO₂) values calculated from the TUV model. NO_x concentrations at this site varied from 1 to 9 nmol mol⁻¹, assuming $\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\approx \mathrm{1}$. During the nighttime, NO_x was in the form of NO₂ (f(NO₂=1) because O₃ concentrations were higher than NO_x; thus the δ(NO₂) values should be identical to δ(NO_x) (Δ(NO₂–NO_x)=0). In the daytime a certain amount of NO was produced by direct NO emission and NO₂ photolysis, but the f(NO₂) was still high (0.73±0.08). Our calculation suggested that the daytime Δ(NO₂–NO_x) values should be only $+\mathrm{1.3}\pm \mathrm{3.2}$ ‰, with a lowest value of −1.3 ‰. These Δ(NO₂–NO_x) values were similar to the observed and modeled summer daytime δ(NO₂) values in West Lafayette, IN (Walters et al., 2018), which suggest that the average daytime Δ(NO₂–NO_x) values at ${\mathrm{NO}}_x=\mathrm{3.9}\pm \mathrm{1.2}$ nmol mol⁻¹ should range from +0.1 ‰ to +2.4  ‰. In this regime, we suggest that the Δ(NO₂–NO_x) values were generally small due to the significant contribution of LCIE and high f(NO₂).

The LCIE should be the dominant factor controlling the NO–NO₂ isotopic fractionation in remote regions, resulting in a completely different diurnal pattern of Δ(NO₂–NO_x) compared with the urban–suburban area. Direct hourly measurements of NO_x at remote sites are rare; thus we used a total NO_x concentration of 50 pmol mol⁻¹ and a daily O₃ concentration of 20 nmol mol⁻¹ at Summit, Greenland (Dibb et al., 2002; Hastings et al., 2004; Honrath et al., 1999; Yang et al., 2002) and assumed $\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\approx \mathrm{1}$ and that the conversion of NO to NO₂ was completely controlled by O₃ to calculate the NO∕NO₂ ratios. Here the isotopes of NO_x were almost exclusively controlled by the LCIE due to the high A values (>110). The Δ(NO₂–NO_x) values displayed a clear diurnal pattern (Fig. 3d), with a maximum value of −0.3 ‰ in the “nighttime” (solar zenith angle >85^∘) and a minimum value of −5.0 ‰ during midday. This suggests that the isotopic fractionations between NO and NO₂ were almost completely controlled by LCIE in remote regions when NO_x concentrations were <0.1 nmol mol⁻¹. However, since the isotopic fractionation factors of nitrate formation reactions (NO₂+OH, NO₃+HC, N₂O₅+H₂O) are still unknown, more studies are needed to fully explain the daily and seasonal variations in δ(${\mathrm{NO}}_{\mathrm{3}}^{-}$) in remote regions.

Nevertheless, our results have a few limitations. First, currently there are very few field observations that can be used to evaluate our model; therefore, future field observations that measure the δ¹⁵N values of ambient NO and NO₂ should be carried out to test our model. Second, more work, including theoretical and experimental studies, is needed to investigate the isotope fractionation factors occurring during the conversion from NO_x to NO_y and nitrate: in the NO_y cycle, EIE (isotopic exchange between NO₂, NO₃ and N₂O₅), KIE (formation of NO₃, N₂O₅ and nitrate) and PHIFE (photolysis of NO₃, N₂O₅, HONO and sometimes nitrate) may also exist and be relevant for the δ¹⁵N of HNO₃ and HONO. In particular, the N isotope fractionation occurring during the ${\mathrm{NO}}_{\mathrm{2}}+\mathrm{OH}\to {\mathrm{HNO}}_{\mathrm{3}}$ reaction needs investigation. Such studies could help us to model the isotopic fractionation between NO_x emission and nitrate and eventually enable us to analyze the δ¹⁵N value of NO_x emission by measuring the δ¹⁵N values of nitrate aerosols and nitrate in wet depositions. Third, our discussion only focuses on the reactive nitrogen chemistry in the troposphere; however, the nitrogen chemistry in the stratosphere is drastically different from the tropospheric chemistry; thus future studies are also needed to investigate the isotopic fractionations in the stratospheric nitrogen chemistry. Last, the temperature dependence of both EIE and LCIE needs to be carefully investigated because of the wide range of temperature in both the troposphere and stratosphere. Changes in temperature could alter the isotopic fractionation factors of both EIE and LCIE as well as contribute to the seasonality of isotopic fractionations between NO_x and NO_y molecules.

5 Conclusions

The effect of NO_x photochemistry on the nitrogen isotopic fractionations between NO and NO₂ was investigated. We first measured the isotopic fractionations between NO and NO₂ and provided mathematical solutions to assess the impact of NO_x level and NO₂ photolysis rate (j(NO₂)) on the relative importance of EIE and LCIE. The EIE and LCIE isotope fractionation factors at room temperature were determined to be 1.0289±0.0019 and 0.990±0.005, respectively. These calculations and measurements can be used to determine the steady-state Δ(NO₂–NO) and Δ(NO₂–NO_x) values at room temperature. Subsequently we applied our equations to polluted, clean and remote sites to model the daily variations in Δ(NO₂–NO_x) values. We found that the Δ(NO₂–NO_x) values could vary from over +20 ‰ to less than −5 ‰ depending on the environment: in general, the role of LCIE becomes more important at low NO_x concentrations, which tend to decrease the Δ(NO₂–NO_x) values. Our work provided a mathematical approach to quantify the nitrogen isotopic fractionations between NO and NO₂ that can be applied to many tropospheric environments, which could help interpret the measured δ¹⁵N values of NO₂ and nitrate in field observation studies.

Appendix A: Chamber descriptions

The chamber is a 10 m³ Teflon bag equipped with several standard instruments including a temperature and humidity probe, NO_x monitor and O₃ monitor. A total of 128 wall-mounted blacklight tubes surrounded the chamber to mimic tropospheric photochemistry, and the photolysis rate of NO₂ (j(NO₂)) when all lights are on has been previously determined to be $\mathrm{1.4}\times {\mathrm{10}}^{-\mathrm{3}}$ s⁻¹, similar to a j(NO₂) coefficient at an 81^∘ solar zenith angle. The irradiation spectrum of the black lights is shown in Fig. A1. The chamber was kept at room temperature and 1 atm. Before each experiment, the chamber was flushed with zero air at 40 L min⁻¹ for at least 12 h to ensure the background NO_x, O₃ and other trace gases were below the detection limit.

Figure A1 Spectral actinic flux versus wavelengths of the UV light source used in our experiments.

Appendix B: Box model assessing the time needed for NO–NO₂ to reach isotopic equilibrium

The time needed to reach NO–NO₂ isotopic equilibrium during light-off experiments was assessed using a 0-D box model. This box model contains only two reactions:

$$\begin{array}{}\text{(BR1)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}{+}^{\mathrm{14}}\mathrm{NO}\to {}^{\mathrm{15}}\mathrm{NO}{+}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\\ & k=\mathrm{8.14000}\times {\mathrm{10}}^{-\mathrm{14}}{\text{cm}}^{\mathrm{3}}{\text{s}}^{-\mathrm{1}}\end{array}\text{(BR2)}& {\displaystyle }\begin{array}{rl}& {}^{\mathrm{15}}\mathrm{NO}{+}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\to {}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}{+}^{\mathrm{14}}\mathrm{NO}\\ & k^{\prime }=\mathrm{8.37525}\times {\mathrm{10}}^{-\mathrm{14}}{\text{cm}}^{\mathrm{3}}{\text{s}}^{-\mathrm{1}}\end{array},\end{array}$$

where k and k^′ are rate constants of the reactions. The differences in rate constants were calculated by assuming an α(NO₂–NO) value of 1.0289. Six simulations were conducted at various initial NO (with δ¹⁵N=0 ‰) and O₃ levels that were similar to our experiment. Then the δ¹⁵N values of NO and NO₂ during the simulation were calculated from the model and are shown in Fig. B1, suggesting that in our experimental condition, all systems should reach isotopic equilibrium within 1 h.

Figure B1 Simulated NO–NO₂ isotopic equilibrium process in the chamber at various NO and O₃ concentrations.

Appendix C: Deriving Eqs. (7) and (8)

When the system (Reactions R1–R6) reaches a steady state, we have

$$\begin{array}{}\text{(C1)}& \mathrm{d}\left[{}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}\right]/\mathrm{d}t=\mathrm{0}.\end{array}$$

Therefore, using Reactions (R1)–(R6)

$$\begin{array}{}\text{(C2)}& \begin{array}{rl}& k_{\mathrm{1}}\left[{}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}\right]\left[{}^{\mathrm{14}}\mathrm{NO}\right]+j\left({\mathrm{NO}}_{\mathrm{2}}\right){\mathit{\alpha }}_{\mathrm{1}}\left[{}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}\right]=\\ & k_{\mathrm{5}}{\mathit{\alpha }}_{\mathrm{2}}\left[{}^{\mathrm{15}}\mathrm{NO}\right]\left[{\mathrm{O}}_{\mathrm{3}}\right]+k_{\mathrm{1}}\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)\left[{}^{\mathrm{15}}\mathrm{NO}\right]\left[{}^{\mathrm{14}}{\mathrm{NO}}_{\mathrm{2}}\right].\end{array}\end{array}$$

From here we refer to ¹⁴NO₂ and ¹⁴NO as NO₂ and NO for convenience. Rearranging the above equation, we get

$$\begin{array}{}\text{(C3)}& {\displaystyle \frac{{[}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}]}{\left[{}^{\mathrm{15}}\mathrm{NO}\right]}}={\displaystyle \frac{k_{\mathrm{5}}{\mathit{\alpha }}_{\mathrm{2}}\left[{\mathrm{O}}_{\mathrm{3}}\right]+k_{\mathrm{1}}\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)\left[{\mathrm{NO}}_{\mathrm{2}}\right]}{j_{{\mathrm{NO}}_{\mathrm{2}}}{\mathit{\alpha }}_{\mathrm{1}}+k_{\mathrm{1}}\left[\mathrm{NO}\right]}}.\end{array}$$

Meanwhile, since the Leighton cycle reaction still holds for the majority of isotopes (NO and NO₂), we have

$$\begin{array}{}\text{(C4)}& j_{{\mathrm{NO}}_{\mathrm{2}}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]=k_{\mathrm{5}}\left[\mathrm{NO}\right]\left[{\mathrm{O}}_{\mathrm{3}}\right].\end{array}$$

Thus,

$$\begin{array}{}\text{(C5)}& {\displaystyle \frac{\left[{\mathrm{NO}}_{\mathrm{2}}\right]}{\left[\mathrm{NO}\right]}}={\displaystyle \frac{k_{\mathrm{5}}\times \left[{\mathrm{O}}_{\mathrm{3}}\right]}{j_{{\mathrm{NO}}_{\mathrm{2}}}}}.\end{array}$$

From the text, when $j_{{\mathrm{NO}}_{\mathrm{2}}}>\mathrm{0}$, we defined $A={\mathit{\tau }}_{\mathrm{exchange}}/{\mathit{\tau }}_{\mathrm{photo}}=j_{{\mathrm{NO}}_{\mathrm{2}}}/(k_{\mathrm{1}}\times [\mathrm{NO}\left]\right)$. Using the above equations, we know

$$\begin{array}{}\text{(C6)}& {\displaystyle \frac{j_{{\mathrm{NO}}_{\mathrm{2}}}}{\left[\mathrm{NO}\right]}}={\displaystyle \frac{k_{\mathrm{5}}\left[{\mathrm{O}}_{\mathrm{3}}\right]}{\left[{\mathrm{NO}}_{\mathrm{2}}\right]}}=A{k}_{\mathrm{1}}\text{(C7)}& {\displaystyle \frac{j_{{\mathrm{NO}}_{\mathrm{2}}}}{k_{\mathrm{1}}\left[\mathrm{NO}\right]}}={\displaystyle \frac{k_{\mathrm{5}}\left[{\mathrm{O}}_{\mathrm{3}}\right]}{k_{\mathrm{1}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]}}=A.\end{array}$$

Next, to calculate δ(NO₂)–δ(NO), we use the definition of delta notation:

$$\begin{array}{}\text{(C8)}& {\displaystyle }\begin{array}{rl}\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\text{–}\mathit{\delta }\left(\mathrm{NO}\right)& =R_{{\mathrm{NO}}_{\mathrm{2}}}/R_{\mathrm{SD}}-R_{\mathrm{NO}}/R_{\mathrm{SD}}\\ & =(R_{{\mathrm{NO}}_{\mathrm{2}}}/R_{\mathrm{NO}}-\mathrm{1})(\mathrm{1}+\mathit{\delta }(\mathrm{NO}\left)\right)\end{array}\text{(C9)}& {\displaystyle }\begin{array}{rl}{\displaystyle \frac{R_{{\mathrm{NO}}_{\mathrm{2}}}}{R_{\mathrm{NO}}}}& ={\displaystyle \frac{\left[{}^{\mathrm{15}}{\mathrm{NO}}_{\mathrm{2}}\right]\left[\mathrm{NO}\right]}{\left[{}^{\mathrm{15}}\mathrm{NO}\right]\left[{\mathrm{NO}}_{\mathrm{2}}\right]}}\\ & ={\displaystyle \frac{k_{\mathrm{5}}{\mathit{\alpha }}_{\mathrm{2}}\left[{\mathrm{O}}_{\mathrm{3}}\right]\left[\mathrm{NO}\right]+k_{\mathrm{1}}\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)\left[{\mathrm{NO}}_{\mathrm{2}}\right]\left[\mathrm{NO}\right]}{j_{{\mathrm{NO}}_{\mathrm{2}}}{\mathit{\alpha }}_{\mathrm{1}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]+k_{\mathrm{1}}\left[\mathrm{NO}\right]\left[{\mathrm{NO}}_{\mathrm{2}}\right]}}.\end{array}\end{array}$$

Divide both sides by k₁[NO][NO₂]:

$$\begin{array}{}\text{(C10)}& {\displaystyle \frac{R_{{\mathrm{NO}}_{\mathrm{2}}}}{R_{\mathrm{NO}}}}={\displaystyle \frac{\frac{k_{\mathrm{5}}{\mathit{\alpha }}_{\mathrm{2}}\left[{\mathrm{O}}_{\mathrm{3}}\right]}{k_{\mathrm{1}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]}+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}{\frac{j_{{\mathrm{NO}}_{\mathrm{2}}}{\mathit{\alpha }}_{\mathrm{1}}}{k_{\mathrm{1}}\left[\mathrm{NO}\right]}+\mathrm{1}}}.\end{array}$$

Rearrange and substitute $\frac{k_{\mathrm{5}}\left[{\mathrm{O}}_{\mathrm{3}}\right]}{k_{\mathrm{1}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]}$ and $\frac{j_{{\mathrm{NO}}_{\mathrm{2}}}}{k_{\mathrm{1}}\left[\mathrm{NO}\right]}$ with A:

$$\begin{array}{}\text{(C11)}& {\displaystyle \frac{R_{{\mathrm{NO}}_{\mathrm{2}}}}{R_{\mathrm{NO}}}}={\displaystyle \frac{{\mathit{\alpha }}_{\mathrm{2}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}{{\mathit{\alpha }}_{\mathrm{1}}A+\mathrm{1}}}\text{(C12)}& {\displaystyle \frac{R_{\mathrm{NO}}}{R_{{\mathrm{NO}}_{\mathrm{2}}}}}={\displaystyle \frac{{\mathit{\alpha }}_{\mathrm{1}}A+\mathrm{1}}{{\mathit{\alpha }}_{\mathrm{2}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}\text{(C13)}& {\displaystyle \frac{R_{\mathrm{NO}}}{R_{{\mathrm{NO}}_{\mathrm{2}}}}}-\mathrm{1}={\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{1}}-{\mathit{\alpha }}_{\mathrm{2}}\right)A-\left(\mathit{\alpha }\right({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO})-\mathrm{1})}{{\mathit{\alpha }}_{\mathrm{1}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}.\end{array}$$

Thus,

$$\begin{array}{}\text{(C14)}& \begin{array}{rl}\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-& \mathit{\delta }\left(\mathrm{NO}\right)=\\ & {\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}\right)A+\left(\mathit{\alpha }\right({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO})-\mathrm{1})}{{\mathit{\alpha }}_{\mathrm{1}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}\\ & (\mathrm{1}+\mathit{\delta }({\mathrm{NO}}_{\mathrm{2}}\left)\right).\end{array}\end{array}$$

Then, using mass balance

$$\begin{array}{}\text{(C15)}& \mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)f\left({\mathrm{NO}}_{\mathrm{2}}\right)+\mathit{\delta }\left(\mathrm{NO}\right)(\mathrm{1}-f({\mathrm{NO}}_{\mathrm{2}}\left)\right)=\mathit{\delta }\left({\mathrm{NO}}_x\right)\end{array}$$

we can derive Eq. (8):

$$\begin{array}{}\text{(C16)}& \begin{array}{rl}\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)-& \mathit{\delta }\left({\mathrm{NO}}_x\right)=\\ & {\displaystyle \frac{\left({\mathit{\alpha }}_{\mathrm{2}}-{\mathit{\alpha }}_{\mathrm{1}}\right)\times A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)-\mathrm{1}}{{\mathit{\alpha }}_{\mathrm{1}}A+\mathit{\alpha }\left({\mathrm{NO}}_{\mathrm{2}}\text{–}\mathrm{NO}\right)}}\\ & \left(\mathrm{1}+\mathit{\delta }\left({\mathrm{NO}}_{\mathrm{2}}\right)\right)\left(\mathrm{1}-f\left({\mathrm{NO}}_{\mathrm{2}}\right)\right).\end{array}\end{array}$$