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the Creative Commons Attribution 4.0 License.

# Interpretation of NO_{3}–N_{2}O_{5} observation via steady state in high-aerosol air mass: the impact of equilibrium coefficient in ambient conditions

### Xiaorui Chen

Steady-state approximation for interpreting NO_{3} and
N_{2}O_{5} has large uncertainty under complicated ambient conditions
and could even produce incorrect results unconsciously. To provide an
assessment and solution to the dilemma, we formulate datasets based on
in situ observations to reassess the applicability of the method. In most of
steady-state cases, we find a prominent discrepancy between *K*_{eq} (equilibrium
coefficient for reversible reactions of NO_{3} and N_{2}O_{5}) and
correspondingly simulated $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\left[{\mathrm{NO}}_{\mathrm{2}}\right]\times \left[{\mathrm{NO}}_{\mathrm{3}}\right]\right)$, especially under high-aerosol conditions in winter. This gap
reveals that the accuracy of *K*_{eq} has a critical impact on the steady-state
analysis in polluted regions. In addition, the accuracy of *γ* (N_{2}O_{5}) derived by steady-state fit depends closely on the
reactivity of NO_{3} (kNO_{3}) and N_{2}O_{5}(*k*N_{2}O_{5}). Based on a complete set of simulations, air mass of
kNO_{3} less than 0.01 s^{−1} with high aerosol and temperature higher
than 10 ^{∘}C is suggested to be the best suited for steady-state
analysis of NO_{3}–N_{2}O_{5} chemistry. Instead of confirming the
validity of steady state by numerical modeling for every case, this work
directly provides appropriate concentration ranges for accurate steady-state
approximation, with implications for choosing suited methods to interpret
nighttime chemistry in high-aerosol air mass.

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_{3}–N

_{2}O

_{5}observation via steady state in high-aerosol air mass: the impact of equilibrium coefficient in ambient conditions, Atmos. Chem. Phys., 22, 3525–3533, https://doi.org/10.5194/acp-22-3525-2022, 2022.

The nitrate radical (NO_{3}), an extremely reactive species prone to buildup
at night, is an ideal candidate for steady-state analysis in combination with
dinitrogen pentoxide (N_{2}O_{5}) due to fast equilibrium reactions
between them (Reaction R1).

Under the steady-state condition, the lifetime of NO_{3} (denoted as *τ*_{ss}(NO_{3})) can be calculated as the
ratio of NO_{3} concentration over the production rate
(${k}_{{\mathrm{NO}}_{\mathrm{2}}+{\mathrm{O}}_{\mathrm{3}}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]\left[{\mathrm{O}}_{\mathrm{3}}\right])$
or over the removal rate of both NO_{3} and N_{2}O_{5}, as indicated
in Eq. (1). A similar representation of N_{2}O_{5} steady-state lifetime
is also shown in Eq. (2). The loss frequencies of various sink pathways of
NO_{3} and N_{2}O_{5} are integrated as total first order in the
following equations, represented by the kNO_{3} and kN_{2}O_{5} terms.
Briefly, kNO_{3} is contributed by the reaction of the NO_{3} radical
with NO and hydrocarbons and uptake on particles at night, ranging from
hundredths of reciprocal seconds to several reciprocal seconds depending on the air mass. Due to
its large rate constant with NO, the concentration usually dominates the
lifetime of the NO_{3} radical in urban areas with fresh NO emission.
Otherwise, the reactions with hydrocarbons, especially unsaturated
hydrocarbons, are preferential for NO_{3} in rural areas. The *K*_{eq} denotes
the equilibrium coefficient for Reactions (R1a) and (R1b), derived by
Eq. (3).

Numerous works have taken advantage of the steady-state calculation to
quantify the total first-order loss rate for NO_{3} or N_{2}O_{5} such
that they drew conclusions about the oxidation capacity and reactive
nitrogen budgets contributed by this chemical system (Allan et al., 1999, 2000; Carslaw et al., 1997; Platt et al., 1984; Vrekoussis
et al., 2007; Wang et al., 2013). Since the steady-state approximation was
used to interpret atmospheric observation of NO_{3}–N_{2}O_{5}
(Brown et al., 2003; Platt et al., 1981), this method was also widely
implemented to quantify the N_{2}O_{5} uptake coefficient (*γ* (N_{2}O_{5})) (Brown et al., 2009, 2003; Li et al., 2020; McDuffie et al., 2019; Phillips et al., 2016; Wang et al., 2017a, c, 2020a).

However, with the influence induced by complicated atmospheric conditions
and emission, steady state in ambient air mass will not always be the
case (as illustrated in Sect. S1 and Fig. S1 in the Supplement). These situations are
prevalent in nocturnal boundary layer (Phillips et al., 2016; Stutz et al., 2004; Wang et al., 2017a, c) and therefore increase the
difficulty of applying steady state directly to NO_{3}–N_{2}O_{5} observation data, whereas few studies have systematically characterized
the error source and application conditions of this method (Brown et al., 2009).

Due to a faster approach to equilibrium than steady state, the application of
*K*_{eq} in the calculation of steady-state equations seems to be reasonable (Brown et al., 2003). For example, the ambient NO_{3} concentration was usually
calculated based on ambient N_{2}O_{5} concentration with
*K*_{eq}×[NO_{2}] when determining their budgets or characterizing the
lifetime or sink attribution of these two reactive nitrogen compounds
(Brown et al., 2011; Osthoff et al., 2006; Wang et al., 2018, 2017c, d; Yan et al., 2019). In addition, the mathematical
conversion between NO_{3} and N_{2}O_{5} concentration via *K*_{eq}
coefficient can simplify the calculation in the iterative box model, which
derives *γ*(N_{2}O_{5}) by iterating its value in the model until
the predicted N_{2}O_{5} concentration matches the observation
(Wagner et al., 2013; Wang et al., 2020b). However, considerable
uncertainty could be associated with the quantification of *K*_{eq} and its
different parameterizations (Cantrell et al., 1988; Pritchard, 1994). The impact of *K*_{eq} value on steady-state fit or
concentration conversion has not been explored to date in the analysis of
NO_{3}–N_{2}O_{5} steady state.

In this study, we formulate a half artificial dataset with expected
properties based on field campaigns. Specifically, most of the species contained
in the dataset are observed values while only NO_{3} and N_{2}O_{5}
were calculated by the steady-state model (illustrated in the Sect. 2.2).
With the dataset, we illustrate the reasons for deviation of parameterized
*K*_{eq} from $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$ in ambient
conditions, the possible uncertainties of linear fit based on steady-state Eqs. (4) and (5) (the related variables are explained in
Sect. 2.1) that resulted from different *K*_{eq} values, and the influence of relevant
atmospheric variables on *γ* (N_{2}O_{5}) derivation via the steady-state method. Furthermore, a series of ambient condition tests specify the
exact ranges suited for steady-state analysis according to not only the
validity of steady state but also *K*_{eq} values, which optimize the validity check by numerical modeling in previous research (Brown et al., 2009, 2003) and develop complete standard for data filtering.

## 2.1 *γ* (N_{2}O_{5}) derivation by steady-state approximation

The framework of steady-state approximation for the NO_{3}–N_{2}O_{5}
system is basically built on its chemical production and removal pathways,
in case of extremely weak physical processes (e.g., transport, dilution, and
deposition) relative to its chemical processes. With simultaneous
measurements of NO_{3}, N_{2}O_{5}, and relevant precursor
concentrations, the steady-state lifetime *τ*_{ss}(NO_{3}) and *τ*_{ss}(N_{2}O_{5}) can be quantified for a targeted period as shown in Eqs. (1) and (2). By substituting the kN_{2}O_{5} with $\mathrm{0.25}\times c\times {S}_{\mathrm{a}}\times \mathit{\gamma}$ (N_{2}O_{5}), the *γ*(N_{2}O_{5}) and the
reactivity of NO_{3} (kNO_{3}, including the reactions of NO_{3} with
NO and hydrocarbons) can therefore be determined by Eqs. (4) and (5).

Here *c* represents the mean molecular velocity of N_{2}O_{5}, *S*_{a}
represents the aerosol surface area, and *K*_{eq} is calculated from the rate
constant of reversible Reactions (R1a) (*k*_{R1a}) and (R1b) (*k*_{R1b}), which is
a temperature-dependent parameter. It should be noted that the photolysis of
NO_{3} is not considered in the kNO_{3} due to weak radiation at night,
and the homogeneous hydrolysis was also ignored due to its small
contribution in comparison to the heterogeneous pathway. A similar presumption was
also implemented in previous studies (Brown et al., 2009; Mentel et al., 1996; Wahner et al., 1998). In the form of these two equations, the potential
covariance between *S*_{a} and NO_{2} concentration can be avoided to decrease
the uncertainty (Brown et al., 2009). By being fit to these two equations,
*γ* (N_{2}O_{5}) can be directly derived from slope of the plot of
${\mathit{\tau}}_{\mathrm{ss}}^{-\mathrm{1}}\left({\mathrm{NO}}_{\mathrm{3}}\right)$ against
0.25*c* *S*_{a}*K*_{eq}[NO_{2}] or
from the intercept of the plot of ${\left(\mathrm{0.25}c\phantom{\rule{0.125em}{0ex}}{S}_{\mathrm{a}}{\mathit{\tau}}_{\mathrm{ss}}\left({\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right)\right)}^{-\mathrm{1}}$ against ${\left(\mathrm{0.25}c\phantom{\rule{0.125em}{0ex}}{S}_{\mathrm{a}}{K}_{\mathrm{eq}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]\right)}^{-\mathrm{1}}$ respectively. In the following analysis, the linear
fit based on Eq. (5) is preferred in steady-state approximation.

## 2.2 Steady-state model and half-artificial datasets

The steady-state model is reformed from a zero-dimensional box model to produce
NO_{3} and N_{2}O_{5}, which are in steady state as far as possible. It
is constrained by measurements of NO, NO_{2}, O_{3}, CO, CH_{4}, VOCs,
HCHO, *S*_{a}, relative humidity (RH), temperature (*T*), and pressure, coupled with the
Regional Atmospheric Chemistry Mechanism, version 2 (RACM2). Each data point
is treated as an independent air mass, aging 10 h and keeping input
constraint unchanged. As NO_{3}–N_{2}O_{5} chemistry, the interest of
this work, usually shows marked impacts during the night, only the time
periods with negligible photolysis frequency are under consideration. In the
standard simulation (herein referred to as Mod0), the uptake coefficient of
N_{2}O_{5} is set to 0.02, as a reasonable value of literature (Brown et al., 2006; Chen et al., 2020; McDuffie et al., 2018; Morgan et al., 2015; Phillips et al., 2016; Wagner et al., 2013; Wang et al., 2017c; Yu et al., 2020).

Two half-artificial datasets are derived from PKU2017 and TZ2018 field
campaigns (see Sect. S2) based on the steady-state model for analysis in the
following sections. The simulated NO_{3} and N_{2}O_{5} and other
observed values used for the constraints of the steady-state model jointly
formulate these half-artificial datasets. Specifically, the NO_{3} and
N_{2}O_{5} concentrations in this dataset are the output of the steady-state model simulation and guaranteed to be in steady state with respect to
other observed precursors. To verify the steady state of NO_{3} and
N_{2}O_{5} for each data point, we filtered the dataset according to the
deviation between the steady-state lifetime of N_{2}O_{5} (${\mathit{\tau}}_{\mathrm{ss}}\left({\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right)=\frac{\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]}{{k}_{\mathrm{R}\mathrm{1}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]\left[{\mathrm{O}}_{\mathrm{3}}\right]})$
and calculated lifetime of N_{2}O_{5} (${\mathit{\tau}}_{\mathrm{calc}}\left({\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right)=({k}_{{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}}+\frac{{k}_{{\mathrm{NO}}_{\mathrm{3}}}}{{K}_{\mathrm{eq}}\left[{\mathrm{NO}}_{\mathrm{2}}\right]}{)}^{-\mathrm{1}})$.
If the deviation exceeds 10 % for a data point, it will be excluded from
the following analysis. We presume that if any data point output from the
model is still out of steady state in terms of NO_{3} and N_{2}O_{5},
the sink rate constant of air mass represented by this data point should be
too weak for steady-state analysis within a reasonable timescale. In
addition, the data higher than 5 ppbv NO are filtered out in the following
calculation, since the resulting large variation in kNO_{3} can bias the
linear fit even though NO_{3} and N_{2}O_{5} approach steady
state rapidly under high NO (discussed in Sect. 3.2). The fraction of excluded
data is less than 8 %, and they are expected to have little influence on our
results. The calculated nighttime loss fraction accounted for by NO_{3} and
N_{2}O_{5} show large discrepancy (see Sect. S3 and Fig. S2) between
these two half-artificial datasets, which provide us a good opportunity to
investigate the factors impacting steady-state approximation across
different conditions.

Rather than using observation data directly, a half-artificial dataset can
provide a larger amount of valid data for steady-state analysis with a known
*γ* (N_{2}O_{5}) value. In addition, this method avoids the impacts
from steady-state deviation, which helps to analyze the factors influencing
*γ* (N_{2}O_{5}) quantification via steady-state approximation
backwards from a known steady-state condition.

## 3.1 Varying equilibrium coefficient under steady state

The rates of NO_{3}–N_{2}O_{5} reversible reactions are expected to be
equal for the steady-state case, so that the equilibrium coefficient *K*_{eq} can
be determined from either the rate constant ratio of Reactions (R1a) and (R1b) or the
ratio of $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$. Although this
approach is reasonable under ideal conditions, the exact same rates
between reversible reactions and the following calculation based on *K*_{eq}
scaling are not so appropriate for ambient atmosphere where the removal
pathway for NO_{3}–N_{2}O_{5} is not negligible, especially under the
high-aerosol-loading condition. NO_{3} and N_{2}O_{5} achieve steady
state after 1.5 h evolution, when concentration and rates remain
constant (Fig. 1). In this simulation, the starting mixing ratios of
NO_{2} and O_{3} are 10 and 23 ppbv, respectively, which is the average
level for the nighttime conditions in PKU2017. The concentrations of these
two precursors are held constant in the simulation to better illustrate the
influence of removal rates. This result will stay almost the same no matter whether
these starting values are initialized to be constant or allowed to vary.
Under steady state, the net equilibrium reaction rate in Fig. 1b and c
stays negative and positive for NO_{3} and N_{2}O_{5}, respectively.
In addition, the absolute values and difference of the forward and backward
reaction rates remain unchanged after achieving steady state. This result is
similar to a previous numerical calculation study (Brown et al., 2003),
while the deviation between reversible reaction rates becomes larger in our
case.

In this case, the original equilibrium is imperfectly realized (a perfect
realization of the original equilibrium condition is that *K*_{eq} and the ratio
of $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$ are equivalent as in Eq. 6), leading to errors on the projection of NO_{3} and N_{2}O_{5}
concentration via *K*_{eq}×[NO_{2}]. In fact, we note that a new
equilibrium between NO_{3} and N_{2}O_{5} is developed with constant
but unequal rates. Under this new equilibrium condition, the ratio of (R1b)
reaction rate (the red dashed line in Fig. 1d) over (R1a) reaction rate (the
black dashed line in Fig. 1d) can be regarded as the degree of approaching
original equilibrium (the blue line in Fig. 1d). In addition, this value
is also the ratio of $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$
against original *K*_{eq}; therefore we defined this ratio as a correction factor
*ε*, implemented to calculate accurate
$\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$ with significant
N_{2}O_{5} removal pathways. The value of *K*_{eq} after scaled by
*ε* can be used for converting the concentration of
NO_{3} and N_{2}O_{5} via Eq. (6):

Sensitivity tests are conducted to demonstrate the dependence of
*ε* on relevant variables based on the steady-state model.
The averages ambient conditions observed at the wintertime Peking University (PKU) site and
summertime Tai Zhou (TZ) site are taken as two basic constraints for sensitivity tests
(Table S2 in the Supplement). By separately altering variables, such as
NO_{2}, O_{3}, kN_{2}O_{5}, kNO_{3}, and *T*, the sensitivity of the
*ε* value can be obtained as shown in Figs. 2 and S4. The *ε* value depends primarily on
kN_{2}O_{5} and *T* in both scenarios, where *ε*
increases with *T* (approaching 1 under relatively high *T*) and decreases with
kN_{2}O_{5}. In comparison, the *ε* value behaves
insensitively to kNO_{3} as well as NO_{2} and O_{3} concentration, at
least within the range of reasonable ambient conditions. High
kN_{2}O_{5} results from high-aerosol events, usually occurring in
winter accompanied by low temperature and high relative humidity in some
populated areas (Baasandorj et al., 2017; Huang et al., 2014; Wang et al., 2017b, 2014), further decreasing the accuracy of original *K*_{eq}
values. It can be inferred that in order to accurately interpret the
relationship of NO_{3} and N_{2}O_{5}, calculation relying on the
equilibrium equation and steady-state approximation should consider the
dependence of *ε* on ambient conditions.

Even if the *K*_{eq} value serves as a good representation of the ratio of
$\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$ or *ε* can be readily quantified in the field, the discrepancy among different
databases in calculating *K*_{eq} still increases the uncertainties of
NO_{3}–N_{2}O_{5} calculation through steady-state approximation or
equilibrium, which has not been carefully considered. Here, we apply a set
of uniform formulas to describe *k*_{R1a} and *k*_{R1b} (see Sect. S4)
from preferred values of several popular atmospheric chemistry mechanisms
(Mozart, CB05, Saprc07, RACM2 and kinetic databases JPL2015 as well as
IUPAC2017) and finally calculating *K*_{eq}. As is shown in Figs. S5 and S6, *K*_{eq} variations derived from these six different databases reflect
considerable discrepancy from each other, especially in colder conditions.
Because parameterized *K*_{eq} values are only dependent on ambient temperature,
they continuously increase with time due to the decrease in temperature. In
addition to the discrepancy between different *K*_{eq} parameterizations, the
*ε* value varies dissimilarly with each *K*_{eq}, ranging from
70 % to 90 %. All these results demonstrate that, in most cases, *K*_{eq}
values simply derived from an existing database would fail to reproduce an
accurate relationship between NO_{3} and N_{2}O_{5}.

To further elucidate the impact of *K*_{eq} on deriving *γ* (N_{2}O_{5})
via steady-state approximation (hereafter defined as *γ*_{ss} (N_{2}O_{5})), Fig. S6 shows the steady-state fit based on all
six database-derived *K*_{eq} values and in the same time periods as Fig. S5 through
Eqs. (4) and (5), respectively (both equations can derive a pair of
*γ*_{ss} (N_{2}O_{5}) and kNO_{3}). *K*_{eq} (corrected with
*ε*) is calculated with NO_{3} and N_{2}O_{5}
concentration simulated based on RACM2. Fits based on Eq. (4) could lead to
11 *%*–46 % underestimation of *γ*_{ss} (N_{2}O_{5}), as indicated by varying slopes in Fig. S7b and d, when using the database-derived *K*_{eq}. Conversely, fit by Eq. (5)
(shown in Fig. S7a and c) biases the result of kNO_{3} as the
slopes without much influence on *γ*_{ss}(N_{2}O_{5}) as
the intercept. Previous research ascribed inconsistent fit results between
two equations to measurement uncertainty (Brown et al., 2009, 2006). However, fit with the original *K*_{eq} might be the primary reason for
such inconsistent results and even differentiates the derived *γ*_{ss} (N_{2}O_{5}) and kNO_{3} from true values. Therefore, steady-state fit based on Eq. (5) might be the best choice for *γ* (N_{2}O_{5}) derivation via steady-state approximation. Similarly, Eq. (4) is preferred to be applied when kNO_{3} is the final objective.

## 3.2 Impacts of NO_{3}–N_{2}O_{5} reactivity on steady state

In order to further explore the impacting factors on the steady-state fit
method, *γ*_{ss} (N_{2}O_{5}) results are derived for each 2 h time period of the PKU2017 and TZ2018 datasets based on output from the steady-state
model. Since the pre-set *γ* (N_{2}O_{5}) in this model is 0.02,
the degree of deviation from this value is supposed to reflect the accuracy
of the fitted result.

It can be noticed from Eq. (5) that the variability of kNO_{3} during the
same time period leads data points to scatter on lines with different
slopes, which could bias the resulted *γ*_{ss} (N_{2}O_{5}) from the
model pre-set value. As is shown in Fig. 3, the absolute percentages of
*γ*_{ss} (N_{2}O_{5}) deviation grow dramatically with the
increase in relative standard deviation of kNO_{3} (kNO_{3} RSD) in both winter and summer datasets. The positive correlation even gives rise to
extreme deviation in the summer dataset, with up to almost 10 times of model
setting *γ* (N_{2}O_{5}). In fact, there remain accurate *γ*_{ss}(N_{2}O_{5}) values derived in each range of kNO_{3} RSD,
indicating a not strictly positive correlation between *γ*_{ss} (N_{2}O_{5}) deviation and kNO_{3} RSD. This implies that large
variation in kNO_{3} only enhances the possibilities of inaccurate results
from steady-state fit rather than hinder the *γ*_{ss} (N_{2}O_{5}) quantification all the time.

In addition to the large variation in kNO_{3} in a short time period, the absolute
level of kNO_{3} and kN_{2}O_{5} could influence the possibilities
of inaccurate *γ*_{ss} (N_{2}O_{5}) from different aspects.
Although the enhancement of kNO_{3} and kN_{2}O_{5} boosts the
approach to steady state (Sect. S5 and Fig. S8), higher levels of
kNO_{3} amplify the bias of *γ*_{ss} (N_{2}O_{5}), contrary to
kN_{2}O_{5}, with the same relative variation in kNO_{3} (Sect. S6 and
Fig. S10). This indicates that the region with plural emissions (e.g., strong
biogenic or vehicular emission) might not be suited for steady-state fit due
to the high kNO_{3}. Therefore, a trade-off between the variation in
kNO_{3} and the high level of kNO_{3} (fast approach to steady state)
should be made when deriving *γ*_{ss} (N_{2}O_{5}).

## 3.3 Implication for accurate steady-state analysis of NO_{3}–N_{2}O_{5}

While a few studies have examined the validity of steady state under certain
conditions via numerical modeling when interpreting the ambient data
(Brown et al., 2009, 2003), a clear range well suited to
steady-state analysis of NO_{3}–N_{2}O_{5}, taking both *K*_{eq} and validity of steady state into consideration, has not been determined to
date.

Here almost 20 000 simulations are displayed in the parallel plot of Fig. 4, where each line connects five constraint parameters to the calculated steady-state time and *ε* (the correction factor for *K*_{eq}
parameterization to match the exact ratio of
$\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$, detailed in Eq. 6). The
gray traces represent the simulations that could not match steady state within
600 s and were defined as less valid cases here. By this definition, we
intend to indicate that it is also viable to apply steady-state
approximation on air mass, which requires more than 600 s to match steady
state, whereas the uncertainty caused therefrom could increase to some
extent. The pink and blue traces together represent the simulations that could
match valid steady state within 600 s without consideration of *K*_{eq} deviation
(in other words the value of *ε*). Furthermore, the
criteria to apply steady-state approximation appropriately is
approach to steady state within 600 s and *ε*
larger than 0.9, which are indicated as pink traces. While the levels of *T*,
NO_{2}, and O_{3} have a minor effect on the approach to steady state,
simultaneous low kN_{2}O_{5} (indicated as low *S*_{a} in the plot) and
kNO_{3} prevent the NO_{3}–N_{2}O_{5} system from developing steady
state. For example, when kNO_{3} is lower than 0.01 s^{−1}, the air mass
will be valid only if *S*_{a} increases to at least 3000 µm^{2} cm^{−3} with *γ* (N_{2}O_{5}) of 0.02. This implies that clean
air mass is not suited for steady state in any cases, whereas the high-aerosol
condition provides more possibilities to approach steady state even with low
kNO_{3}. However, in order to interpret NO_{3}–N_{2}O_{5} chemistry with an accurate *K*_{eq} coefficient, the *ε* larger
than 0.9 is additionally taken into consideration, which excludes 50 % of
valid steady-state cases mainly with high aerosol and lower than
10 ^{∘}C. These cases could bias $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$ from original *K*_{eq} (also indicated in Fig. 2), leading to inaccurate results of calculation based on *K*_{eq}.

In this study, we found that the parameterized *K*_{eq} coefficient deviates much
from the ratio of $\left[{\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{5}}\right]/\left(\right[{\mathrm{NO}}_{\mathrm{2}}]\times [{\mathrm{NO}}_{\mathrm{3}}\left]\right)$ in some
cases where steady state is valid. The indicator of the deviation,
*ε*, is relatively sensitive to N_{2}O_{5}
reactivity and ambient temperature. It implies that conditions suited for
steady-state analysis should be determined according to not only the
validity of steady state but also *K*_{eq}, especially under high-aerosol
conditions, like some regions in India, China, Europe, and the US
(Baasandorj et al., 2017; Cesari et al., 2018; Huang et al., 2014; Mogno et al., 2021; Petit et al., 2017; Wang et al., 2017b). Considering that a high
level of kNO_{3} might amplify the bias of *γ*_{ss} (N_{2}O_{5}) yield from steady-state fit and appears to be
accompanied by fast variations, air mass of kNO_{3} less than 0.01 s^{−1} with high aerosol and *T* higher than 10 ^{∘}C are therefore the
best suited for steady-state analysis of NO_{3}–N_{2}O_{5} chemistry, which indicates that this method would be more applicable in
polluted regions with high aerosol loading during summertime. If the
restriction of *ε* is relaxed to 30 %, some winter
conditions will also be applicable. Our results provide an insight into
improving the accuracy of the steady-state approximation method and finding suited
areas to interpret nighttime chemistry. Further improvement of in situ
NO_{3}–N_{2}O_{5} budget quantification might rely on the direct
measurements via flow tube system or machine learning prediction based on
ancillary parameters.

The datasets used in this study are available from the corresponding author upon request (wanghch27@mail.sysu.edu.cn; k.lu@pku.edu.cn).

The supplement related to this article is available online at: https://doi.org/10.5194/acp-22-3525-2022-supplement.

KL and HW designed the study. XC and HW analyzed the data and wrote the paper with input from KL.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Thanks for the data contributed by field campaign team.

This research has been supported by the National Natural Science Foundation of China (grant nos. 21976006 and 42175111), the Beijing Municipal Natural Science Foundation (grant no. JQ19031), the State Key Joint Laboratory of Environmental Simulation and Pollution Control (grant no. 21K02ESPCP), the National Key Basic Research Program For Youth (grant nos. DQGG0103-01 and 2019YFC0214800), and the National State Environmental Protection Key Laboratory of Formation and Prevention of Urban Air Pollution Complex (grant no. CX2020080578).

This paper was edited by Qiang Zhang and reviewed by two anonymous referees.

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_{2}, NO

_{3}, and N

_{2}O

_{5}) under various conditions. A previously neglected bias for the coefficient applied for interpreting their effects is disclosed, and the relevant ambient factors are examined. We therefore provide a good solution to an accurate representation of nighttime chemistry in high-aerosol areas.