Resolving systematic errors in sulfate source apportionment: a field-validated kinetic isotope fractionation framework

Sulfates are critical constituents of atmospheric fine particulate matter (PM_2.5) that significantly influence air quality and climate dynamics. While stable isotope fractionation analysis is a powerful tool for tracing atmospheric sulfate formation mechanisms, conventional isotopic models rely on idealized complete SO₂ oxidation scenarios. This oversimplification introduces systematic errors and neglects vital kinetic isotope effects generated during incomplete SO₂ processing. To address this knowledge gap, we established a field-validated analytical framework combining seasonal PM_2.5 observations in Nanjing, China, with Bayesian isotope mixing and process-specific Rayleigh fractionation modeling. Our kinetic fractionation-corrected approach accounts for actual atmospheric oxidation processes, revealing that transition-metal ion (TMI)-catalyzed and NO₂-mediated pathways dominate secondary sulfate production. Conversely, comparative analysis demonstrates that traditional complete-oxidation models disproportionately diminish TMI pathway contributions. Furthermore, implementing kinetic fractionation corrections successfully resolves systematic biases in source apportionment. We demonstrate that traditional models misrepresent source contributions, overestimating coal combustion by 10 % and underestimating traffic emissions by 8 % during summer photochemical episodes. These findings establish a refined isotopic tracing framework that resolves long-standing calculation discrepancies. Ultimately, this framework delivers essential constraints for atmospheric sulfur cycle modeling and underscores the necessity for multi-pollutant regulation strategies targeting vehicular emissions and co-emitted transition meals.

1 Introduction

Sulfate aerosols constitute a critical fraction of atmospheric particulate matter, with their formation mechanisms representing a pivotal yet unresolved aspect of air pollution control (He et al., 2012; Wang et al., 2018; Dao et al., 2019; Tao et al., 2016; Chen et al., 2023; Yu et al., 2026). The oxidation processes of sulfur dioxide (SO₂) exhibit remarkable mechanistic complexity stemming from its dual physicochemical nature (Kang et al., 2016; Mukai et al., 2001; Ma et al., 2026). The exceptionally high hydration equilibrium constant drives near-instantaneous formation of S(IV) species (HSO${}_{\mathrm{3}}^{-}$/SO${}_{\mathrm{3}}^{\mathrm{2}-}$) (Zhang et al., 2026), while the thermodynamically favorable oxidation potential creates a multidimensional reaction pathways including both homogeneous radical processes and heterogeneous interfacial reactions (He et al., 2018; Harris et al., 2013b; Tanaka et al., 1994; Zhang and Chan, 2023). These intrinsic reaction dynamics, when coupled with the extreme spatiotemporal variability of atmospheric conditions, create substantial obstacles for mechanistic elucidation under ambient environments. Current investigative approaches, including controlled laboratory simulations, field measurement, and computational modeling, each face inherent limitations when applied in isolation. Laboratory studies inevitably simplify the unbounded nature of atmospheric systems, while field observations struggle to isolate specific chemical processes amid complex real-world interactions. Modeling studies (Yang et al., 2025; Zheng et al., 2024; Lin et al., 2022, 2025), though valuable for hypothesis testing, remain constrained by incomplete mechanistic understanding and parameterization challenges. This methodological trilemma has resulted in persistent knowledge gaps regarding the dominant pathways and quantitative contributions of various sulfate formation mechanisms in different atmospheric regimes.

The application of stable isotope techniques in sulfate source attribution has emerged as a powerful analytical approach, leveraging the distinct isotopic fingerprints characteristic of different pollution sources and their remarkable stability during atmospheric transport (Yang et al., 2025; Lin et al., 2022; Harris et al., 2012b; Guo et al., 2024). This methodology capitalizes on the system-specific fractionation patterns inherent to various sulfate formation pathways, which serve as critical tracers for reconstructing oxidation mechanisms. Contemporary research has quantitatively established the sulfur isotopic signatures (δ³⁴S) (Han et al., 2022; Yang et al., 2025) associated with major atmospheric oxidation routes, including transition metal ion-catalyzed (TMI), ozone-mediated (O₃), nitrogen oxide-driven (NO_x), and hydrogen peroxide-initiated (H₂O₂) pathways, through systematic analysis of particulate matter samples (Yang et al., 2025; Fan et al., 2020a; Harris et al., 2013a; Guo et al., 2019; Ruan et al., 2025; Wang et al., 2016, 2020). Advanced analytical tools, particularly Bayesian isotopic mixing and Rayleigh fractionation models, have demonstrated substantial utility in deconvoluting complex sulfate formation mechanisms. Yet, their diagnostic potential is often overshadowed by a fundamental theoretical flaw: the reliance on complete SO₂ oxidation paradigms. Because genuine atmospheric regimes rarely achieve such idealized conversion, this oversimplification severely penalizes the accuracy of sulfur transformation kinetics. Ultimately, conventional Rayleigh approaches introduce systematic uncertainties into source apportionment by neglecting the vital kinetic isotope effects and partial intermediates generated during incomplete SO₂ processing.

Building upon these insights, the present study establishes a comprehensive multi-model analytical framework that synergistically couples ambient field observations with a suite of diagnostic tools, ranging from backward trajectory analysis to Bayesian isotope mixing and process-specific Rayleigh fractionation modeling. This integrated approach achieves precision in quantifying sulfate formation pathways while systematically addressing the critical but often neglected kinetic limitations inherent to incomplete SO₂ oxidation under real atmospheric conditions. By applying sulfur and oxygen isotope analyses to seasonally resolved aerosol samples from Nanjing, we demonstrate significant divergence between conventional complete-oxidation Rayleigh models and our kinetic fractionation-corrected framework. Our systematic model intercomparison highlights the essential calibration requirements for stable isotope fractionation, successfully resolving the systematic biases that have long plagued sulfate source attribution. This methodological leap provides a refined method for tracking urban sulfate pollution, particularly within complex environments characterized by incomplete oxidation. Ultimately, the seasonally partitioned data from Nanjing serves as a quantitative benchmark, explicitly detailing how traditional complete-oxidation frameworks misrepresent source contributions during both summer photochemical and winter coal-combustion episodes.

2 Materials and methods

2.1 Sampling Sites

The atmospheric sampling protocol employed a monitoring site (118°43^′ E, 32°12^′ N) (Fig. S1 in the Supplement) capturing seasonal extremes through January (winter) and July (summer) observations. A modified TH1000H high-volume sampler (Wuhan Tianhong Instrument) incorporated a dual-layer filtration assembly enabling concurrent PM_2.5 and SO₂ collection through co-laminated quartz (203 mm × 254 mm) and alkali-impregnated glass fiber filters (203 mm × 254 mm). Pre-treatment protocols involved muffle furnace combustion (723 K for 2 h) followed by immersion in 2 % K₂CO₃+2 % glycerol solution for glass fiber filters. Sampling operations maintained 1.05 m³ min⁻¹ flow rates during 24 h cycles with meteorological parameters (wind speed/direction, temperature, pressure, humidity) continuously logged. Post-sampling handling included foil-wrapping (pre-combusted 723 K) followed by 24 h desiccation and dark refrigeration. The final dataset comprised samples after excluding precipitation events, with quartz filters positioned upstream for particulate collection and treated glass fiber membranes downstream for gaseous SO₂ capture.

2.2 Sample Collection

The ionic composition analysis of PM_2.5 samples was performed using Dionex ICS-3000 and ICS-2000 ion chromatography systems (Thermo Fisher Scientific) for simultaneous determination of major water-soluble ions. The analytical suite comprised five cations (Na⁺, NH${}_{\mathrm{4}}^{+}$, K⁺, Mg²⁺, Ca²⁺) and three anions (Cl⁻, NO${}_{\mathrm{3}}^{-}$, SO${}_{\mathrm{4}}^{\mathrm{2}-}$). For the determination of sulfate δ³⁴S values, water-soluble sulfate within the filter extracts was initially isolated as BaSO₄ through the addition of an excess BaCl₂ solution. Because the sampling matrix inevitably captures both sulfate and sulfite, the latter originating from ambient SO₂, a targeted purification protocol was introduced; specifically, treating the co-precipitate with 1 M HCl selectively dissolved any residual BaSO₃ while leaving the acid-insoluble BaSO₄ intact. Following collection via filtration, the purified precipitate was rinsed thoroughly with deionized water until completely free of chloride ions and subsequently dried. To eliminate potential organic interferences prior to isotopic interrogation, the dried BaSO₄ underwent thermal treatment in a muffle furnace at 1073 K for 2 h. Sulfur and oxygen isotope ratios were determined by elemental analyzer (EA, Flash 2000, Thermo) combined with isotope mass spectrometer (IRMS, Delta V Plus, Finningan). Since BaSO₄ is an analytical sample for the determination of sulfur and oxygen isotope values in sulfate, the determination of sulfur isotope is to thermally decompose BaSO₄ and Cu₂O in a vacuum state, and use the generated SO₂ to analyze in a mass spectrometer (Yanagisawa and Sakai, 1983). In the determination of oxygen isotope, BaSO₄ was pyrolyzed by graphite furnace at 1723 K, and the oxygen isotope value of CO produced was measured in continuous flow mode (Mizutani and Rafter, 1969). The oxygen isotope in water was determined by the automatic carbon dioxide-water equilibrium method.

To ensure robust data quality, the analytical precision for all major water-soluble ions was rigorously verified via replicate measurements, consistently yielding values better than 5 %. For the isotopic composition, results are reported in standard delta notation relative to international reference scales, specifically, Vienna Canyon Diablo Troilite (V-CDT) for δ³⁴S and Vienna Standard Mean Ocean Water (V-SMOW) for δ¹⁸O, as formulated in Eqs. (1) and (2). Notably, our analytical system demonstrated high stability, with long-term reproducibility constrained to better than 0.2 ‰ and 0.3 ‰ for sulfur and oxygen isotopes, respectively.

$$\begin{array}{}\text{(1)}& {\displaystyle }\begin{array}{rl}{\mathit{\delta }}^{\mathrm{34}}\mathrm{S}\left(\mathrm{‰}\right)& =\left[{(}^{\mathrm{34}}\mathrm{S}{/}^{\mathrm{32}}\mathrm{S}{)}_{\mathrm{sample}}/{(}^{\mathrm{34}}\mathrm{S}{/}^{\mathrm{32}}\mathrm{S}{)}_{\mathrm{reference}}-\mathrm{1}\right]\\ & \times \mathrm{1000}\end{array}\text{(2)}& {\displaystyle }\begin{array}{rl}{\mathit{\delta }}^{\mathrm{18}}\mathrm{O}\left(\mathrm{‰}\right)& =[{(}^{\mathrm{18}}\mathrm{O}{/}^{\mathrm{16}}\mathrm{O}{)}_{\mathrm{sample}}/{(}^{\mathrm{18}}\mathrm{O}{/}^{\mathrm{16}}\mathrm{O}{)}_{\mathrm{reference}}-\mathrm{1}]\\ & \times \mathrm{1000}\end{array}\end{array}$$

Meteorological parameters and criteria air pollutant concentrations were acquired from two authoritative national databases during the sampling campaign. The China Meteorological Administration Data Service Center (https://data.cma.cn, last access: 16  August 2025) provided comprehensive weather records, while real-time air quality metrics were retrieved from the China National Environmental Monitoring Centre's online platform (http://www.aqistudy.cn/, last access: 20 August 2025), representing the official monitoring network for atmospheric composition surveillance.

2.3 Air Mass Back Trajectories and Identification of Source Regions

We conducted analysis of air mass back trajectories (Figs. S3 and S4). The potential source regions were identified with (a) cluster analysis, (b) fractional cluster contributions, and (c) concentrated weighted trajectory (CWT) analysis of sulfate concentrations (Lin et al., 2022; Dasari et al., 2020; Fan et al., 2023) (methodological details in Sect. S2 in the Supplement).

2.4 Bayesian Stable Isotope Mixing Model

The source apportionment of sulfate aerosols was conducted through Stable Isotope Analysis in R (SIAR) modeling, a Bayesian framework that probabilistically quantifies contribution distributions and associated uncertainties from multiple emission sources by processing ambient δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ measurements (Eq. 4) alongside source-specific δ³⁴S signatures of precursor sulfur gases with all environmental samples maintaining consistent natural properties to eliminate cross-data variability particularly seasonal δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ fluctuations where the winter dataset (n=16) and summer dataset (n=15) were separately processed through SIAR to establish seasonally resolved sulfate source profiles for the Nanjing region.

The SIAR model determines the probability distribution of source contributions to mixtures and uncertainties associated with multiple sources (Fan et al., 2020a; Zheng et al., 2024; Harris et al., 2012a). In the SIAR framework, the isotopic value j in species i (X_ij) is expressed as:

$$\begin{array}{}\text{(3)}& {\displaystyle }{X}_{ij}={\displaystyle \frac{\sum _{k=\mathrm{1}}^k{p}_k{q}_{jk}(S_{jk}+C_{jk})}{\sum _{k=\mathrm{1}}^k{p}_k{q}_{jk}}}+{\mathit{\epsilon }}_{ij}\text{(4)}& {\displaystyle }{S}_{jk}\sim N\left({\mathit{\mu }}_{jk}{\mathit{\omega }}_{jk}^{\mathrm{2}}\right)\text{(5)}& {\displaystyle }{C}_{jk}=\left({\mathit{\lambda }}_{jk}{\mathit{\tau }}_{jk}^{\mathrm{2}}\right)\text{(6)}& {\displaystyle }{\mathit{\epsilon }}_{ij}\sim (\mathrm{0},{\mathit{\sigma }}_j^{\mathrm{2}})\end{array}$$

Where q_jk is isotopic value j in source; S_jk is source k for isotope j, normally distributed with mean (μ_jk) and variance (${\mathit{\omega }}_{jk}^{\mathrm{2}}$); C_jk is isotopic enrichment factor for isotope j in source k, normally distributed with mean (λ_jk) and variance (${\mathit{\tau }}_{jk}^{\mathrm{2}}$); p_k is proportion of source k; ε_ij is residual error representing unquantified variation between individual species, normally distributed with mean (0) and variance (${\mathit{\sigma }}_j^{\mathrm{2}}$).

2.5 Relative Contributions of SO₂ Oxidation Pathways to Sulfate Aerosols

Four sulfate formation pathways were considered: OH, ${\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}$, TMI, and NO₂ oxidation. Given the similar fractionation factors of H₂O₂ and O₃ (Harris et al., 2012a, b; Lin et al., 2022), we used α(${\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}$) to represent their combined fractionation. The temperature-independent α³⁴S fractionation factors are calculated as (Lin et al., 2022; Fan et al., 2020b):

$$\begin{array}{}\text{(7)}& {\displaystyle }{\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{OH}}\left(\mathrm{‰}\right)=-(\mathrm{0.004}\pm \mathrm{0.015})\times T+(\mathrm{10.60}\pm \mathrm{0.73})\text{(8)}& {\displaystyle }\begin{array}{rl}{\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}}\left(\mathrm{‰}\right)& =-(\mathrm{0.085}\pm \mathrm{0.004})\\ & \times T+(\mathrm{16.51}\pm \mathrm{0.15})\end{array}\text{(9)}& {\displaystyle }\begin{array}{rl}{\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{TMI}}\left(\mathrm{‰}\right)& =-(\mathrm{0.237}\pm \mathrm{0.004})\\ & \times T+(-\mathrm{5.039}\pm \mathrm{0.044})\end{array}\text{(10)}& {\displaystyle }\begin{array}{rl}{\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{{\mathrm{NO}}_{\mathrm{2}}}\left(\mathrm{‰}\right)& =\exp \left[\right(\mathrm{0.2437}\pm \mathrm{0.0457})/T\\ & +(\mathrm{0.0008}\pm \mathrm{0.019})]\end{array}\end{array}$$

Where T is ambient temperature (°C). The total α³⁴S fractionation is computed as (Fan et al., 2020b; Harris et al., 2013a):

$$\begin{array}{}\text{(11)}& \begin{array}{rl}{\mathit{\alpha }}^{\mathrm{34}}\mathrm{S}& =f_{\mathrm{OH}}\times {\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{OH}}+f_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}}\times {\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}}\\ & +f_{\mathrm{TMI}}\times {\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{TMI}}+f_{{\mathrm{NO}}_{\mathrm{2}}}\times {\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{{\mathrm{NO}}_{\mathrm{2}}}\end{array}\end{array}$$

Where f_OH, $f_{{\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}}$, f_TMI and $f_{{\mathrm{NO}}_{\mathrm{2}}}$ denote relative fractions of sulfate production from respective pathways, constrained by $\sum f_x=\mathrm{1}$.

2.6 Sulfur Isotope Fractionation Modeling via Rayleigh Fractionation

Due to the absence of sulfur isotope equilibrium between atmospheric SO₂ and SO${}_{\mathrm{4}}^{\mathrm{2}-}$, the Rayleigh Fractionation model can be applied to calculate the sulfur isotope fractionation effects between them. Within this framework, the δ³⁴S value of residual atmospheric SO₂ (δ³⁴S-SO₂) is described by (Zheng et al., 2024; Lin et al., 2022):

$$\begin{array}{}\text{(12)}& \begin{array}{rl}{\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}}& ={\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}-\mathrm{emission}}\\ & +{\mathit{\alpha }}^{\mathrm{34}}{\mathrm{S}}_{g\to p}\times \ln (\mathrm{1}-\mathrm{SOR})\end{array}\end{array}$$

Where α³⁴S_g→p denotes the fractionation factor for SO₂ (gaseous) → SO${}_{\mathrm{4}}^{\mathrm{2}-}$ (particles) conversion through gas and aqueous phases, ${\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}-\mathrm{emission}}$ represents the δ³⁴S value of emission sources, and SOR is the SO₂ oxidation ratio, which can be calculated by the molar mass of SO₂ and SO${}_{\mathrm{4}}^{\mathrm{2}-}$:

$$\begin{array}{}\text{(13)}& \mathrm{SOR}={\displaystyle \frac{m\left({\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right)/M\left({\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right)}{m\left({\mathrm{SO}}_{\mathrm{2}}\right)/M\left({\mathrm{SO}}_{\mathrm{2}}\right)+m\left({\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right)/M\left({\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right)}}\end{array}$$

The source δ³⁴S value can be estimated from isotopic mass balance:

$$\begin{array}{}\text{(14)}& \begin{array}{rl}{\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}-\mathrm{emission}}& ={\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}}\times (\mathrm{1}-\mathrm{SOR})\\ & +{\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\times \mathrm{SOR}\end{array}\end{array}$$

Where δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ is the observed δ³⁴S in sulfate. When SOR approaches 1 (complete SO₂ oxidation), ${\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}-\mathrm{emission}}$ equals δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$. The fractionation effects α is derived by combining Eqs. (14) and (15):

$$\begin{array}{}\text{(15)}& {\mathit{\alpha }}_{g\to p}={\displaystyle \frac{({\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{2}-\mathrm{emission}}-{\mathit{\delta }}^{\mathrm{34}}\mathrm{S}-{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-})\times \mathrm{SOR}}{(\mathrm{1}-\mathrm{SOR})\times \ln (\mathrm{1}-\mathrm{SOR})}}\end{array}$$

The Complete Oxidation (CO) Process acts as an idealized baseline, operating on the premise that all gaseous SO₂ is exhaustively converted to particulate sulfate, yielding a sulfur oxidation ratio (SOR) of 1. Consequently, this framework directly equates the calculated source isotopic signature (δ³⁴S-source) to the measured sulfate value (δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$), intentionally omitting the isotopic fractionation kinetics that govern the oxidation pathway. In contrast, the Incomplete Oxidation (InCO) Process reflects actual atmospheric conditions, specifically the reality that SO₂-to-sulfate conversion is inherently partial (SOR <1). Grounded in the Rayleigh fractionation model, this framework synchronously integrates the observed isotopic values of gaseous SO₂ and particulate SO${}_{\mathrm{4}}^{\mathrm{2}-}$ with the corresponding SOR. By doing so, it inversely derives more realistic source isotopic signatures alongside the kinetic fractionation factors (α), a conceptual advancement that forms the core innovation of our study.

2.7 Estimation of Primary Sulfate and Secondary Sulfate

The underestimation of primary sulfate emissions can partially account for the phenomenon known as missing sulfates (Ding et al., 2021). The anthropogenic primary sulfate is estimated as 3 % of anthropogenic sources of sulfur emissions in the model (Alexander et al., 2009). Assuming all the detected concentrations of SO₂ and precursors of secondary sulfate are anthropogenic, we can deduce Eq. (17). Considering the influence of anthropogenic sulfate sources on primary sulfate (He et al., 2018), the method for estimating primary sulfate can be expressed as follows:

$$\begin{array}{}\text{(16)}& {\displaystyle }{n}_{\mathrm{ps}}=n_{\mathrm{ss}}+n_{\mathrm{ts}}+n_{\mathrm{ap}}\text{(17)}& {\displaystyle }{n}_{\mathrm{ap}}=\mathrm{3}\mathit{\%}\times (n_{{\mathrm{SO}}_{\mathrm{2}}}+n_{\mathrm{sas}})\text{(18)}& {\displaystyle }{n}_{\mathrm{sas}}=n_{\mathrm{tos}}-n_{\mathrm{ss}}-n_{\mathrm{ts}}-n_{\mathrm{ap}}\end{array}$$

Where n_tos, n_ps and n_sas represent the molar concentrations of the total primary, and secondary atmospheric sulfate, respectively, with the primary fraction further resolved into sea-salt (n_ss), terrestrial (n_ts), and anthropogenic primary (n_ap) components. To constrain these natural backgrounds, we employed companion tracer concentrations and mass ratios; specifically, sea-salt sulfate was quantified from soluble Na⁺ assuming a purely marine origin (n_ss=k[Na⁺], where $k=\left[{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right]/\left[{\mathrm{Na}}^{+}\right]=\mathrm{0.25}$) (Zheng et al., 2024; He et al., 2018), while terrestrial sulfate was evaluated based on crustal Ca²⁺ abundances ($\mathrm{0.18}\times \left[{\mathrm{Ca}}^{\mathrm{2}+}\right]$) (Zheng et al., 2024; He et al., 2018; Dasari et al., 2022; Dasari and Widory, 2024). Building upon this physical allocation, the stable sulfur isotope signatures (δ³⁴S) are integrated via an isotopic mass-balance framework to isolate the secondary formation pathways (Zheng et al., 2024):

$$\begin{array}{}\text{(19)}& {\displaystyle }\begin{array}{rl}{\mathit{\delta }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{obs}}& =f_{\mathrm{ss}}\times {\mathit{\delta }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{ss}}+f_{\mathrm{ts}}\times {\mathit{\delta }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{ts}}+f_{\mathrm{ap}}\\ & \times {\mathit{\delta }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{ap}}+f_{\mathrm{sas}}\times {\mathit{\delta }}^{\mathrm{34}}{\mathrm{S}}_{\mathrm{sas}}\end{array}\text{(20)}& {\displaystyle }{f}_{\mathrm{ps}}=f_{\mathrm{ss}}+f_{\mathrm{ts}}+f_{\mathrm{ap}}\end{array}$$

Here, δ³⁴S_obs denotes the bulk observed δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ value, while the terms prefixed with f represent the corresponding molar fractions of each discrete source relative to the total sulfate burden, matched with their respective endmember isotopic signatures where f_ps, f_ss, f_ts, f_ap, and f_sas are the proportion of primary sulfate, sea salt sulfate, terrestrial sulfate, anthropogenic primary sulfate, and secondary sulfate, respectively (δ³⁴S_obs, δ³⁴S_ss, δ³⁴S_ts, δ³⁴S_ap, and δ³⁴S_sas).

Figure 1 Proportion of different air mass periods.

3 Results and Discussion

Chemical profiling of fine particulate matter in Nanjing reveals that sulfate dictates the water-soluble inorganic fraction, accounting for 27.9 % to 42.3 % of the total ionic mass. This pronounced dominance explicitly underscores the region's persistent susceptibility to sulfate-driven aerosol pollution. This pattern aligns with observations from other Chinese urban centers including Hangzhou (33.7 %) (Lin et al., 2022), Guangzhou (40.2 %–61.0 %) (Tao et al., 2014b), and Beijing (33.0 %) (Wei et al., 2018), collectively highlighting the prevalence of sulfate-rich PM_2.5 across China. Pollution episodes were categorized by PM_2.5 concentrations into clean (≤35 µg m⁻³), moderate (35–75 µg m⁻³), heavy (75–150 µg m⁻³), and haze (>150 µg m⁻³) periods, with haze frequency (20 %, Fig. 1) doubling that of clean periods (10 %). Sulfate concentrations exhibited pronounced pollution-dependence, peaking at (27.55±14.36 µg m⁻³) during haze events 3.2 times higher than clean-phase levels (8.56±3.19 µg m⁻³).

Figure 2 Time series of concentrations (values) of (a) PM_2.5 and ions, (b) SO${}_{\mathrm{4}}^{\mathrm{2}-}$ and δ³⁴S−SO${}_{\mathrm{4}}^{\mathrm{2}-}$, δ¹⁸O−SO${}_{\mathrm{4}}^{\mathrm{2}-}$, (c) SO₂ and SOR, and (d) ambient temperature (T) and RH.

NH${}_{\mathrm{4}}^{+}$ and NO${}_{\mathrm{3}}^{-}$ jointly accounted for 50.1 %–57.9 % of water-soluble ions (Fig. 2a), while their strong covariation with SO${}_{\mathrm{4}}^{\mathrm{2}-}$ (Fig. 2a, b) substantiates secondary sulfate formation as the principal PM_2.5 component and implicates combustion-derived emissions as its dominant source. The significant positive correlation between SO${}_{\mathrm{4}}^{\mathrm{2}-}$ and PM_2.5 concentrations (Fig. S2) further establishes sulfate's direct mechanistic role in particulate pollution escalation. The atmospheric source apportionment can be quantitatively assessed through the $\left[{\mathrm{NO}}_{\mathrm{3}}^{-}\right]/\left[{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}\right]$ mass ratio, where elevated values reflect increasing dominance of mobile versus stationary emission sources (Xiao et al., 2014; Arimoto et al., 1996). Our measurements yielded ratios spanning 0.15–1.95 (mean 0.98±0.42), exceeding Shanghai's 2002 baseline (0.43) (Yao et al., 2002) yet remaining below Jiaozuo's 2017 levels (1.36) (Zheng et al., 2024), collectively demonstrating Nanjing's intermediate position in mobile-source contribution among Chinese cities. These secondary aerosols originate through atmospheric oxidation of precursor gases, with SO₂ and NO_x principally derived from coal combustion and vehicular emissions respectively, while NH₃ predominantly stems from agricultural and livestock activities (Tao et al., 2014a). The strong inter-species correlations (Fig. S3a, b) confirm the co-formation of (NH₄)₂SO₄ and NH₄NO₃ through gas-to-particle conversion processes. Revealed across distinct pollution episodes, the unique sulfur and oxygen isotopic signatures (Fig. 2b) provide unequivocal evidence of temporal shifts in sulfate formation pathways. To accurately deconvolute the relative contributions of these specific reaction routes, rigorous mechanistic modeling becomes indispensable. Crucially, the persistent incomplete oxidation of sulfur dioxide to sulfate under all pollution conditions (Fig. 2c) underscores the imperative for oxidation-process-constrained model optimization to elucidate the authentic atmospheric sulfate formation mechanisms.

Figure 3 Pie charts of relative abundances of ions to TWSIIs and the values in Nanjing for the different air clusters. The concentrations of TWSIIs and the δ³⁴S–SO${}_{\mathrm{4}}^{\mathrm{2}-}$ values are shown in the parentheses (a: winter b: summer).

The oxygen isotopic composition of atmospheric sulfate serves as a reliable indicator of oxidation pathways, with sulfate in PM_2.5 predominantly originating from SO₂ oxidation, consistent with established atmospheric chemistry. The δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ values (+1.7 ‰ to +6.3 ‰) of Nanjing sulfate aerosols, indicating coal combustion (+4.6 ‰ to +6.6 ‰) as the dominant source, thereby reflecting a greater coal-derived contribution relative to other Chinese regions. Cluster analysis of air mass trajectories reveals distinct seasonal patterns. During summer (Fig. 3b), trajectories originating locally (54.55 %), southwest China (13.64 %), southeast China (18.18 %), and the eastern Yellow Sea (13.64 %) exhibit higher SO${}_{\mathrm{4}}^{\mathrm{2}-}$ concentrations locally (19.81±5.61 µg m⁻³) versus southwest (7.34±2.35 µg m⁻³), southeast (15.6±4.31 µg m⁻³), and Yellow Sea (16.98±5.29 µg m⁻³) sources, while δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ values show an inverse trend with lower values locally (2.88±0.73 ‰) and in the southwest (2.70±0.93 ‰) compared to southeast (3.33±0.9 ‰) and Yellow Sea regions (4.55±0.25 ‰), suggesting heterogeneous sources and formation mechanisms. Winter measurements (Fig. 3a) display significant sulfate concentration disparities across back trajectories of air mass (e.g., cluster 1: 23.84±15.28 µg m⁻³; cluster 3: 7.34 µg m⁻³), but statistically invariant δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ values (cluster 1: 4.77±0.99 ‰; cluster 3: 5.04±0.46 ‰); northeastern province (5.48±0.29 ‰), with both concentration and δ³⁴S elevated versus summer, attributable to long-range transport of primary sulfates and coal-combustion emissions enriched in heavy isotopes. The significant negative correlation (Fig. 2b, d) between δ³⁴S and temperature arises as higher temperatures promote OH-mediated uniform oxidation, facilitating isotopic equilibrium fractionation during ${\mathrm{HSO}}_{\mathrm{3}}^{-}/{\mathrm{SO}}_{\mathrm{4}}^{\mathrm{2}-}$ exchange, which favors light-sulfur-isotope incorporation into sulfate and lowers summer δ³⁴S (Han et al., 2016; Novák et al., 2001). Thus, secondary sulfate formation pathways and source variations primarily drive the observed winter-high/summer-low δ³⁴S pattern in PM_2.5, underscoring the mechanistic study's critical role in sulfate pollution control.

Figure 4 Footprints of sulfate aerosols at the receptor site during the (a) winter and (b) summer computed by the FLEXPART model. The color scale denotes the residence time (s) of sulfate aerosols in the grid cell.

Besides, by synergistically combining AMBTs (Fig. S4), CWT (Fig. S5) analysis, and FLEXPART footprint modeling (Fig. 4a, b), our integrated framework now successfully bridges the gap between empirical trajectory data and actual emission source strengths. This robust spatial validation explicitly supports our overarching objective to resolve long-standing discrepancies in atmospheric sulfate source attribution.

Figure 5 (a, b) Rayleigh fractionation models for secondary SO${}_{\mathrm{4}}^{\mathrm{2}-}$ formation in different seasons. (c, d, e, f) Relative contributions of primary, OH, TMI, NO₂, and ${\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}$ pathways to secondary SO${}_{\mathrm{4}}^{\mathrm{2}-}$ generation across different seasons. (c) and (d) represent the fractionation effects under InCO Process; (e) and (f) represent the fractionation effects under CO Process.

The atmospheric sulfate formation pathways exhibit distinct sulfur isotope fractionation signatures, as demonstrated by theoretical calculations of δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ evolution under different oxidation mechanisms. Complete oxidation of SO₂ via ${\mathrm{H}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}/{\mathrm{O}}_{\mathrm{3}}$ pathways would drive δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ depletion from 15.3 ‰ to 2.6 ‰ (Fig. 5a), while OH-mediated oxidation would produce isotopic depletion from 11.8 ‰ to 3.4 ‰ (Fig. 5a), both scenarios reflecting progressive sulfur oxidation (SOR 0 to 1). Conversely, exclusive S(IV) + O₂ (TMI) pathway participation would generate δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ enrichment from −6.4 ‰ to −1.0 ‰ (Fig. 5a). These differential fractionation patterns confirm the coexistence of multiple sulfate production routes in the atmosphere, with all observed δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ values falling within the predicted theoretical ranges. However, our field measurements demonstrate that total particulate sulfate concentrations peak at 27.55±14.36 µg m⁻³ during severe haze events, which predominantly occur in winter. Consequently, when translating these relative fractions into absolute mass concentrations, the NO₂ pathway produces a substantially higher burden of particulate sulfate in winter compared to summer.

Figure 6 Linear regression between fractionation factor α1 (InCO Process) and fractionation factor α2 (CO Process) (the red line represents y=x [1:1 line], and the dark teal-blue line represents the fitted regression line between them).

The isotopic fractionation factor α for sulfur isotopes was derived via Rayleigh fractionation models applied to δ³⁴S-SO_2−emission calculations based on atmospheric δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ and SOR, with synchronous field measurements (InCO Process) of δ³⁴S-SO₂ averaging 1.7±1.2 ‰ (range: −1.80 ‰ to +4.05 ‰), yielding $\mathit{\alpha }=\mathrm{2.0}\pm \mathrm{1.8}$ ‰ (range: −1.2 ‰ to +6.7 ‰) (Fig. 6b) per designated formula. Under complete oxidation (CO Process) (SOR = 1), δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ demonstrated a significant negative correlation with SOR (δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}=-\mathrm{3.16}\times \text{SOR}+\mathrm{5.03}=\mathrm{5.03}$; $r=-\mathrm{0.36}$, p≤0.05) (Fig. S6), resulting in δ³⁴S-SO_2−emission of 1.8 ‰ at full conversion, which exceeds values from Shanghai (1.3 ‰) (Li et al., 2020), Hangzhou (1.5 ‰) (Lin et al., 2022), and Jiaozuo (1.7 ‰) (Zheng et al., 2024) and represents the pre-transformation source signature. An alternative α calculation (formula 9) generated a mean of 2.8±1.7 ‰ (range: −0.2 ‰ to +6.1 ‰), higher than Guangzhou (<1.5 ‰) (Fan et al., 2020b) but lower than Beijing (4.2±1.2 ‰) (Lin et al., 2018) with elevated α in eastern China attributable to temperature-dependent fractionation increases (Fig. S7). The discrepancies arose from oxidation-rate variations under full-oxidation assumptions that alter pathway quantification.

By systematically evaluating the reaction-specific fractionation factors intrinsic to distinct SO₂ oxidation mechanisms, we successfully apportioned the diverse pathways governing sulfate formation. This evaluation reveals that TMI-catalyzed and NO₂-mediated reactions collectively dictate atmospheric sulfate production across all considered scenarios. Crucially, as corroborated by isotopic mass balance calculations, imposing idealized complete-oxidation assumptions systematically suppresses the calculated contribution of the TMI pathway, a bias successfully resolved through the application of kinetic partial-oxidation models. This systematic bias originates from neglecting kinetic fractionation effects during intermediate oxidation stages, particularly under high aerosol acidity conditions where TMI chemistry prevails. The persistent dominance of these two pathways persists despite varying meteorological conditions and emission profiles, suggesting their fundamental role in sulfur cycling.

Figure 7 (a) Seasonal variations of source apportionments to sulfate aerosols in Nanjing. (b) Source apportionments of sulfate aerosols in Nanjing resolved by the SIAR model with consideration of fractionation effects under InCO Process and CO Process.

Quantitative source apportionment of atmospheric sulfate was achieved through coupled δ³⁴S-SO${}_{\mathrm{4}}^{\mathrm{2}-}$ measurements and Bayesian isotopic mixing models, requiring emission source-specific δ³⁴S signatures as critical input parameters. Correction for sulfur isotopic fractionation during SO₂– to –SO${}_{\mathrm{4}}^{\mathrm{2}-}$ conversion is essential, as this process enriches δ³⁴S values by 1.2 ‰–7.2 ‰ (Lin et al., 2022), necessitating subtraction of pathway-specific fractionation coefficients prior to source quantification. Neglecting fractionation effects distorts source contributions, yielding winter estimates of 19 % (anthropogenic primary sulfate), 4 % (terrestrial sulfate and sea salt), 32 % (coal), 4 % (oil), 15 % (biomass), and 26 % (vehicle), alongside summer values of 8 %, 2 %, 62 %, 3 %, 9 %, and 16 % (Fig. 7a) respectively–results that inaccurately emphasize traffic emissions contrary to regional emission inventories and chemical transport modeling. Incorporating fractionation factors resolves this discrepancy, aligning sulfate sources with inventory data where coal combustion dominates annually, while idealized complete-oxidation models underestimate summer vehicle contributions by 8 % and overestimate coal combustion proportions (about 10 %) by comparable margins despite minimal winter deviations (Fig. 7b). This systematic validation underscores the requirement for fractionation correction in isotope-based atmospheric sulfate sourcing.

4 Conclusions

This study advances atmospheric sulfate source apportionment by transcending the idealized complete SO₂ oxidation assumptions that constrain conventional isotopic models. By integrating ambient sulfur and oxygen isotope field observations with Rayleigh fractionation models, we successfully derived kinetic partial-oxidation fractionation factors that reflect genuine atmospheric conditions. Our integrated framework reveals the predominance of TMI-catalyzed and NO₂-mediated pathways in atmospheric sulfate production across varying seasonal extremes. By comparing our kinetic fractionation-corrected framework against traditional complete-oxidation models, we demonstrate that idealized frameworks systematically obscure chemical realities. Neglecting kinetic fractionation effects during intermediate oxidation stages disproportionately diminishes the calculated contributions of the TMI pathway, particularly under high aerosol acidity conditions where TMI chemistry prevails. By applying our corrected framework, we align our source apportionment more accurately with regional emission inventories, successfully resolving calculation discrepancies that have long plagued isotopic and transport-modeling approaches. While our multi-model analytical framework corrects systematic biases in pathway quantification, certain caveats remain. The empirical validation in this study is currently constrained to seasonal extremes, specifically January and July observations, at a single monitoring site in Nanjing. Future work is needed to expand this kinetic fractionation-corrected framework across diverse geographical regions and continuous temporal scales to further validate oxidation-process-constrained model optimizations. Correcting these systematic isotopic biases carries profound implications for our understanding of atmospheric pollution states and climate regulation. Our findings expose a significant seasonal misalignment in current evaluations, demonstrating that traditional approaches overestimate summer coal combustion contributions by 10 % while underestimating summer traffic emissions by 8 %. Consequently, effective sulfate mitigation cannot rely solely on blanket SO₂ reductions. Regulatory frameworks must pivot toward multi-pollutant, seasonally differentiated strategies that dynamically target summer vehicular dynamics and the co-emitted transition metals driving catalytic sulfate formation.