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# Technical note: Determination of binary gas-phase diffusion coefficients of unstable and adsorbing atmospheric trace gases at low temperature – arrested flow and twin tube method

### Torsten Carstens

### Dirk Hupperich

### Silke Schweighoefer

### Ulrich Schurath

Gas-phase diffusion is the first step for all heterogeneous reactions under atmospheric conditions. Knowledge of binary diffusion coefficients is important for the interpretation of laboratory studies regarding heterogeneous trace gas uptake and reactions. Only for stable, nonreactive and nonpolar gases do well-established models for the estimation of diffusion coefficients from viscosity data exist. Therefore, we have used two complementary methods for the measurement of binary diffusion coefficients in the temperature range of 200 to 300 K: the arrested flow method is best suited for unstable gases, and the twin tube method is best suited for stable but adsorbing trace gases. Both methods were validated by the measurement of the diffusion coefficients of methane and ethane in helium and air as well as nitric oxide in helium. Using the arrested flow method the diffusion coefficients of ozone in air, dinitrogen pentoxide and chlorine nitrate in helium, and nitrogen were measured. The twin tube method was used for the measurement of the diffusion coefficient of nitrogen dioxide and dinitrogen tetroxide in helium and nitrogen.

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The critical role of heterogeneous reactions in atmospheric chemistry is widely accepted. The diffusion of gas molecules towards the surface is the first step in a heterogeneous reaction, and it can influence and sometimes even control the overall rate of the uptake of a trace gas onto the surface (Kolb et al., 2010; Tang et al., 2014a). Diffusion also plays a role in atmosphere–biosphere interactions: the incorporation of trace gases like ozone and nitrogen dioxide into leaves and isoprene out through stomata is diffusion controlled (Laisk et al., 1989; Eller and Sparks, 2006; Fall and Monson, 1992).

Marrero and Mason (1972), Massman (1998), Tang et al. (2014a, 2015), and Gu et al. (2018) compiled and evaluated the available experimental data on the diffusion coefficients of atmospheric trace gases. However, the existing compilations focus on stable gases; experimental diffusion coefficients of ozone, nitrogen dioxide, chlorine nitrate and dinitrogen pentoxide are still missing. They cannot be predicted with the required accuracy because detailed kinetic theory requires intermolecular potentials that are not generally available for atmospherically relevant compounds.

Poling et al. (2004)Poling et al. (2004)Poling et al. (2004)Poling et al. (2004)Poling et al. (2004)Poling et al. (2004)Brokaw and Svehla (1966)Massman (1998)Brokaw and Svehla (1966)Patrick and Golden (1983)Patrick and Golden (1983)The letter v – obtained from viscosity data; b – obtained from *T*_{b} and *V*_{b} using Eq. (4).

Chapman and Enskog derived the following equation from the kinetic theory of gases for the molecular binary diffusion coefficient:

where *m* is the mass of the molecules, *k* is the Boltzmann constant, *p* is the pressure and *T* is the absolute temperature. *σ*_{AB} is
the characteristic length of the intermolecular force law, and Ω_{D} is the dimensionless collision integral of diffusion. It depends on
the temperature and the characteristic energy *ϵ*_{AB} of the Lennard–Jones potential describing the intermolecular force
(Poling et al., 2004; Marrero and Mason, 1972). Ω_{D} as a function of temperature is expressed by the fit function

where $\mathrm{\Theta}=kT/{\mathit{\u03f5}}_{AB}$, *A*=1.06036, *B*=0.15610, *C*=0.19300, *D*=0.47635, *E*=1.03587, *F*=1.52996, *G*=1.76474 and *H*=3.89411 (Neufeld et al., 1972; Poling et al., 2004). The equations

are usually employed to relate the interaction parameters of the Lennard–Jones potential between components *A* and *B* to the interaction potential
parameters of the individual components. A tabulation of the potential parameters of the species considered in this work is given in
Table 1. The Lennard–Jones parameters *σ* and *ϵ* are generally not available for unstable atmospheric trace gases.
Patrick and Golden (1983) estimated them by using the equations

from *T*_{b} the normal boiling point temperature and *V*_{b} the molar volume at boiling point. In cases in which *V*_{b}
cannot be determined experimentally, it is obtained from tables of atomic volumes using the LeBas method. Patrick and Golden (1983) assumed the systematic
errors of *σ* and *ϵ* obtained by this method to be ≤20 *%*.

The diffusion coefficient as a function of pressure in a narrow temperature range close to the reference temperature *T*_{0} is usually expressed as

where *T*_{0}=273.15 K is the standard temperature and *p*_{0}=101 325 Pa is standard pressure (STP). Close to the reference
temperature *T*_{0}, the temperature coefficient *b* can be calculated as follows (Poling et al., 2004):

From Eq. (2) it is obtained by derivation

Fuller et al. (1966) developed a simple correlation equation for the estimation of gas-phase diffusion coefficients using additive
atomic volumes *V*_{A} and *V*_{B} for each species. With the molar masses *M*_{A} and *M*_{B} ([*M*]= g mol^{−1}) of each species and [*p*]= bar,
the diffusion coefficient ([*D*]= cm^{2} s^{−1}) is given by

Tabulations of atomic volume increments are summarized by Poling et al. (2004) and Tang et al. (2014a).

In the atmosphere, for typical submicron-sized aerosol particles, gas-phase diffusion does not usually limit uptake. Therefore, for modeling atmospheric processes, it is sufficient to use diffusion coefficients obtained using the Fuller method. However, in many laboratory experiments for the measurement of mass accommodation coefficients, conditions are such that gas-phase diffusion limitations need to be taken into account (Kirchner et al., 1990; Müller and Heal, 2002; Davidovits et al., 2006).

## 2.1 Arrested flow method

The arrested flow (AF) method was first described by Knox and McLaren (1964) and McCoy and Moffat (1986): the diffusion coefficient of a given trace gas is derived
from the broadening of width *ς*_{t} of trace gas plugs arrested for different times in a long void gas chromatography glass column
(length *l*=2.8 m, radius *r*=0.189 cm). A plug is generated by injecting a small amount of dilute trace gas into a steady flow of
carrier gas by means of computer-controlled solenoid valves. The flow is arrested when the plug has traveled halfway down the tube; see
Fig. 1. In the absence of turbulence, the initial plug profile spreads out along the tube by molecular diffusion only. Until the flow is
arrested, the box profile of the trace gas is reshaped to Gaussian by Taylor diffusion (Taylor, 1953, 1954) if the condition

is fulfilled, where $\dot{V}$ is the carrier gas flow rate. After a given arrest time *t*_{a}, the trace gas is eluted with
≈ 20 sccm (1 sccm = 1 mL min^{−1} at 273.15 K and 1013 hPa) and the concentration profile is
measured with a suitable gas chromatography detector. This procedure is repeated for different arrest times *t*_{a}. The experimental peak
profiles are fitted to Gaussians to determine the peak variance ${\mathit{\varsigma}}_{\mathrm{t}}^{\mathrm{2}}$. According to theory based on Fick's second law of
diffusion,

a plot of ${\mathit{\varsigma}}_{\mathrm{t}}^{\mathrm{2}}$ versus arrest time *t*_{a} should be linear. The slope of the plot of *ς*_{z} vs. *t*_{a}
is given by

Since the variance is measured in units of time, it has to be converted to units of length using the gas flow speed *v* in the column

From the carrier gas mass flow $\dot{n}$, temperature *T* and pressure *p* in the column that approximately equals atmospheric pressure, the flow
speed can be determined by

The column is embedded in an aluminum block cooled by a recirculating cryostat (Lauda RLS6). The aluminum block is mounted in a plastic box insulated by Styrodur. The column temperature homogeneity is monitored with two Pt-100 sensors connected to the upper and lower parts of the column coil. The solenoid valves are connected by 1∕16” Teflon tubes and controlled by a computer using the software Asyst 3.1 (Keithley). At each temperature, 12 to 20 peaks are recorded at different arrest times.

The systematic error of the determined diffusion coefficients using this method primarily depends on the systematic error of measuring the inner
diameter of the column and the systematic error of the mass flow rate. A Teflon tube pushed through the column was used to determine the length of
the column. The void volume of the column was determined by filling the column with water and measuring the weight of the water. From volume and
length, the cross-sectional area and radius are calculated, yielding a mean radius with a systematic error of 0.5 %. After the experiments, the
column was cut into small fragments. The inner diameter of these fragments was measured using a caliper gauge. We found that the inner diameter
synchronously changes with column winding with a variability of 1 %. When using Eq. (14) to transform Δ*ς*_{t} to
Δ*ς*_{z}, not the mean velocity but the actual velocity *v* and radius *r* at the location where the peak is arrested are
relevant. Therefore, the actual systematic error of the radius is about 1 %. The mass flow controllers were calibrated using a soap bubble flow
meter. Thus, the systematic error of the mass flow rate is about 1.5 %. This sums up to a total theoretical systematic error for the AF method of
about 7 %. The random error of the method is about >0.4 *%*, twice the repeatability >0.2 *%* of the flow rate.

## 2.2 Twin tube method

The twin tube (TT) method is a steady-state technique for diffusion coefficient measurements over a wide temperature range using a diffusion bridge
(Marrero and Mason, 1972). It is insensitive to wall adsorption effects, which may invalidate AF measurements at low temperature. Our apparatus consists of
two parallel horizontal flow tubes (length 2 m, inner diameter 10 mm) connected by a bunch of *n*=220 carefully
thermostatted fused silica capillaries of radius $r=(\mathrm{39.2}\pm \mathrm{0.4})\phantom{\rule{0.125em}{0ex}}\mathrm{\mu}\mathrm{m}$ and length $l=(\mathrm{20.8}\pm \mathrm{0.3})\phantom{\rule{0.125em}{0ex}}\mathrm{mm}$; see
Fig. 2. The capillaries are embedded in a block made of brass. The cooling liquid of a cryostat (Lauda RLS6) circulates through the brass
block, thereby covering the range from ambient temperature down to 198 K. Close to the diffusion bridge, the temperature in the block is
measured with two Pt-100 sensors. The capillaries are pasted into two parallel slits in a short section of the parallel flow tubes that is made of
stainless steel. Upstream and downstream of the brass block, the flow tubes consist of glass. The entire apparatus consisting of the flow tubes and the
diffusion bridge is housed in a large insulated box that can be cooled down to 260 K. After changing the setting of the recirculating
thermostat by an increment of 10 K it takes about 1 h until the temperature of the diffusion bridge has equilibrated.

Pure carrier gas is flown through one of the flow tubes, while a constant trace gas concentration *c*_{0} is maintained in the other. A concentration
gradient is established along the capillaries. This gives rise to a constant flux *J*_{D} by molecular diffusion through the diffusion bridge
described by Fick's first law of diffusion:

where *c* is the trace gas concentration at the low concentration end. Pressure differences between the flow tubes are carefully eliminated to
suppress trace gas transport by viscous flow through the capillaries. This requires that both flow tubes are totally symmetric. The difference
pressure is monitored using a differential high-accuracy pressure transducer (MKS model 398, measuring range 10^{−4} to 1 Torr). By
measuring the concentration ratio in both flow tubes, downstream of the diffusion bridge the diffusion coefficient is obtained by

when *c*≪*c*_{0}, where $\dot{V}$ is the volume flow rate of the carrier gas. The ratio of mass transport by viscous flow to diffusion flow through
the capillaries is given by (viscosity *η*)

Therefore, the ratio of interfering viscous trace gas to mass flow by diffusion was minimized by using narrow capillaries. For the diffusion of
NO_{2} in N_{2} at standard pressure and temperature, the fraction of viscous flow can be held <1 *%* when keeping the differential
pressure $\mathrm{\Delta}p<\mathrm{2}\times {\mathrm{10}}^{-\mathrm{4}}$ Torr. During the TT experiments the differential pressure was maintained low so that the fraction of viscous
flow was less than 0.3 *%*.

A trace gas detector is required that is linear over a wide concentration range and stable over time. Depending on the trace gas and the detector properties, the trace gas can either be detected by continuous mode (Fig. 3) or by peak mode (Fig. 4). The low concentration is determined with a random error of about 1 %. The signal of the trace gas detector is fed into an A/D converter with 16 bit resolution (Data Translation DT2705/5715A).

The supplier of the capillary columns used in this work as raw material for assembling the diffusion bridge reports the inner diameter with
a systematic error of 10 %. This is too much for the measurement of diffusion coefficients. Therefore, two segments of the column were used to
determine the inner diameter by weighing an empty and a water-filled section of the capillary column. Thereby, the radius of the diffusion bridge
capillaries was determined with a systematic error of 1 %. We tried to validate the result with electron micrography of two cross sections of the
column material. However, the systematic error of the inner diameter measured by electron micrography is about 5 %. When assuming a systematic
error of the flow rate of 1.5 %, this results in a total systematic error of the method of <4 *%*. The random error of the method depends on the
random error of about 1 % of the determination of the lower concentration *c*.

Later on during the experiments it was found that some capillaries of the diffusion bridge became blocked by dust or condensed matter. Fortunately, the TT method can be utilized to measure the diffusion coefficients of several species simultaneously when using the peak mode and a gas chromatograph as a detector. If the diffusion coefficient of one of the trace gases has been determined with another reliable technique at one temperature, this diffusion coefficient can be used as an internal standard; see Fig. 5. It is assumed that the effective area of the capillaries is independent of temperature.

To determine *D*_{0} and *b*, the diffusion coefficients obtained at different temperatures were fitted by nonlinear regression to Eq. (5).
They were weighted by the inverse of their statistical error where available. The results are summarized in Table 3 together with the
diffusion coefficients calculated by the Chapman–Enskog theory using Eq. (1) and the Fuller method using Eq. (10). The input
parameters used are summarized in Table 1.

## 3.1 Method validation

### 3.1.1 Diffusion of methane (CH_{4}) and ethene (C_{2}H_{4}) in helium and air

These gases were investigated with both methods for validation purposes. They are stable and non-adsorbing, and there are reference data on the diffusion coefficients in the literature. The diffusion coefficients in helium have been measured previously over a wider temperature range with high precision and accuracy by Dunlop and Bignell (1987) for methane and Dunlop and Bignell (1990) for ethene. Evaluated diffusion coefficients of hydrocarbons in air at 298 K are reported in the review of Tang et al. (2015). In addition, the diffusion coefficients can be calculated using the Lennard–Jones model and the Chapman–Enskog theory using Eq. (1).

We used a flame ionization detector (Carlo Erba FID 40 with EL980 control unit), which is fast, sensitive and linear over a wide concentration range, to measure the hydrocarbons. For the AF experiments, 0.1 % methane or ethane in helium or air was injected as a 300 ms pulse. About 20–26 sccm was used as a flow rate. The arrest time was varied from 0 to 200 s. For the TT experiments 0.5 and 1 % methane in air, 0.4 % methane in He, 0.4 % ethene in He, and 0.5 % ethene in air were admitted into the flow tube. Downstream of the diffusion bridge, the trace gas was analyzed using the peak mode setup; see Fig. 4.

Source: ^{a} Dunlop and Bignell (1992), ^{b} Dunlop and Bignell (1987), ^{c} Dunlop and Bignell (1990), ^{d} Tang et al. (2015).

The results are summarized in Tables 2 and 3 as well as in Figs. 6 and 7. Higher diffusion coefficients were found for the AF method compared to the TT method.

### 3.1.2 Diffusion of nitric oxide (NO) in helium

NO was monitored by a chemiluminescence detector (Marić et al., 1989), which was adapted to the lower flow rates of the diffusion
experiments. In the detector, NO reacts with O_{3} in a low-pressure reaction chamber (0.9–2 mbar) in a chemiluminescence reaction:

The emitted photons were detected using a Hamamatsu R562 photomultiplier tube.

For the AF experiment 100 ppm NO was injected as a 300 ms pulse into the flow tube with a flow rate of 22.5 sccm. The valves
were connected using stainless-steel tubes. For the TT experiment 30–70 ppm NO in He was admitted into the diffusion bridge. The setup
displayed in Fig. 3 was used to monitor NO in the continuous mode. It was found that after measuring the high-concentration *c*_{0},
several hours are needed until the baseline has stabilized when measuring a clean carrier gas. Therefore, it is not possible to measure *c* and *c*_{0} in
succession. Thus, first *c*_{0} was measured at room temperature. Then the detector was switched to clean carrier gas until the baseline
stabilized. Then *c* was measured after lowering the temperature until the signal stabilized. After arriving at 200 K, *c*_{0} was measured
again. Then the measurement was repeated by raising the temperature stepwise. Therefore, the complete measurement extended over several days.

The diffusion coefficients for NO obtained by the two methods are in fair agreement with the reference data from Dunlop and Bignell (1992); see Fig. 8 and Table 2. In contrast to the diffusion coefficients of methane and ethane, the diffusion coefficients obtained by the TT method are slightly larger than the diffusion coefficients obtained by the AF method.

Comparing all diffusion coefficients obtained for stable gases to reference data, it is found that the deviation of the AF and TT method is less than 8 % compared to the reference data. However, for the TT method this is a little more, as expected because of theoretical systematic error, which can be explained by decreasing effective areas of the diffusion capillaries.

## 3.2 Diffusion of atmospheric trace gases

### 3.2.1 Diffusion of ozone (O_{3}) in air

The diffusion coefficient of ozone in air has never been measured before. Ivanov et al. (2007) reported $D=(\mathrm{0.53}\pm \mathrm{0.03})$ cm^{2} s^{−1} for
the diffusion of O_{3} in He at 298 K. Since ozone is an unstable but non-adsorbing species, only the AF method was used for the
determination of the diffusion coefficient. A fast and sensitive ozone detector is required to record the ozone peaks leaving the column. A suitable
detection technique is chemiluminescence arising from the reaction of ozone with coumarin 47 (Lambda Physik; 7-diethylamino-4-methylcoumarin)
adsorbed on silica-gel plates (Schurath et al., 1991). Chemiluminescence is emitted in the range *λ*= 440–550 nm, which is detected by
a photomultiplier (Hamamatsu 931 B). The anode current was admitted through a 100 kΩ resistor and measured as voltage by a microvoltmeter
(Keithley model 155).

Ozone-containing air was generated in an aluminum block enclosing an elliptically shaped polished chamber. A rod-shaped low-pressure Hg UV lamp and a quartz tube with air running through are mounted parallel in the focal lines of the elliptically shaped chamber. Thereby, the UV radiation is focused to the air flowing through the quartz tube (Becker et al., 1975). Five blocks arranged in series were used, yielding an ozone concentration of about 40 ppm.

The injection time was varied from 250 to 500 ms, and the arrest time was varied from 0 to 360 s. It was found that the maxima of the
eluted peaks did not coincide: peaks arrested longer were eluted later. Later it was found that this was caused by leaking neoprene seals on the solenoid
valves. Therefore, at some temperatures only arrest times of less than 100 s were included in the fit of
${\mathit{\varsigma}}_{z}^{\mathrm{2}}$ vs. *t*_{a}. The statistical error of the slope of ${\mathit{\varsigma}}_{z}^{\mathrm{2}}$ vs. *t*_{a} was up to 3.7 %.

With regard to the systematic error of 7 % in the AF method for the diffusion coefficients (Section 3.1.1), the obtained value with error
ranges is ${D}_{\mathrm{0}}=\mathrm{0.15}\pm \mathrm{0.01}$ cm^{2} s^{−1}. This value is in accordance with the value of *D*_{0}=0.1444 cm^{2} s^{−1} estimated by
Massman (1998) from critical constants using the model of Chen and Othmer (1962).

### 3.2.2 Diffusion of nitrogen dioxide (NO_{2}) and dinitrogen tetroxide (N_{2}O_{4}) in helium and nitrogen

NO_{2} is in equilibrium with its dimer,

Therefore, a pure sample of NO_{2} for determinations of *D* is not available. Chambers and Sherwood (1937) assumed that the ratio of $D\left({\mathrm{NO}}_{\mathrm{2}}\right)/D\left({\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{4}}\right)=\mathrm{1.43}$, yielding ${D}_{\mathrm{0}}\left({\mathrm{NO}}_{\mathrm{2}}\right)=\mathrm{0.121}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ in nitrogen from their value of ${D}_{\mathrm{0}}\left({\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{4}}\right)=(\mathrm{0.0845}\pm \mathrm{0.0005})$ cm^{2} s^{−1}. Massman (1998) estimated *D*_{0}(NO_{2})= 0.146 cm^{2} s^{−1} from ${D}_{\mathrm{0}}\left({\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{4}}\right)=\mathrm{0.101}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{s}}^{-\mathrm{1}}$ in nitrogen reported by Sviridenko et al. (1973). Since NO_{2} is an adsorbing species, the diffusion coefficient can only be measured using the TT method. The total flux of the pseudo-species N_{IV} = NO_{2} + 2 N_{2}O_{4} through the capillaries is given by

At higher temperatures and when keeping the concentration of NO_{2} low, the diffusion of N_{2}O_{4} can be neglected. The degree of dissociation,

can be calculated from the equilibrium constant

where ${p}^{\ominus}=\mathrm{1}$ bar. The equilibrium constant close to 250 K is estimated from JANAF Thermochemical Tables (NIST, 1998):

Actually, *D*(N_{IV}) is determined using the TT experiment, depending on *D*(NO_{2}) and *D*(N_{2}O_{4}) (see Supplement S1):

A thermostatted permeation tube consisting of a PTFE tube (2.5 cm length, 4.3 mm diameter) closed with two Swagelok connectors filled
with liquid N_{2}O_{4} is used as an NO_{2} source. The measurements were performed in the temperature range from 200 to
300 K. NO_{2} was measured as NO using the chemiluminescence analyzer preceded by a thermal converter. The converter, which consisted of
a gold wire in a thin quartz tube heated to 540 K, was run with 1.5–3 % methanol vapor instead of the more commonly used CO reagent
(Langenberg et al., 1998). This eliminated poisoning of the gold wire, which occurs when metal carbonyl impurities are present in the CO gas. The total conversion
rate is unknown. However, for the TT experiment, it is only required that the conversion rate is independent of concentration, which was checked for
the concentration range of 3 to 100 ppm NO_{2} by a dynamic dilution experiment. NO_{2} was monitored using the continuous mode in
Fig. 5. In order to compensate for aging effects of the capillary bridge, CH_{4} was admixed as an internal standard to the carrier
gas flowing through the NO_{2} permeation source. The four-port valve in front of the detector was replaced by a six-port valve (Valco UC10W,
125 µm sample loop), which enables the detection of CH_{4} by the flame ionization detector. For CH_{4} in He ${D}_{\mathrm{0}}=(\mathrm{0.582}\pm \mathrm{0.003})$ cm^{2} s^{−1} at 273.15 K (Dunlop and Bignell, 1987) and for CH_{4} in N_{2} $D=(\mathrm{0.216}\pm \mathrm{0.001})$ cm^{2} s^{−1} at
298 K (Mueller and Cahill, 1964) were chosen as reference diffusion coefficients for the internal standard. The measurement was performed in a similar
manner as the NO measurement with the TT experiment by lowering and raising the temperature stepwise. Above 250 K the statistical error of
the low concentration is about 3 %. However, below 250 K signal stability was low. In contrast to the measurements above 250 K,
apparent *D*(N_{IV}) measured at a certain temperature by lowering the temperature stepwise does not reproduce the value of
*D*(N_{IV}) measured by raising the temperature stepwise.

Figure 10 displays the obtained diffusion coefficients of the pseudo-species N_{IV} as a function of temperature. It is obvious that
below 250 K the plot deviates. Above 250 K diffusion of N_{2}O_{4} can be neglected because N_{2}O_{4} is mostly dissociated
in the concentration range of our study. To estimate the diffusion coefficient of NO_{2} only data points with a dissociation degree *α*>0.95 were included in the fit of Eq. (5). Thus, regarding the errors of the diffusion coefficients of the internal standards, the
diffusion coefficients for NO_{2} at STP are ${D}_{\mathrm{0}}=(\mathrm{0.520}\pm \mathrm{0.004})$ cm^{2} s^{−1} and ${D}_{\mathrm{0}}=(\mathrm{0.145}\pm \mathrm{0.002})$ cm^{2} s^{−1} in
helium and nitrogen, respectively.

To determine *D*_{0}(N_{2}O_{4}), Eq. (23) was fitted to experimental data on *D*(N_{IV}) vs. *T* and *p*(N_{IV}) by
nonlinear regression. The temperature dependency of *D*(N_{IV}) was described by Eq. (5) and *α* as a function of temperature,
and *p*(N_{IV}) was calculated using Eq. (21). *D*_{0}(NO_{2}) and *b*(NO_{2}) were taken as fixed input parameters from
the fit of Eq. (5) as described above. However, an independent determination of *D*_{0}(N_{2}O_{4}) and *b*(N_{2}O_{4}) was not
possible. Therefore, *b*(N_{2}O_{4})=1.75 was set arbitrarily, yielding the diffusion coefficients listed in Table 3. Since
*b*(N_{2}O_{4}) is expected in the range 1.5 to 2, the fit was repeated setting *b*=2 and *b*(N_{2}O_{4})=1.5 to estimate the upper and lower
limit of *D*_{0}(N_{2}O_{4}) listed in Table 3. Compared to the diffusion coefficient of N_{2}O_{4} in N_{2}, the diffusion
coefficient in He determined by the fit is much lower than the diffusion coefficient calculated by the Lennard–Jones model. In addition, our values
are lower than the values of Sviridenko et al. (1973). We therefore consider our diffusion coefficients for N_{2}O_{4} to be unreliable. However, we
can explain the observed temperature dependency of *D*(N_{IV}) in the transition to lower temperatures.

### 3.2.3 Diffusion of chlorine nitrate (ClONO_{2}) in helium and nitrogen

Chlorine nitrate is an unstable compound. Therefore, the diffusion coefficient can only be measured using the AF method and not by the TT method. Chlorine nitrate was prepared by the reaction (Davidson et al., 1987)

Cl_{2}O was prepared by admitting chlorine into a column filled with Raschig rings covered with freshly precipitated HgO (Schmeisser et al., 1967):

The formed Cl_{2}O was condensed over N_{2}O_{5} in a cold trap cooled with liquid nitrogen. Then the cold trap was cooled with ethanol at
193 K, which was allowed to warm up to 253 K within about 15 h.

The identity of the product was checked by an FTIR spectrometer (Nicolet, model Protégé 460) with a 10 m absorption path. The spectrum
was recorded with 1 cm^{−1} resolution. ClONO_{2} was characterized by typical bands at 535–580, 750–825, 1270–1320 and
1695–1770 cm^{−1} compared to reference spectra measured by Davidson et al. (1987) and Orphal et al. (1997). No contamination of N_{2}O_{5},
NO_{2} and HCl was found.

Chlorine nitrate dissociates by the equilibrium reaction

The half-life of chlorine nitrate with respect to the thermal decomposition is 11 min at 300 K and about 7.8 h at 273 K. In addition to homogeneous dissociation, chlorine nitrate is lost by heterogeneous reaction with adsorbed water on the column surface (Tang et al., 2016):

To minimize this interference, the column was preconditioned with chlorine nitrate to remove moisture. During the first series of measurements with He
as a carrier gas, chlorine nitrate was continuously admitted into the column for 10–15 min prior to the experiments. However, due to desorption
the baseline stabilized only after some time. Therefore, during the second series of measurements with N_{2} as a carrier gas, series of peaks of
chlorine nitrate were admitted into the column until the peak size stabilized. During one series of measurements, 16 peaks covering
arrest times from 0 to 100 s and carrier gas flow rates of 19.6 sccm N_{2} and 28.6 sccm He were measured. The diffusion
coefficient was measured in a temperature range of 235 to 300 K.

The detection of chlorine nitrate was performed as described by Anderson and Fahey (1990): an excess of NO (30–75 ppm) was added as a constant flow
of 6.5–8 sccm in N_{2} before the detector by a T-tube. Behind the T-tube a glass capillary of 8 cm length and 0.1 cm
inner diameter was mounted. The capillary was inserted in a stainless-steel tube, which was heated on a length of 1.7 cm to 433 K by
two heating resistors. In the heating zone, chlorine nitrate is dissociated to ClO by Reaction R5. By subsequent scavenging
reactions, NO is irreversibly removed.

A complete conversion of chlorine nitrate with NO is assumed. The drop in NO concentration equals the ClONO_{2} concentration and was
monitored using the chemiluminescence detector.

It is presumed that the chlorine nitrate loss processes during peak arrest is a pure first-order process, which is a requirement for the application
of the AF method. To check the first-order kinetics, logarithmic peak areas as a measure for the chlorine nitrate concentration were plotted against the
arrest time, yielding a straight line. This validated the first-order characteristic of the chlorine loss process. The first-order loss constants
ranged from $\mathrm{8.8}\times {\mathrm{10}}^{-\mathrm{4}}$ to $\mathrm{4.9}\times {\mathrm{10}}^{-\mathrm{3}}$ s^{−1}. During the experiments with N_{2} as a carrier gas, loss rates increased
with decreasing temperature. During the experiments with He, loss rates increased with increasing temperature. However, for the He experiments,
a less effective preconditioning technique was applied, as described above.

The diffusion coefficients obtained at different temperatures are displayed in Fig. 11 (a). When taking a systematic error of 7 %
for the AF method into account, the diffusion coefficients at STP are ${D}_{\mathrm{0}}=(\mathrm{0.31}\pm \mathrm{0.03})$ and ${D}_{\mathrm{0}}=(\mathrm{0.085}\pm \mathrm{0.007})$ cm^{2} s^{−1} in
helium and nitrogen, respectively.

### 3.2.4 Diffusion of dinitrogen pentoxide (N_{2}O_{5}) in helium and nitrogen

Crystalline N_{2}O_{5} was synthesized as described by Davidson et al. (1978) and Tang et al. (2014b): a small flow of pure NO is mixed with
O_{3}∕O_{2} in a glass reactor, trapping the product at 193 K using a cold trap immersed in a cold ethanol bath. O_{3} was
generated in pure dry O_{2} with a silent discharge ozone generator (Sorbios, model GSG). After mixing NO with O_{3}∕O_{2} in the
glass reactor, a brown color appeared initially, indicating the formation of NO_{2}:

After about 3 h, the addition of NO is stopped and the cooling bath of the cold trap is removed. To remove traces of NO_{2}, the product is
transferred into a second cold trap using an O_{3}∕O_{2} stream. By means of a cryostat, the synthesized N_{2}O_{5} crystals were stored
in an ethanol bath kept at 193 K. The identity of the product was checked by infrared spectroscopy. The spectrum was recorded with
1 cm^{−1} resolution. N_{2}O_{5} was characterized by typical bands at 750, 860, 1250, 1340 and 1725 cm^{−1} reported by
a reference spectrum measured by Cantrell et al. (1988).

The cold trap filled with N_{2}O_{5} crystals was immersed in the bath of a cryostat (Lauda RLS 6) thermostatted at 235 to 250 K. Dried
carrier gas was admitted through the cold trap. An upper limit of the partial pressure of N_{2}O_{5} in contact with the solid can be estimated
with data from McDaniel et al. (1988). This results in an upper limit of the mole fraction of about 0.1 to 0.6 %. The trace gas was admitted into the
AF experiments using short Teflon tubes.

N_{2}O_{5} was detected as already described by Fahey et al. (1985): N_{2}O_{5} is thermally decomposed to NO_{2} and NO_{3} radicals,
which are then titrated by NO to form NO_{2}:

The drop in NO concentration is equal to the N_{2}O_{5} concentration. NO was measured again by the chemiluminescence detector. Downstream of the
AF experiment, a constant flow of 6–8 sccm of 30–45 ppm NO was added.

The diffusion coefficient of N_{2}O_{5} was measured in the temperature range 245 to 298 K. Below 245 K no measurement was possible
because N_{2}O_{5} was totally adsorbed in the column. At one fixed temperature 16 peaks were measured using arrest times between 0 and
100 s. At the beginning of a measurement series, the column was preconditioned with carrier gas containing N_{2}O_{5} to remove moisture
for 15–20 min. Prior to a measurement with arrest time, a peak without arrest time was pushed trough the column. During one series of
measurements 16 peaks covering arrest times from 0 to 100 s and carrier gas flow rates of 19.3 sccm N_{2} and
28.8 sccm He were measured.

Equation (5) was used to obtain ${D}_{\mathrm{0}}=(\mathrm{0.276}\pm \mathrm{0.003})$ cm^{2} s^{−1}, $b=(\mathrm{1.0}\pm \mathrm{0.2})$ in He, and ${D}_{\mathrm{0}}=(\mathrm{0.0709}\pm \mathrm{0.0006})$ cm^{2} s^{−1} and $b=(\mathrm{1.1}\pm \mathrm{0.1})$ in N_{2}. Thus, the observed temperature coefficient is much lower than expected from
Chapman–Enskog theory. As long as the N_{2}O_{5} loss process is purely first order, the peak variance should not be affected by loss processes. To
check this, peak areas were determined by integration. Plots of log (peak area) versus arrest times yielded apparent first-order loss constants *k*_{1}
ranging from $\mathrm{4}\times {\mathrm{10}}^{-\mathrm{3}}$ to $\mathrm{2}\times {\mathrm{10}}^{-\mathrm{2}}$ s^{−1}. To check if *D* depends on *k*_{1}, the diffusion coefficient was expressed by

as a function of *T* and *k*_{1}. It was found that *D* not only significantly depends on *T*, but also on *k*_{1}. With *P*, the probability of the null
hypothesis, $a=-(\mathrm{8}\pm \mathrm{4}$) s (*P*<0.05) and $a=-(\mathrm{21}\pm \mathrm{9})$ s (*P*<0.06) were found for He and N_{2}, respectively. One
reason for this may be that the order of the loss process of N_{2}O_{5} is less than first order. The other fit parameters are displayed in
Table 3. For He as a carrier gas, a temperature coefficient of *b*<1.5 was found. Due to the small temperature range investigated, the
temperature coefficient is rather uncertain. As a final result, for He ${D}_{\mathrm{0}}=\left({\mathrm{0.30}}_{-\mathrm{0.06}}^{+\mathrm{0.03}}\right)$ and for N_{2} ${D}_{\mathrm{0}}=\left({\mathrm{0.08}}_{-\mathrm{0.02}}^{+\mathrm{0.01}}\right)$ cm^{2} s^{−1} are obtained when considering the systematic error of 7 % for the AF method and the possible
influence of dinitrogen pentoxide degradation on the diffusion coefficients obtained. Wagner et al. (2008) reported a diffusion coefficient of
0.085 cm^{2} s^{−1} for N_{2}O_{5} in N_{2} at 760 Torr and 296 K, which is within the error limits of our result.

The AF method is best suited for the measurement of the diffusion coefficients of volatile non-adsorbing trace gases, even if the trace gas is unstable like ozone. However, it is required that the trace gas loss process is first order. Otherwise, the Gaussian peak shape is distorted and the variance of the peaks depends on species reactions. The TT method is best suited for stable but adsorbing species.

For stable nonpolar gases, diffusion coefficients can be estimated from viscosity data using the Lennard–Jones model with a systematic error
of <5 *%*, which is smaller than the systematic errors of less than 7 % of the AF and TT methods. For unstable atmospherically relevant trace
gases and polar gases, the Lennard–Jones model parameters cannot be obtained by viscosity measurements. They can only be estimated from critical
temperatures and volumes. Where dipole–induced-dipole interactions come into play the systematic errors of the diffusion coefficients obtained in
this way are of the same order as the errors of the diffusion coefficients of the unstable and reactive trace gases investigated in this study.

For the species investigated in this study, it is found that Fuller's method overestimates the diffusion coefficients of inorganic compounds with a systematic error of typically less than 35 % and underestimates the diffusion coefficients of organic compounds with a systematic error of less than 15 %.

Raw data on temperature-dependent diffusion coefficients are included in the Supplement. The sample code for the calculation of diffusion coefficients using the Lennard–Jones and the Fuller model written in the language R (R Core Team, 2017) is also included.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-20-3669-2020-supplement.

Conceptualization and methodology were undertaken by SS, investigation was done by TC, DH and SS, formal analysis and visualization were performed by TC, DH, SL and SS, SL wrote the draft, and supervision and funding acquisition were carried out by US. All the authors have read and approved the final paper.

The authors declare that they have no conflict of interest.

Torsten Carstens acknowledges a doctoral grant from the Karlsruher Institut für Technologie. We thank Ralf Rubröder and Birgit Walter for setting up the AF experiment; Pete Boecker for the assistance measuring FT-IR spectra; Harald Saathoff for providing infrared spectra of N_{2}O_{5}, HCl and NO_{2}; Ewald Hild for the preparation of electron micrographs of the fused silica columns; and Dieter Gauer for technical assistance.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. 457/8-1, 457/8-2, and 457/8-3) within the priority program “Basics of the Impact of Air and Space Transportation on the Atmosphere”.

This paper was edited by Daniel Knopf and reviewed by two anonymous referees.

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