Binary Diffusion Coefficients for Methane and Fluoromethanes in Nitrogen

Abstract

Binary gas-phase diffusion coefficients, of interest in physical models of atmospheric and combustion chemistry, have been measured in ${\text{N}}_2$ for the homologous series of refrigerant-related (fluoro)methanes: methane (${\text{CH}}_4$), fluoromethane (${\text{CH}}_3\text{F}$), difluoromethane (${\text{CH}}_2{\text{F}}_2$), and trifluoromethane (${\text{CHF}}_3$). Values have been determined by reverse-flow gas chromatography, which has been previously demonstrated to provide accurate results over a wide range of temperatures. Coefficients were measured at temperatures of (300 to 550) K for all species and extending up to 650 K and 723 K for ${\text{CH}}_2{\text{F}}_2$ and ${\text{CHF}}_3$, respectively, and down to 250 K for ${\text{CH}}_4$. We also performed measurements for ${\text{CH}}_4$ in air at temperatures of (250 to 350) K, obtaining values the same as in ${\text{N}}_2$ within 0.3 %, well within our experimental uncertainty. We report the first measurements for ${\text{CH}}_3\text{F}$ and compare with the limited literature data for the other compounds. Our results agree broadly with earlier measurements in both ${\text{N}}_2$ and air. The greater temperature ranges reported in this work lead to temperature dependences that differ from most previous experiments, although they are consistent with several literature estimates and are similar to temperature exponents found for small hydrocarbons in ${\text{N}}_2$. Comparison of the present work with a recent study that found different diffusion coefficients for methane when determined in a typical arrested flow apparatus and a novel “twin tube” method unaffected by sample adsorption shows much better agreement with the the arrested flow results over all common temperatures.

Introduction

Knowledge of the diffusivity of gaseous molecules is required for many physical models, including those used in atmospheric chemistry and in the modeling of flames and other combustion phenomena. The present studies were prompted primarily by recent work at NIST to develop flammability models of fluorinated refrigerants, including R32 (difluoromethane). Such models play a role in the development of next-generation compounds and blends of compounds that are safe and effective refrigerants while having reduced global warming potentials (GWPs). In the course of our flammability model work it was noted that diffusivity data for many fluorinated species of interest were sparse and sometimes conflicting. Much of the extant data was based on work performed decades ago with uncertain reliability and a limited temperature range. We have previously demonstrated the application of reverse-flow gas chromatography to the accurate measurement of the binary diffusion coefficients of hydrocarbons at temperatures ranging from ambient to 723 K.^1–3 The present report extends those studies to the homologous series of (fluoro)methanes, methane (${\text{CH}}_4$), fluoromethane (${\text{CH}}_3\text{F}$), difluoromethane (${\text{CH}}_2{\text{F}}_2$), and trifluoromethane (${\text{CHF}}_3$).

The current work was carried out in nitrogen (${\text{N}}_2$), although one would expect very similar results in air. Matsunaga,⁴ for example, measured diffusion coefficients for ${\text{CHF}}_3$ in ${\text{N}}_2$, oxygen (${\text{O}}_2$), and air and obtained values equal within about 1 %. There are few previous experimental measurements of the diffusion coefficients of the compounds considered here, and the majority of the existing work pertains to ${\text{CH}}_4$. Even for that fundamental compound there exists a surprisingly large uncertainty in the literature data. There is a more than 10 % spread in the available measurements and review values for methane, as well as some inconsistency in the relative values in air and ${\text{N}}_2$. The value of the diffusion coefficient of ${\text{CH}}_4$ in ${\text{N}}_2$ recommended in the 1998 review of Massman⁵ is about 3 % larger than suggested by Marrero and Mason⁶ in their 1972 evaluation. However, Massman recommends an even larger value in air, while simple mass considerations would suggest diffusion in air would be similar or slightly slower than in ${\text{N}}_2$. The Massman value in air is close to that derived by Yaws⁷ using an estimation technique and is similar to those typically used in combustion modeling. However, Langenberg et al.⁸ in 2020 reported measurements for methane in air using both an arrested flow and a novel “twin tube” methodology that is not affected by sample adsorption, with a focus on measurements at ambient temperature down to ≈50 K below ambient. The “twin tube” methodology was estimated to have a lower systematic uncertainty than the arrested flow technique and those results were more consistent with the 1972 values of Marrero and Mason than the earlier work of Coward and Georgeson⁹ and later recommendation of Massman. In addition to the experimental measurements for ${\text{CH}}_4–{\text{N}}_2$, there exist various direct calculations of interest, including a 2020 correlation developed by Hellmann¹⁰ based on an ab initio potental energy surface, and earlier values from Chae et al.¹¹ using molecular dynamics simulations.

Dramatically less data on the diffusion of the fluorinated methanes is present in the literature. The only direct measurements in ${\text{N}}_2$ or air appear to be those reported by Matsunaga and coworkers, for ${\text{CH}}_2{\text{F}}_2$ in 2002¹² and ${\text{CHF}}_3$ in 2011,⁴ both obtained via Taylor dispersion.¹³ We are unaware of any previous measurement for diffusion of ${\text{CH}}_3\text{F}$ in either ${\text{N}}_2$ or air, although Yaws⁷ has reported estimated values in air. Presently, we have used a reverse-flow gas chromatography technique to determine the coefficients of interest. In previous work¹ we have carefully vetted the methodology against established reference methods and values, including measurements of the diffusion coefficients for dilute argon (Ar) in helium (He), dilute He in Ar, dilute ${\text{N}}_2$ in Ar, and dilute ${\text{CH}}_4$ in He. Excellent agreement with standard reference values was found in all cases. The present data should provide a validated and self-consistent set of results directly applicable to the (fluoro)methanes, and serve as reference data to verify estimation methods for unstudied fluorocarbons.

Experiment and Determination of Diffusion Coefficients

Apparatus

Diffusion coefficients were measured using a reverse-flow gas chromatography apparatus described previously^1–3 with minor modifications. A brief summary of the apparatus and the modifications to the instrument will be provided here. The apparatus employs two gas chromatographs (GCs), a “primary” GC, used to control valves and gas flows and to detect sample, and a “secondary” GC, used as a stable software-controlled oven whose temperature can be precisely set. The measurement is accomplished with two 316 stainless steel columns of 0.457 cm inner diameter (0.635 cm outer diameter) joined perpendicularly. One column, called the diffusion column and shown vertically in diagram a of Figure 1 with length $L$, is deadheaded at one end and precision-welded via a machined tee fitting at the opposite end to the center of the second column, called the analytical column (horizontal columns in Figure 1). During the measurement, a constant flow of bath gas is established from the inlet ($\mathbf{\text{C}}$ in the figure) through the analytical column to a detector $\mathbf{\text{D}}$, and a small aliquot of sample is injected via a custom fitting at the deadheaded end of the diffusion column, located opposite the connection to the analytical column. Laminar flow conditions are maintained in the analytical column at the interface with the quiescent diffusion column. In this way, the injected sample diffuses through the entirety of the stagnant diffusion column and is entrained in the analytical column flow at the diffusion tube endpoint. The time-dependent concentration of sample in the analytical column flow is recorded and analyzed to determine the binary diffusion coefficients. Injected samples are typically (1 to 3) % mixtures of analyte in bath gas, and concentrations in the diffusion tube rapidly decrease. Our previous work¹ found no dependence of the measured diffusion coefficient on the concentration of the injected gas within similar limits, and we assume throughout that coefficients are determined in the dilute limit of the analyte in the bath gas.

Figure 1:

Analytical column flow and origin of reversal lineshapes. Schematic showing the flow through the analytical column across the outlet of the vertical diffusion column and the corresponding detector signal profile. Arrows indicate direction of flow and shading depicts sample concentration. $\mathbf{\text{C}}$ is the carrier gas inlet, and $\mathbf{\text{D}}$ represents the detector. For clarity, only a single reversal feature is shown, although reversals are carried out repeatedly during the experiment, typically every (150 to 180) s. The features are not to scale and small concentration gradients are not depicted. The full diffusion column of length $L$ is shown in diagram a and is truncated in the remaining diagrams. The region of the holdup time, $t_h$, is indicated by the line in diagram c, and the peak time, $\tau$, and baseline-corrected peak height $s$ are shown in diagram f. Note that the peak undergoes a small amount of diffusion during $t_h$, so the actual data show a Gaussian-like lineshape. The reader is referred to Ref. 1 for further detail.

Analytical column flow comes from the inlet of the primary GC and passes to a flame ionization detector (FID) on the same GC. To account for variation of the baseline signal over the long (up to 3 h) experiment times, a heated four-port rotary valve is inserted in the analytical column flow path. Switching of this valve causes the inlet and outlet of the analytical column to switch, thereby reversing the flow direction, as shown in Figure 1 (a and b). During the reversal, a small slug of gas with already entrained sample passes again by the diffusion column outlet, effectively concentrating the sample in that volume (b), followed by an additional concentration upon reintroduction of the original flow path after a short interval (c), 12 s at low temperatures and 4 s at high temperatures. Reversals are performed at evenly spaced intervals throughout the experimental runs, typically every (150 to 180) s, to produce a series of reversal lineshapes on top of a broad diffusion signal. Because these reversal peaks are defined relative to the the envelope signal rather than the absolute baseline signal, they correct for any variation in the latter.^1,14 The peak heights are used to determine a concentration versus time profile of the injected sample at the outlet of the diffusion column relative to the injection time at the head of the column. This concentration variation can then be used to determine the dilute binary diffusion coefficient in the bath gas used. The need to use a long analytical column to allow reversals introduces a time lag between sample exiting the diffusion column and the detector (holdup time, $t_h$, in Figure 1). The extraction of the binary diffusion coefficient from these data and the implications of the time lag in its determination is later discussed in detail.

The measurement temperature is set by placing both the analytical and diffusion columns in the oven of a secondary GC, which serves only as a precise oven with temperatures settable from (250 to 723) K. At temperatures lower than 300 K, the oven was cooled with a GC-controlled flow of cold gas from a liquid nitrogen tank. For reasons related to the linearization analysis (vide infra), it is desirable to have a constant high flow rate of gas through the analytical column. Since flow rate depends on temperature, we have placed a needle valve in the oven of the primary GC held at a fixed temperature of 453 K and adjusted it to provide a constant flow as high as possible without affecting the stability of the flame in the FID, approximately 40 cm³ · min⁻¹ (101.3 kPa, 273 K) at the set pressure of the experiment.

The source of largest uncertainty at high temperatures in our previous work was due to the vertical temperature gradient in the GC oven and the need to orient the diffusion column vertically to fit. To minimize this error, the present diffusion column was formed from a (43.93 ± 0.01) cm tube bent backward and forward multiple times with a series of six 45° bends within two 90° bends to form an angular S-shape. This shape allows us to keep the entire diffusion column in a single horizontal plane of the oven at near-uniform temperature. Temperatures were determined from the nominal set temperature of the oven. Previous work¹ has shown average combined relative uncertainties (0.95 confidence level), $u_r(D)=0.003$ from the nominal temperature. In addition, combined relative uncertainties due to the vertical gradient varied from $u_r(D,298\text{K})=0.001$ to $u_r(D,723\text{K})=0.007$ (both 0.95 confidence). Since the current diffusion tube maintains a horizontal plane in the approximate center of the oven to mitigate the gradient uncertainty, we conservatively estimate a combined relative uncertainty (0.95 confidence), $u_r(D)=0.004$ for all temperatures.

Because diffusion has an $\approx T^{1.75}$ dependence on temperature, this significantly reduces the measurement error at high temperatures, where the vertical temperature gradient of the oven can be several degrees. Stretch and compression of the metal in the region of the bends, however, increases the systematic length uncertainty. To minimize this error, we presently calibrated the effective length by comparing diffusion of $n$-butane in He at 300 K in the current tube with our previously reported values² thereby deriving an effective tube length of (44.75 ± 0.25) cm, where the $2\sigma$-uncertainty was derived from the uncertainty in the measured diffusion coefficients. The net effect of the current design is a marked reduction of uncertainty at high temperatures at the expense of a slightly increased uncertainty at low temperatures. The systematic error in the diffusion coefficient $D$ is expected to be dominated by this length uncertainty, for which $D\propto L^2$, where $L$ is the diffusion column length (see below). We estimate the systematic expanded relative uncertainty in $D$, $u_r(D)=0.011$, (0.95 confidence level) and use this value in the determination of combined expanded uncertainties in $D$ reported below.

The pressure in the columns is set via the inlet pressure of the primary GC and is chosen such that the measured diffusion coefficient (as opposed to the diffusion coefficient scaled to 101.3 kPa reported here and throughout the literature) is less than 0.1 cm² · s⁻¹. The limitation on the in situ diffusion coefficient is necessary to ensure that the analysis discussed below (Linearization Analysis) is valid based on the assumptions required to obtain an analytical solution of Fick’s Law. Measurement pressures range from (403.5 to 679.3) kPa and are chosen to provide convenient measurement times of (60 to 180) min while maintaining the $D<0.1{\text{cm}}^2\cdot {\text{s}}^{-1}$ restriction.

The inlet pressure set in the GC software is a gauge pressure and thus varies with ambient pressure to ensure a constant differential pressure across an installed column at low flow rates. By using a dual-gauge silicon sensor manometer¹⁵ (Mensor, Model CPG2300, San Antonio, TX) we have simultaneously monitored the tube pressure via the injection port and the ambient laboratory pressure, and determined that variations in atmospheric pressure are only partially compensated at the large flow rates used in the present experiments. Standard uncertainty in pressure is provided by the manufacturer to be u(D)=1 hPa. Since the measurement is taken at the head of the static diffusion tube, we expect this uncertainty to dominate the combined uncertainty, resulting in a combined expanded uncertainty (0.95 confidence level), $U(D)=2\text{hPa}$. From these results, for purposes of defining the absolute measurement pressure, we apply an empirical correction to the nominal tube pressure set for the inlet carrier gas.

However, differences between the set pressure and the absolute tube pressure were found to be small, at most 0.3 %, and varied linearly with the ambient pressure as well. Therefore, we have chosen to correct the tube pressures using hourly altimeter readings from the closest airport, KGAI. These reported measurements represent the station pressure scaled to mean sea level in a 288 K standard atmosphere¹⁶ and a lapse rate of 6.5 K · km⁻¹ and are then scaled to the approximate laboratory altitude of 134 m. These barometric pressure readings were found to correlate extremely well to the laboratory manometer readings. Final corrected tube pressures varied by up to ±0.6 kPa for the highest pressures, and in all cases, the observed differences in scaled diffusion coefficients and temperature dependence parameters were well within the uncertainties presented in Tables 2 and 3.

Table 2:

Measured binary diffusion coefficients at 101.3 kPa for dilute fluorocarbons in ${\text{N}}_2$. $T$ is temperature, $D$ is the binary diffusion coefficient, and $U(D)$ is the combined expanded uncertainty in $D$ (0.95 level of confidence); compound names may be found in Table 1.

Compound	$T/\text{K}$ a	$D/\left({\text{cm}}^2\cdot {\text{s}}^{-1}\right)$	$U(D)/{\text{cm}}^2\cdot {\text{s}}^{-1}$
${\text{CH}}_4$	250	0 170	0.002
${\text{CH}}_4$	273	0 200	0.002
${\text{CH}}_4$	300	0 237	0.003
${\text{CH}}_4$	350	0 310	0.004
${\text{CH}}_4$	400	0 391	0.004
${\text{CH}}_4$	450	0 475	0.005
${\text{CH}}_4$	500	0 577	0.007
${\text{CH}}_4$	550	0 673	0.007
${\text{CH}}_3\text{F}$	300	0 186	0.002
${\text{CH}}_3\text{F}$	350	0 245	0.003
${\text{CH}}_3\text{F}$	400	0 311	0.005
${\text{CH}}_3\text{F}$	450	0 385	0.005
${\text{CH}}_3\text{F}$	500	0 463	0.007
${\text{CH}}_3\text{F}$	550	0 549	0.009
${\text{CH}}_2{\text{F}}_2$	300	0 156	0.003
${\text{CH}}_2{\text{F}}_2$	350	0 207	0.003
${\text{CH}}_2{\text{F}}_2$	400	0 259	0.003
${\text{CH}}_2{\text{F}}_2$	450	0 321	0.005
${\text{CH}}_2{\text{F}}_2$	500	0 386	0.004
${\text{CH}}_2{\text{F}}_2$	550	0 454	0.005
${\text{CH}}_2{\text{F}}_2$	600	0 528	0.006
${\text{CH}}_2{\text{F}}_2$	650	0 613	0.010
${\text{CHF}}_3$	300	0 145	0.003
${\text{CHF}}_3$	350	0 191	0.002
${\text{CHF}}_3$	400	0 240	0.005
${\text{CHF}}_3$	450	0 289	0.007
${\text{CHF}}_3$	500	0 346	0.004
${\text{CHF}}_3$	550	0 409	0.005
${\text{CHF}}_3$	600	0 475	0.012
${\text{CHF}}_3$	650	0 549	0.013
${\text{CHF}}_3$	700	0 623	0.012
${\text{CHF}}_3$	723	0 661	0.016

^aCombined relative expanded uncertainty $U_r(D)=0.004$

Table 3:

Temperature dependence parameters for diffusion coefficients of dilute fluorocarbons in ${\text{N}}_2$ at 101.3 kPa of the form $D/\left({\text{cm}}^2\cdot {\text{s}}^{-1}\right)=a{(T/\text{K})}^n$, where $D$ is the binary diffusion coefficients, $T$ is temperature, and $u(a)$ and $u(n)$ are the expanded uncertainties (0.95 confidence level) of parameters a and $n$, respectively, derived from the fits. Compound names may be found in Table 1.

Compound	$T/\text{K}$	$a/\left({\text{cm}}^2\cdot {\text{s}}^{-1}\right)$	$u(a)/\left({\text{cm}}^2\cdot {\text{s}}^{-1}\right)$	$n$	$u(n)$
${\text{CH}}_4$	250– 550	1.22 · 10−5	5.8 · 10−7	1.731	0.004
${\text{CH}}_3\text{F}$	300– 550	7.14 · 10−6	3.0 · 10−7	1.783	0.007
${\text{CH}}_2{\text{F}}_2$	300– 650	6.90 · 10−6	3.0 · 10−7	1.759	0.008
${\text{CHF}}_3$	300– 723	8.08 · 10−6	5.6 · 10−7	1.718	0.010

In the original apparatus,¹ the Agilent ChemStation GC control software parameters were changed automatically by an external script with mouse and keyboard control. It was, at the time, an effective but fragile means to reverse the analytical column flow. We have replaced this with a microcontroller-actuated relay system built in-house and placed directly inline with the solenoid that actuates the reversal valve pneumatic driver. Reversal frequency and timing are stored in programmable read-only memory of the microcontroller, which is externally triggered by the primary GC upon initiation of an experiment. The time-resolution and stability of the microcontroller is far inferior to that of the GC, and we have observed differences in the reversal valve actuation times to be as large as 0.6 s run-to-run, although they are typically 0.2 s. In any case, such time variations result only in shifts of the reversal peaks that are readily observed in the GC chromatograms and will have no impact on the analysis or the observed diffusion coefficients.

Chemicals

Chemicals used in the present work are shown in Table 1.

Table 1:

Chemicals used in this study

Chemical Formula	Source	Stated Mole Fraction Purity	Observed Mole Fraction Purity	Analysis Technique
${\text{CH}}_4$ a	Matheson	0.99995		none
${\text{CH}}_3\text{F}$ b	Lancaster	0.98	>0.998	GC/FIDg
${\text{CH}}_2{\text{F}}_2$ c	DuPont	none	>0.998	GC/FID
${\text{CHF}}_3$ d	PCR, Inc.	none	>0.998	GC/FID
${\text{N}}_2$ e	Roberts Oxygen, Rockville, MD	0.99999		none
airf	Roberts Oxygen	0.999975h		none

^amethane, CAS 74–82–8

^bfluoromethane, CAS 593–53–3

^cdifluoromethane, CAS 75–10–5

^dtrifluoromethane, CAS 75–46–7

^enitrogen, CAS 7727–37–9

^fbreathing air, CAS 132259–10–0

^gGas Chromatography with Flame-Ionization Detection

^hSince the exact composition of breathing air is not standardized, this quantity is based on the manufacturer’s specification of total hydrocarbon content <25 μL · L⁻¹. The dew point is specified to be <319 K.

Analysis

As discussed below, determination of the diffusion coefficient from the measured spectra was accomplished by a linear fit of ln $s$ ${\tau }^{3/2}$ vs. $1/\tau$, where $s$ are the baseline-corrected peak heights and $\tau =t-t_h-t_R/2$, where $t$ is the time after sample injection corresponding to an individual peak maximum, $t_h$ is the holdup time, or the time required for the sample to travel from the diffusion tube exit to the detector, and $t_R$ is the reversal time. The peak heights $s$ and peak times $t$ are determined by fitting a parabola to the peak maximum and 5 points (5.5 s) on either side for each reversal.

The slope of this line is $-L^2/(4D)$, where $L$ is the effective length of the diffusion column, and $D$ is the binary diffusion coefficient. Typical data and linear fits for individual runs are shown in Figure 2 for the diffusion of ${\text{CHF}}_3$ in ${\text{N}}_2$ at 679.1 kPa with their respective fits for several temperatures. This fitting technique is derived from a Laplace transform analysis of Fick’s Law, $D\frac{{\partial }^{2}C}{\partial z^2}=\frac{\partial C}{\partial t}$, where $C$ is the sample concentration and $z$ is the position variable. The Laplace analysis for a finite-length tube open at one end with a sample aliquot injected at the closed end was originally presented by Crank¹⁷ and was solved by Katsanos and Karaiskakis with boundary conditions reflecting the removal of sample due to the analytical column flow.¹⁴ The reader is referred to those references and our previous studies^1–3 for additional detail.

Figure 2:

Typical fitted data using linearization analysis. Data are presented for the diffusion of ${\text{CHF}}_3$ in ${\text{N}}_2$ at 679.1 kPa at the indicated temperatures. Plots are ln $s$ ${\tau }^{3/2}$ vs $1/\tau$ with a slope of $-L^2/(4D)$, where $s$ is the fitted reversal peak height, and $\tau$ is the post-injection time, corrected for holdup and reversal as described in the text.

The holdup time, $t_h$ in the calculation of $\tau$ above, represents the time required to pass through the half of the analytical column between the outlet of the diffusion tube and the detector. In our previous work, these were measured independently for each temperature, pressure, and flow setting. However, we now use each individual reversal to obtain holdup times throughout each experiment. As shown in Figure 1, the flow reversal (prior to b in the figure) elicits a rapid drop of signal to the bath gas baseline over (1 to 2) s. The switch back to the initial flow direction (c) occurs prior to analyte reaching the detector in the reverse direction. The signal initially stays at the noise baseline after the flow returns to its previous direction because of the slug of pure bath gas introduced immediately after the reversal (left side of b and c). During the reversal, not all of the pre-reversal sample in the analytical column is exposed a second time to the diffusion tube outlet, and after reversal completion, the signal rises to a point near the original envelope signal (between c and d). Diffusion in the analytical column during the reversal is small but present, and this is reflected in a narrow half-Gaussian lineshape to the rise back to the original envelope signal. At that point, the concentrated peak region reaches the detector, and the reversal peak appears in the detector signal (e and f).

The holdup time is determined as the difference between the inflection points of the rising edge of the reversal peak and the rising edge of the peak. Within individual runs, the standard deviation of the holdup time is typically (0.2 to 0.3) s and always well under the GC data collection period of 500 ms. Run-to-run variations over several days can differ by up to 1.5 s, likely due to the slight variations in tube pressure due to changes in atmospheric pressure. Even for those data with the lowest uncertainty, variations in the measured diffusion coefficients due to the holdup time variation are minimal compared to other experimental uncertainties.

Alternatively, $\tau$ could have been determined by $t-t_h^{\prime }$, where $t_h^{\prime }$ is the total time the sample is present in the analytical column during a reversal, determined by the difference between the reversal peak location and the rising edge from the bath gas baseline (i.e. disregarding the set reversal time, $t_R$). However, we have chosen to utilize $t_R/2$ and calculate the holdup from the second inflection point to cancel errors. The use of both inflection points accounts for diffusion that occurs during the reversal and flow from diffusion outlet to detector that would be expected to affect the locations of the inflection points similarly.

At low temperatures, where observed diffusion coefficients, namely those at the measurement pressure, are low, diffusion in the analytical tube during the reversal is small and the sample concentration at the diffusion tube outlet changes little over the interval. Under these conditions, the linearization strategy is effective. However, at high temperatures where diffusion is fast and experiments correspondingly shorter, diffusion during the reversal is larger, the sample concentration at the outlet is changing rapidly, and the 12 s reversal is a larger fraction of $\tau$. In these cases, linearization can fail, leading to a constantly varying slope of the linearized spectrum. In these cases, we have chosen to decrease the reversal time to 4 s. Although this necessarily decreases the height of the peak induced by the reversal, the output concentrations are sufficiently high at high temperatures due to the large observed diffusion coefficient that adequate signal-to-noise ratios are achieved. Under these conditions a constant slope was observed in fitting the linearized spectra. The reduced reversal time was used for temperatures 600 K and greater in both ${\text{CH}}_2{\text{F}}_2$ and ${\text{CHF}}_3$. Our previous study¹ evaluated the effects of reversal time on systems that satisfied the linearization strategy criteria and found no effect on the measured diffusion coefficients. We verified this in the present work in experiments with ${\text{CHF}}_3$ at 550 K, conditions where the 12 s reversal spectra were linear, by collecting additional data using a 4 s reversal. We found that the diffusion coefficients derived with the two data sets agreed to within experimental uncertainty. While confirmatory, the results with the reduced reversal time had increased scatter at this temperature due to the smaller signal-to-noise ratio, so those data are not reported.

Results and Discussion

Averaged values for the diffusion coefficients in ${\text{N}}_2$ at 101.3 kPa for all compounds in this study are given in Table 2. The combined expanded uncertainty in the diffusion coefficient, $U(D)$, shown in the table is derived from a random expanded uncertainty (0.95 level of confidence) from run-to-run variations and an estimate of the systematic uncertainty. We believe that the systematic uncertainty is dominated by the uncertainty in our determination of the effective column length (Apparatus section), which corresponds to a systematic expanded relative uncertainty $u_r(D)=0.011$, where $D$ is the derived diffusion coefficient, and we have used this value to derive $U(D)$ in the table. Note that the systematic uncertainty in column length affects all measurements equally and that relative values of the compounds are better defined from the run-to-run variations. In all cases, the linearization analysis was used to determine the diffusion coefficients at the measurement pressure with linear scaling to 101.3 kPa. Unscaled values and conditions for all experiments are reported in the supporting information. As discussed below, measurements at high temperatures for small rapidly-diffusing compounds show nonlinearity in the ln $s$ ${\tau }^{3/2}$ vs $1/\tau$ plots under the conditions used presently and yield inaccurate results when using the linearization analysis. As a consequence, we do not report values obtained at temperatures above 550 K for ${\text{CH}}_4$ and ${\text{CH}}_3\text{F}$, or above 650 K for ${\text{CH}}_2{\text{F}}_2$.

The temperature dependence parameters and uncertainties for each compound are reported in Table 3 and were determined using a power law fitting of all data points of the form $a{T}^n$, as implemented by lmfit-py.¹⁸

Literature

Comparisons of the present results to the literature are presented in Figure 3 and Figure 4. Figure 3 highlights and compares the present work to experimental measurements and reviews in the existing literature. Figure 4 is a deviation plot comparing the present measurements with literature values derived from various estimation procedures and direct calculations.

Figure 3:

Comparison of the present work to available literature. Shapes denote compounds as shown in legend. Present measurements in ${\text{N}}_2$: open symbols with 2 sigma error bars. Literature measurements: symbols as listed below, bath gas, authors, and year are indicated parenthetically. $\mathbf{\text{C}}{\mathbf{\text{H}}}_4$: plus symbols⁹ (air, Coward and Georgeson, 1937); x symbols¹⁹ (${\text{N}}_2$, Wakeham and Slater, 1973); Y symbols²⁰ (${\text{N}}_2$, Engel and Knapp, 1973); fully-filled hexagons²¹ (${\text{N}}_2$, Mueller and Cahill, 1964); bottom-filled hexagon⁵ (${\text{N}}_2$, review, Massman, 1998); left-filled hexagons⁸ (air, Langenberg et al., 2020); right-filled hexagon²² (air, Cowie and Watts, 1971) (obstructed by solid hexagon at 298 K); top-filled hexagons²³ (air, 1961, 1961). $\mathbf{\text{C}}{\mathbf{\text{H}}}_2{\mathbf{\text{F}}}_2$: filled circles¹² (air, Matsunaga et al., 2002). $\mathbf{\text{CHF}}$: filled squares⁴ (fully filled ${\text{N}}_2$ and left-filled air, see text, Matsunaga, 2011). Solid lines show fits of Table 3 to our data; dot-dash⁸ line is for ${\text{CH}}_4$ (air, Langenberg et al., 2020) derived from the “twin tube” methodology; dotted line⁶ is the recommended fit for ${\text{CH}}_4$ (${\text{N}}_2$, Marrero and Mason, 1972 review). The dashed region is shown in more detail in the inset plot.

Figure 4:

Deviation plot of present work to literature estimates and calculations. Open symbols are the present work. Solid lines are the presently recommended temperature dependences with parameters from Table 3. Dashed lines are from Yaws, Ref. 7; dotted lines are from the method of Fuller et al., Ref. 25 using diffusion volumes from Fuller et al., Ref. 26. Dot-long dash lines are derived from Lennard-Jones parameters (see text for references), and the bold widely spaced dashed line that lies near the Lennard-Jones estimates for ${\text{CH}}_4$ is a molecular dynamics simulation from Chae et al., Ref. 11. The dense dot-dashed line for ${\text{CH}}_4-{\text{N}}_2$ is a correlation from Hellmann¹⁰ derived from an ab initio potential energy surface calculated by Hellmann et al.²⁷

Diffusion of ${\text{CH}}_4$ in ${\text{N}}_2$ has been measured directly by Mueller and Cahill for $T=\left(298.0\text{to}398.5\right)\text{K}$,²¹ and values are shown as solid hexagons in Figure 3. The present values are larger across the temperature range but show a very similar temperature dependence. In their review of diffusion coefficients, Marrero and Mason⁶ report a temperature dependence of $\left(1\cdot {10}^{-5}\right)T^{1.75}$ based on the data of Mueller and Cahill²¹ at low temperatures combined with extremely high temperature molecular beam data. Their recommendation is shown in the figure as a dotted line from 298 K and higher. Wakeham and Slater used Taylor dispersion to measure ${\text{CH}}_4-{\text{N}}_2$ diffusion coefficients from (313.7 to 671.3) K and values are shown as x symbols on the figure.¹⁹ Those authors suggested an uncertainty of ±(3 to 4) % for their measurements; the present work deviates (Figure S1 in supporting information) approximately 5 % from those values, but with no obvious bias high or low. Engel and Knapp measured diffusion coefficients of ${\text{CH}}_4$ in ${\text{N}}_2$ from (173 to 273) K and (151 to 594) kPa, and found little deviation from a linear variation with pressure at constant temperature.²⁰ We have reported their diffusion coefficients scaled to 101.3 kPa for comparison to the present data. Although the present work shows limited overlap with those measurements, in general the present work would predict diffusion coefficients 10 % higher than observed by Engel and Knapp. For diffusion of ${\text{CH}}_4$ in ${\text{N}}_2$, Massman⁵ reports $D=0.1892{\text{cm}}^2\cdot {\text{s}}^{-1}$ at $T=273\text{K}$ based on a literature review.

Due to limited measurements in ${\text{N}}_2$, we have elected to additionally compare to values in air, which are more extensive in the literature. We note that Massman’s comprehensive literature review⁵ reports the diffusion coefficient of ${\text{CH}}_4$ in air to be 0.1952 cm² · s⁻¹ at 273 K, ≈ 3 % larger than the value recommended in the same study for diffusion in ${\text{N}}_2$, while finding virtually identical temperature dependences in the two media. Based solely on mass considerations, one might expect the differences to be smaller and in the opposite direction, although recent first principles calculations with ${\text{H}}_2\text{O}$ in ${\text{O}}_2$ and air has shown the opposite effect with a slightly larger diffusion coefficient predicted for diffusion in ${\text{O}}_2$ from (250 to 2000) K.²⁴ We have performed experiments with our apparatus for ${\text{CH}}_4$ in air from (250 to 350) K for comparison to the ${\text{N}}_2$ results. Attempts to determine values in air at higher temperatures were not made in order to avoid high-temperature oxidation of the interior of the diffusion and analytical tubes that might affect future measurements. The results are listed in Table 4. In air, we find diffusion coefficients to vary less than 0.3 % compared to those we observed in ${\text{N}}_2$, well within the reported combined expanded uncertainties. Since these measurements are expected to be subject to the same systematic errors, the differences are better compared to the random expanded uncertainties alone. Again, no difference is observed within these random uncertainties. Based on our previous comparisons with standard reference data,^1,2 the present method should be precise enough to detect differences in the values in air and ${\text{N}}_2$ if they were as large as indicated in the Massman review.

Table 4:

Measured binary diffusion coefficients at 101.3 kPa for dilute methane (${\text{CH}}_4$) in air, where $T$ is temperature, $D$ is the binary diffusion coefficient, and $U(D)$ is the combined expanded uncertainty in $D$ (0.95 confidence level). Differences between the values in air and in ${\text{N}}_2$ were all within the respective $U(D)$ and also within the random expanded uncertainties alone, when considering that both measurements are subject to the same systematic relative expanded uncertainty $u_r(D)=0.011$.

Compound	$T/\text{K}$ a	$D/\left({\text{cm}}^2\cdot {\text{s}}^{-1}\right)$	$U(D)/{\text{cm}}^2\cdot {\text{s}}^{-1}$
${\text{CH}}_4$	250	0.170	0.004
${\text{CH}}_4$	273	0.200	0.005
${\text{CH}}_4$	300	0.237	0.005
${\text{CH}}_4$	350	0.311	0.005

^aCombined relative expanded uncertainty $U_r(D)=0.004$

For methane, the presently reported diffusion coefficients are quite consistent with the measurements of Coward and Georgeson⁹ in air (plus symbols in Figure 3) from (289 to 295) K. Gwertsiteli et al.²³ reported 0.21 cm² · s⁻¹ in air at 101.3 kPa and 293 K, which is about 8 % smaller than our value of 0.227 cm² · s⁻¹ derived from the fit of Table 3 at that temperature. Cowie and Watts²² observed (0.2168 ± 0.0032) cm² · s⁻¹ at 295 K using infrared spectrophotometry, shown as a right-filled hexagon that is obstructed by a solid hexagon from the ${\text{N}}_2$ measurements of Mueller and Cahill²¹ at a similar temperature and is about 5.5 % smaller than the present work. Recent work by Langenberg et al.⁸ examined diffusion of ${\text{CH}}_4$ in air at temperatures from (197 to 294) K using two methods: an “arrested flow” method, where a pulse of gas is introduced into a column and allowed to broaden, and a “twin tube” steady-state method with a capillary diffusion bridge and continuous sampling. Those authors estimated the systematic error of the twin tube method to be ≈40 % better than the arrested flow. The present values are intermediate, but nonetheless agree better with the arrested flow results, as shown in Figure 3 by comparing the open hexagons (present work) to the left-shaded hexagons (arrested flow), and dot-dashed line (twin tube).

For the fluorinated compounds, literature data for diffusion coefficients in any gas is limited, and, to our knowledge, only one measurement for ${\text{CHF}}_3-{\text{N}}_2$ has been reported, that by Matsunaga⁴ (as HFC23). That work reported diffusion coefficients at $T=\left(303\text{to}363\right)\text{K}$ and found $a=1.74\cdot {10}^{-5}$ and $n=1.58$. At the lowest reported temperature, the present work would predict 0.148 cm² · s⁻¹, slightly larger than the Matsunaga value of 0.144 cm² · s⁻¹. He noted no difference in observed diffusion coefficient to 0.001 cm² · s⁻¹ between air, ${\text{N}}_2$, and ${\text{O}}_2$ but reported that measurements at higher temperatures in air and ${\text{N}}_2$ yielded “abnormally small temperature coefficients.” The temperature exponent, n, found in the present work is significantly larger than suggested by the Matsunaga data; as a result, our values increasingly deviate from those reported in that work at the upper studied temperatures, and substantial differences will be obtained if results are extrapolated to e.g. combustion conditions. Since no difference was observed for diffusion coefficients in air, we have also included the measurement of ${\text{CHF}}_3$ in air at 393 K from the same study in Figure 3.

Matsunaga et al. have reported diffusion coefficients at $T=\left(273\text{to}453\right)\text{K}$ of ${\text{CH}}_2{\text{F}}_2$ (as HFC32) in air, which should be comparable as discussed above.¹² At 303 K, those authors observed a diffusion coefficient of 0.157 cm² · s⁻¹, which is slightly smaller than the 0.160 cm² · s⁻¹ value we find in ${\text{N}}_2$. However, their observed temperature dependence, $a=2.72\cdot {10}^{-6}$ and $n=1.92$, differs substantially from ours and leads to diffusion coefficients much larger than ours at higher temperatures. We^1,2 and others^28–31 have found that the temperature coefficient displays a small systematic variation with the bath gas but otherwise shows little dependence on the diffusing compound. The 12 other compounds measured by Matsunaga et al. show temperature exponents of 1.70 to 1.80 that are in line with expectations,²⁹ and it is likely the 1.92 exponent measured for HFC32 is anomalous.

In the absence of experimental data, empirical estimates based on the masses of the analyte and bath gas, partial molar volumes of the respective molecules, and temperature are often used.²⁹ We compare the complete set of present results to various estimates in Figure 4.

The empirical method of Fuller et al.²⁵ (FSG) is popular and utilizes dimensionless atomic “diffusion volumes” for the analyte and bath gases with adjustments for aromatic and heterocyclic ring structure and a fixed temperature exponent:

$$D_{AB}=\frac{0.001T^{1.75}}{{\mu }^{1/2}{\left(v_A^{1/3}+v_B^{1/3}\right)}^2},$$ (1)

where $D_{AB}$ is the binary diffusion coefficient of $A$ in $B$ at 101.3 kPa pressure in cm² · s⁻¹, $T$ is temperature in K, $\mu$ is the reduced molar mass of A and B in g · mol⁻¹, and $v_A$ and $v_B$ are the diffusion volumes of $A$ and $B$.

As is evident in Figure 4 from the dotted lines, the FSG estimates lie well below the measured coefficients across the entire temperature range. Since the temperature exponent in Equation 1 of $n=1.75$ is very similar to the exponents found in the present work (Table 3), we attribute the differences to inaccuracies of the model or the diffusion volumes themselves. In particular, the model does not explicitly treat polarity, which is certainly important in fluorinated compounds and can significantly affect diffusion coefficients.²⁹

Diffusion volumes for the estimates in the figure are taken from Fuller et al.²⁶ (FEG), a follow-up to the original FSG study with updated diffusion volumes with numerous diffusion coefficient measurements of halocarbons in He. It is common^8,32,33 to cite the excellent work of Reid, Prausnitz, and Poling²⁹ (RPP) for these diffusion volumes. However, the values presented in RPP are identical to those in FEG with the lone exception of ${\text{CO}}_2$, which is 26.9 in RPP and 26.7 in FEG. Since RPP makes no mention of a reanalysis, we suspect this is a typographical error.

Gu et al.³³ have evaluated experimental diffusion coefficients of halogenated organic compounds in He at 300 K and compared them to their FSG estimates. In particular, ${\text{CH}}_2{\text{F}}_2$–He was found to have an estimated ($D_e$) to measured ($D_m$) diffusion coefficient ratio of $D_e/D_m=0.95$, which is consistent in direction but much closer in magnitude than the $D_e/D_m=0.80$ reported in this work for ${\text{CH}}_2{\text{F}}_2-{\text{N}}_2$. Although those authors report the data scaled to 298 K, the experimental data were obtained at 430.9 K by Fuller et al.,²⁶ where $D_e/D_m=0.95$. The FSG estimates use a fixed temperature exponent of 1.75 (Equation 1), and that exponent was used to scale the experimental data for ${\text{CH}}_2{\text{F}}_2-\text{He}$ in the evaluation. However, in general, diffusion coefficients of small hydrocarbons in He scale with a lower temperature exponent, 1.65 to 1.67.^{1–3,28–31} Utilizing a $T^{1.66}$ scaling for the Fuller, Ensley, and Giddings result at 298 K yields an $D_e=0.468{\text{cm}}^2\cdot {\text{s}}^{-1}$ and $D_e/D_m=0.92$. Note that the FSG estimate in the evaluation appears to have been calculated at 300 K rather than 298 K, although the experimental data were scaled to 298 K. Since the value for the fluorine diffusion volume was obtained entirely from fluorocarbon measurements in He with a different temperature exponent, it is not surprising the estimates of fluorinated compounds do not scale well outside the corpus.

Yaws⁷ has published diffusion coefficient estimates for all of the compounds in this study in air developed using “literature methods” and, as far as the present authors can ascertain, unpublished “empirical procedures.”³⁴ Although in general such numbers should be regarded with extreme care, the wide use of these values in the literature requires that we present a comparison, which is provided as the dashed lines in Figure 4. In general, these estimates increasingly deviate from experiment near and below ambient temperatures, but reproduce most data well at temperatures above approximately 360 K, although we reiterate Yaws’ caution in extending the estimates to the 1500 K maximum temperature in the absence of experimental data.

We have also included in Figure 4 diffusion coefficients from the literature as derived from Chapman-Enskog theory with Lennard-Jones potential parameters. These diffusion coefficients were calculated as described in Reid et al.²⁹ for an ideal gas and without correction for highly unequal molecular masses. In all cases, the required collision integrals were calculated from the parameters of Neufeld et al..³⁵ In the figure, the long dash-dotted lines represent methane,³⁶ difluoromethane,³⁷ and trifluoromethane.³⁸ The methane estimates overlap at the lowest temperatures and are closest at low temperatures for all three species, but the Lennard-Jones estimates all underestimate the temperature dependence exponent and thus show increasing deviation with increasing temperature. Polar species are well-known to be poorly modeled by single-site Lennard-Jones potentials and should be modeled with a more descriptive potential³⁹ or with additional correction.²⁹ This need is demonstrated by the particularly poor ability of the Lennard-Jones estimates to reproduce the experimental data for ${\text{CH}}_2{\text{F}}_2$ and ${\text{CHF}}_3$.

Bold long dashes in Figure 4 show a 2011 calculation by Chae et al.¹¹ of ${\text{CH}}_4-{\text{N}}_2$ diffusion coefficients using molecular dynamics to probe configuration space with Lennard-Jones interaction potentials. As expected, the molecular dynamics simulations of methane closely match the diffusion coefficients calculated from a simple Chapman-Enskog analysis due to the tetrahedral geometry of the ${\text{CH}}_4$. Those authors calculated diffusion coefficients for numerous longer-chain alkanes and observed that longer-chain alkanes show a larger deviation attributable to their deviations from sphericity.¹¹

Also shown in the figure as a densely packed dot-dash line is a 2020 correlation by Hellmann¹⁰ for ${\text{CH}}_4-{\text{N}}_2$ derived from site-site fits to a detailed ab initio potential energy surface.²⁷ The diffusion coefficients were calculated from the correlation given in the supplementary material of Hellmann¹⁰ assuming an ideal gas at 101.325 kPa pressure and show excellent agreement with the present data across the entire temperature range. We note that the ab initio correlation shows a slight difference from the present extrapolated values at high temperatures, and additional data for ${\text{CH}}_4$ at temperatures greater than reported in this work would be valuable.

Linearization Analysis

Implicit in the derivation of the analytical expression for the linearization strategy is the assumption that diffusion in the analytical column can be neglected, requiring the magnitude of the analytical flow be large enough that diffusion of the sample is small between the diffusion tube exit and the detector.¹⁴ Deviations from this flow regime lead to nonlinearity in the fit, and these effects are most relevant at high temperatures with large $D$.¹

We noted in our previous work that using a 1D Crank-Nicolson (1DCN) simulation of only the diffusion tube provides an excellent fit to reversal data for which the linearization analysis is effective on the timescale of our typical experiments. The excellent reproduction of the data by the simple simulation implies that at low temperatures (i.e. low in situ D) the detector signal reflects a direct map of the diffusion tube outlet concentration scaled by an approximately constant fraction due to the concentration scaling induced by the reversal. For larger in situ D, where the outlet concentration is changing rapidly, this scaling can no longer be considered to be approximately constant during the reversal and holdup times, which manifests as nonlinearity in the linearization analysis plots.

To evaluate the linearization procedure at higher temperatures we have performed a simplex fit of the reversal data with a 1DCN simulation with the diffusion coefficient as the varying parameter and the average measured holdup times. Best-fit results are shown in Figure 5 for ${\text{CH}}_4$ in ${\text{N}}_2$ at 400 K, 550 K, and 700 K at (679.3 ± 0.1) kPa. For the lower temperatures, the numerical simulations very closely reproduce the experimental data; however, the shape of the best-fit curve at 700 K is incorrect, rising and falling too fast. Although the differences appear to be subtle, highly accurate measurements of the diffusion coefficients determined using the present analysis demand better adherence to the 1D model used to describe the diffusion column. The relatively poor reproduction of the shape of the experimental data indicates that the signal at higher temperatures (i.e. large in situ diffusion coefficients) does not represent a straightforward mapping of the concentration of analyte at the outlet of the diffusion column. In these cases, we have elected not to report a diffusion coefficient, which is reflected in the absence of data at higher temperatures for ${\text{CH}}_4$, ${\text{CH}}_3\text{F}$, and ${\text{CH}}_2{\text{F}}_2$.

Figure 5:

Results of fitting data with simplex-minimized one-dimensional Crank-Nicolson simulations for ${\text{CH}}_4/{\text{N}}_2$ at 679.3 kPa

At moderate in situ diffusion coefficients that lie above the 0.1 cm² · s⁻¹ criterion that we have set in this work, curvature in the linearized spectrum is caused by a breakdown of the linearization strategy but can be fitted by a 1DCN model,¹ unlike the high- temperature data in Figure 5. In this regime, fitting of the early data prior to the onset of curvature can provide reasonably accurate results and may provide a means to extend the temperature range for a given diffusion tube length. In these cases, a slight bias toward higher diffusion coefficients may be expected due to the possible effects of the falloff region on the slope of the linearization curve. We have not presently attempted this strategy, however, and rather use a set criterion to limit our analyses to data regimes where curvature is unimportant. Use of a longer diffusion tube would allow measurement of accurate diffusion coefficients at higher temperatures by decreasing the magnitude of the change in the concentration at the diffusion tube outlet and ensuring the linearization strategy assumptions were met. However, use of a longer tube would come at the expense of extremely long run-times at low temperatures, as well as an inability to maintain the tube in a single horizontal plane of the current oven, thus exposing it to the vertical temperature gradient and thereby increasing the uncertainty.

Conclusion

Despite extensive use as refrigerants, data describing the diffusion of fluoromethanes are scant and inconsistent. In this study, we have determined the binary diffusion coefficients for the homologous series of (fluoro)methanes, ${\text{CH}}_4$, ${\text{CH}}_3\text{F}$, ${\text{CH}}_2{\text{F}}_2$, ${\text{CHF}}_3$ in ${\text{N}}_2$ for temperatures (300 to 550) K for all species and extending up to 650 K and 723 K for ${\text{CH}}_2{\text{F}}_2$ and ${\text{CHF}}_3$, respectively, and down to 250 K for ${\text{CH}}_4$. For methane, we have also measured the diffusion coefficient in air at (250 to 350) K, finding no discernible differences with values obtained in ${\text{N}}_2$ over that range of temperatures. The present work provides the first experimental measurements of the diffusion coefficient for ${\text{CH}}_3\text{F}$, as well as an internally consistent data set for methane and fluoromethanes obtained with a single apparatus. Our results broadly reproduce the limited literature data, which mostly pertain to air rather then ${\text{N}}_2$, and are consistent with some published estimates. The present studies, however, encompass a wider temperature range than previous work and extend measurements to higher temperatures. We find that our temperature dependences differ somewhat from most published values but have temperature exponents similar to those we and others have found for small hydrocarbons in ${\text{N}}_2$.