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Anal Chem. Author manuscript; available in PMC 2010 July 15.

Published in final edited form as:

PMCID: PMC2722946

NIHMSID: NIHMS127436

Department of Chemistry, Indiana University 800 Kirkwood Ave. Bloomington, IN 47405

Correspondence can be directed to either author: S. J. Valentine (Email: ude.anaidni@tnelavts) or D. E. Clemmer (Email: ude.anaidni@remmelc).

The ability to separate isotopes by high-resolution ion mobility spectrometry techniques is considered as a direct means for determining mass at ambient pressures. Calculations of peak shapes from the transport equation show that it should be possible to separate isotopes for low mass ions (<200) by utilizing heavy collision gasses and high resolution ion mobility analyzers. The mass accuracy associated with this isotopic separation approach based on ion mobility separation is considered. Finally, we predict several isotopes that should be separable.

Despite interest in ion populations under ambient conditions over the last century,^{1}^{-}^{5} no direct measurements of masses have been made at high pressures. This is because mass spectrometry measurements are based on determination of the motion of ions in a vacuum (usually < 10^{−4} Torr). As ions are extracted from a high-pressure environment into the low pressure regions of mass analyzers the population evolves due to shifts in equilibrium as pressures, temperatures, and species concentrations change. A direct method for measuring the fundamental property of mass at high pressures would find applications in a range of problems, from understanding and monitoring ions in the environment^{6}^{-}^{8} to delineating details of ion formation in a range of new sources that operate in ambient conditions.^{9}^{-}^{13}

Although direct measurements of mass at high pressure are not feasible with conventional mass analyzers, it is possible to record ion mobilities at high pressure. The mobility of an ion through a gas (*K* = *v*/E) is a measure of the ion's velocity (*v*) under the influence of an applied electric field (*E*). Thus, *K* is a fundamental property of the ion in a defined buffer gas. It is generally understood that an ion's mobility is inversely related to its collision cross section (Ω) with the gas as shown in equation 1,^{14}

$$K=\frac{1}{\Omega}\frac{{\left(18\pi \right)}^{1\u22152}}{16}\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}{\left[\frac{1}{{m}_{I}}+\frac{1}{{m}_{B}}\right]}^{1\u22152}\frac{760}{P}\frac{T}{273.2}\frac{1}{N}.$$

(1)

Here, the parameters of temperature (*T*), pressure (*P*), and average buffer gas mass (*m _{B}*) define properties of the buffer gas; the variables

Equation 1 shows that for the ion, there are two fundamental properties that are not known in the measurement of mobility, Ω and m_{I}. Ω depends on the ion's shape, the nature of the ion-buffer gas interaction and to a lesser extent, the masses of the colliding pair. If Ω were defined (e.g., from theoretical calculations),^{15}^{-}^{18} then, *m _{I}* could be determined directly from a measurement of mobility, because equation 1 can be rewritten as:

$${m}_{I}=\frac{{\left[\frac{1}{K}\right]}^{2}{\left[\frac{{\left(18\pi \right)}^{1\u22152}}{16}\right]}^{2}{\left[\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}\right]}^{2}{\left[\frac{760}{P}\frac{T}{273.2}\frac{1}{N}\right]}^{2}{m}_{B}}{{m}_{B}{\Omega}^{2}-{\left[\frac{1}{K}\right]}^{2}{\left[\frac{{\left(18\pi \right)}^{1\u22152}}{16}\right]}^{2}{\left[\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}\right]}^{2}{\left[\frac{760}{P}\frac{T}{273.2}\frac{1}{N}\right]}^{2}}.$$

(2)

An interesting case arises when we consider isotopes. Neglecting small variations in scattering that arise from the differences in mass, different isotopes of the same ion will have identical collision cross sections with the buffer gas; however, the mobilities of isotopes must differ because of differences in *m _{I}* (equation 1). Thus, if the resolving power of ion mobility measurements was sufficient to distinguish these isotopes, then one need not know the cross section

In the present paper, we consider a simple theoretical treatment for the separation of isotopes and propose that this could be used to rigorously determine ion masses under ambient conditions. The result of these considerations is an estimate of the ion mobility resolving power required to accurately measure the masses of different ions. The present work is closely related to efforts to describe the theory of ion mobility separations^{14}^{-}^{19} and is inspired by recent attempts to substantially improve mobility resolving power.^{20}^{-}^{26} This is the first derivation of a mass scale from mobility measurements at high pressure. The results of these considerations suggest that it may already be possible to resolve some isotopes for light ions at high pressures; moreover, as significant improvements in mobility resolving power are made, the present considerations indicate that isotopic structure could become a routine feature of ion mobility data sets, analogous to the patterns observed from mass spectrometry measurements.

The theory of ion motion through gasses has an extensive history,^{1}^{-}^{5}^{,}^{27} dating to the early work of Langevin.^{4} Most of the insight derived here is found from simple manipulations of the ion transport equation.^{14} Discussions of this equation are presented elsewhere.^{14}^{,}^{19}^{,}^{28}^{-}^{30} The derivations described below are based on IMS theory. Therefore, the methodology applies for different types of mobility-based separation devices, provided that the separation is carried out in the low-field limit where ion drift velocities are defined as the product *KE*. Additionally, it should be appropriate for any type of ion source. Figure 1 shows distributions of drift times (*t _{D}*) that are calculated for drifting ions through a buffer gas. Here, we consider two hypothetical ions at ion mobility resolving powers of 25, 75, 225 and 675 [

Ion mobility distributions for two isomers (I_{1} and I_{2}) of a hypothetical ion having a mass of 100 Da. Shown are traces for conditions exhibiting different resolving powers. The insets show a blow up of the drift time region containing the two features **...**

It is worthwhile to comment about these assumed resolving powers. At this point in the development of instrumentation, a value of *R _{IMS}* = 25 is attainable by almost any reasonable ion mobility instrument and would be considered a relatively low resolving power. A value of

Figure 2 shows calculated ion distributions for two isotopes of the isomers that were considered above. To illustrate the separation of different masses having the same cross section, consider M and M+2 isotopes, such that the ion mass is 100 or 102, respectively. A 2.0 Da shift would be typical of a chlorinated molecule and in this example we use the relative isotopic composition for that of ^{35}Cl:^{37}Cl (i.e., 3:1). We have deliberatively chosen an isotopic shift of 2.0 Da, and a light ion to consider first, because the experimental realization of the separation of isotopes should be most easily obtained for this type of ion. Still, as shown in Figure 2, when He is used as the buffer gas we calculate only slight shifts in the positions of peak centers: 50.00 and 50.02 ms for the M and M+2 isotopes, respectively (for isomer 1); and, 51.25 and 51.27 ms for the M and M+2 peaks, respectively (for isomer 2). While the centers of the M and M+2 pairs of peaks shift slightly for each isomer in He, the resolving power of 675 is insufficient to resolve these isotopes. Instead, peaks in the measured drift time distributions in He will appear broadened, compared with what is expected for a single isotope (or, a single peak that is calculated using an isotopically averaged mass).

A more deliberate evaluation of equation 1 indicates that He is a poor choice for the buffer gas because *m _{He}* = 4.0 Da is a value that dominates the reduced mass term. One can see, however, that isotope separation can be favored by use of a heavy collision gas (e.g., Xe with m = 131.29 Da). As shown in Figure 2, when Xe is used as the collision gas, the transport equation predicts that the M and M+2 isotopic peaks are not only apparent but are well resolved at

The calculated distributions presented above suggest that in theory it should be possible to develop a mass scale based on ion mobility separations. To illustrate the separation of different masses because of differences in mobilities, we considered the M and M+2 isotope pair, such as would be found in ions containing a chlorine atom. In order to develop the idea that it may be possible to develop a general scale, we proceed by considering pairs of M and M+1 isotopes, i.e., ions that are separated by 1.0 Da as in the case of the ^{12}C:^{13}C isotopic pairs. Because the collision cross sections of the isotopic peaks are equal and because the relationship of the masses is known, by rearranging equation 1 and substituting the drift times (*t _{D1}* for the M ion and

$$\frac{{\mathit{Et}}_{D1}}{L}\frac{{\left(18\pi \right)}^{1\u22152}}{16}\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}{\left[\frac{1}{{m}_{I}}+\frac{1}{{m}_{B}}\right]}^{1\u22152}\frac{760}{P}\frac{T}{273.2}\frac{1}{N}=\frac{{\mathit{Et}}_{D2}}{L}\frac{{\left(18\pi \right)}^{1\u22152}}{16}\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}{\left[\frac{1}{{m}_{I+1}}+\frac{1}{{m}_{B}}\right]}^{1\u22152}\frac{760}{P}\frac{T}{273.2}\frac{1}{N}.$$

(3)

Here, the left and right side of the equation corresponds to *Ω _{1}* and

$$\frac{{{t}_{D2}}^{2}}{{{t}_{D1}}^{2}}\approx \frac{\frac{1}{{m}_{I}}+\frac{1}{{m}_{B}}}{\frac{1}{{m}_{I}+1}+\frac{1}{{m}_{B}}}.$$

(4)

Solving equation 4 for *m _{I}* yields:

$${m}_{I}\approx \frac{-{m}_{B}{{t}_{D1}}^{2}-{{t}_{D1}}^{2}+{m}_{B}{{t}_{D2}}^{2}+{{t}_{D2}}^{2}\pm \sqrt{{\left({m}_{B}{{t}_{D1}}^{2}+{{t}_{D1}}^{2}-{m}_{B}{{t}_{D2}}^{2}-{{t}_{D2}}^{2}\right)}^{2}-4{m}_{B}{{t}_{D1}}^{2}({{t}_{D1}}^{2}-{{t}_{D2}}^{2})}}{2\left({{t}_{D1}}^{2}-{{t}_{D2}}^{2}\right)}.$$

(5)

Thus, the difference in the drift times of the ion pairs are all that is required to determine *m _{I}*. This quadratic form of this equation yields two solutions. However, because the denominator is negative, the only meaningful solution is the one for which the numerator is negative as well. For example, if values of

Equation 2 can be rewritten such that *K* is expressed in terms of *E*, the drift tube length (*L*), and *t _{D}*. This provides the means to obtain an expression that relates the mass resolving power (

$$\frac{\partial {m}_{I}}{\partial {t}_{D}}=\frac{2{t}_{D}{\Omega}^{2}{\left({m}_{B}\right)}^{2}{\left[\frac{{\left(18\pi \right)}^{1\u22152}}{16}\right]}^{2}{\left[\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}\right]}^{2}{\left[\frac{760}{\mathit{LP}}\frac{\mathit{ET}}{273.2}\frac{1}{N}\right]}^{2}}{{\left[{m}_{B}{\Omega}^{2}-{{t}_{D}}^{2}{\left[\frac{{\left(18\pi \right)}^{1\u22152}}{16}\right]}^{2}{\left[\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}\right]}^{2}{\left[\frac{760}{\mathit{LP}}\frac{\mathit{ET}}{273.2}\frac{1}{N}\right]}^{2}\right]}^{2}}$$

(6)

Equation 6 can be rewritten in terms of *m _{I}* by multiplying both sides by

$${R}_{\mathit{mass}}=\frac{{m}_{B}{\Omega}^{2}-{{t}_{D}}^{2}{\left[\frac{{\left(18\pi \right)}^{1\u22152}}{16}\right]}^{2}{\left[\frac{\mathit{ze}}{{\left({k}_{b}T\right)}^{1\u22152}}\right]}^{2}{\left[\frac{760}{\mathit{LP}}\frac{\mathit{ET}}{273.2}\frac{1}{N}\right]}^{2}}{2{m}_{B}{\Omega}^{2}}{R}_{\mathit{IMS}}.$$

(7)

From this it can be seen that the ability to resolve different masses is directly proportional to *R _{IMS}*.

Additionally, equation 7 makes it possible to estimate values of *R _{IMS}* that would be required to determine the masses of different ions. For example, if it is desirable to resolve isotopic peaks differing by 1.0 Da, then

Finally, it is important to consider the mass accuracy associated with the measurements that are described above. In order to illustrate how we arrive at the mass accuracy of this measurement, we consider an extremely high value for the resolving power, *R _{IMS}* = 10

$${\left[\epsilon \left({m}_{I}\right)\right]}^{2}={\left[\frac{{\mathit{dm}}_{I}}{{\mathit{dt}}_{D1}}\right]}^{2}{\left[\epsilon \left({t}_{D1}\right)\right]}^{2}+{\left[\frac{{\mathit{dm}}_{I}}{{\mathit{dt}}_{D2}}\right]}^{2}{\left[\epsilon \left({t}_{D2}\right)\right]}^{2},$$

(8)

where *ε*(*m _{I}*),

Ion mobility distributions for isotopic peaks (M and M+1) for a hypothetical molecule. The dashed line corresponds to the isotopic peak of mass 100 Da and the solid line corresponds to the isotopic peak of mass 101 Da. The peak width (FWHM) of the first **...**

This approach can also be used to determine the dependence of mass accuracy on *R _{IMS}* and

A plot of the mass accuracy of IMS mass determinations as a function of *R*_{IMS} values. Traces are shown for *m*_{I} values of 200 (solid triangles), 100 (solid circles), and 50 (solid squares) Da. The dashed line shows the mass accuracy determined for *R*_{IMS} = **...**

The above treatment is only applicable to isotopically related ions. The approach becomes more complicated for large mixtures of ions where drift times for different species may overlap. However, we note that the same problem is encountered in MS analysis. Finally this treatment does not consider any technical boundaries associated with the IMS measurement^{35}^{,}^{36} and assumes that high-resolution measurements will be possible.

A simple theory for directly determining the masses of ions from ion mobility measurements is described. This theory predicts the resolution of isotopes at high pressures. The first set of isotopes that should be measurable are light ions with large isotopic spacings. For example, the M and M+2 ions of Cl^{+} and other small molecules containing Cl should be resolvable based on mobility differences that are associated only with mass differences. The theory predicts that many peaks associated with isomer separation may appear to be broader than expected at relatively high IMS resolutions because isotopes are not quite separated fully. This is not completely unlike what has been observed in the development of mass spectrometers when the resolving power was on the cusp of distinguishing isotopes. It is important to know of this phenomenon as the presence of broader peaks than would be expected at the operating resolving power may be a signature of a transition to much higher resolving power instruments. We also note that tremendous advances have been made in the resolving power of these technologies, with a number of groups now recording high-resolution spectra.^{20}^{-}^{25} Additionally, recent developments of circular drift tubes,^{25} and a technique called overtone mobility spectrometry^{33}^{,}^{34} open new avenues for high-resolution measurements. The ability to directly measure mass at high pressures should make it possible to directly assess ions under ambient pressure conditions, something that is currently impossible with all existing mass analyzers that utilize low pressures.

Finally, we have not discussed any specific application for this technology in part because no mass measurments have actually been carried out at high pressures. Thus from a fundamental point of view a first question would be what are the masses of ions under ambient conditions? It is interesting, for example, to note that ions are relatively abundant in air and estimated concentrations for negative ions range from 2,000 to as high as 100,000 ions per cubic centimeter depending on environmental conditions.^{37}

We gratefully acknowledge partial support of this work from grants that support instrumentation development. These include grants from the NIH (P41-RR018942 and AG-024547-04) and funds from the Indiana University METAcyte initiative that is funded by a grant from the Lilly Endowment.

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