GAS CHROMATOGRAPHIC RETENTION INDICES

Thomas J. Bruno

The interpretation of results from chromatographic measurements can often be augmented with an appropriate mathematical treatment of the solute retention that is observed.  The goal of the treatment is to make the resulting metric as independent of the instrument as possible.  A typical situation that arises from analysis by gas chromatography with mass spectrometry is that the library search routine produces “hits” that are ambiguous if not nonsensical (Ref. 1).  The correct interpretation of the mass spectrum must then be done manually (Refs. 2, 3), and the mass spectral data should be augmented by additional analytical techniques.  The specific techniques that should be used must be determined on a case-by-case basis by a qualified person.  One additional datum that is typically already present in gas chromatography with mass spectrometry detection is chromatographic retention.  The raw datum from a chromatographic measurement is a retention time, t_r, of each eluted peak, and a corresponding intensity.  Here, we will not treat other aspects of the output, such as the width and shape of the chromatographic signal.  The retention time (for a given stationary phase) is dependent on the column temperature, column pressure, column geometry (length and inside diameter; phase ratio), and ambient (atmospheric) pressure.

If the volumetric carrier gas flow rate (at the column exit) is measured and multiplied by the retention time, the retention volume, V_R, is obtained.  The adjusted retention volume, V_R′, is the retention volume corrected for the void volume (or mobile phase holdup) of the column.  It is obtained by simply subtracting the retention volume of an unretained solute (V_M):

V_R′ = V_R − V_M    (1)

While it is possible to calculate the corresponding adjusted retention time, t_r′, by subtracting the retention time of the unretained peak, t_m, it is better to work with volumes because average flow-rate variations between individual analyses are then accounted for.  Note that with each level of refinement beyond the raw retention time, a facet of instrument dependence from the resulting parameter is removed.  V_R is independent of the flow rate; V_R′ is, further, independent of column geometry.  Continuing with this approach, the net retention volume, V_N, is defined, by applying a factor, j, to account for the pressure drop across the column:

V_N = jV_R′    (2)

where j is usually the Martin-James compressibility factor³:

$j=\frac{\text{3}}{\text{2}}\left[\frac{{{\displaystyle \left(\frac{{{\displaystyle P}}_i}{{{\displaystyle P}}_o}\right)}}^{\text{2}}-\text{1}}{{{\displaystyle \left(\frac{{{\displaystyle P}}_i}{{{\displaystyle P}}_o}\right)}}^{\text{3}}\text{-1}}\right]$     (3)

In Eq. 3, P_i is the inlet pressure (absolute) and P_o is the outlet pressure (usually atmospheric pressure).  The net retention volume is important because it is independent of the inlet and outlet pressures, as well as being independent of the carrier gas flow rate and column geometry.

The specific retention volume, V_g, corrects the net retention volume for the amount of stationary phase actually on the column, and the column temperature is adjusted or corrected to 0 °C:

${{\displaystyle V}}_{\text{g}}=\text{ (273}\text{.15)}\frac{{{\displaystyle V}}_{\text{N}}}{\text{(}{{\displaystyle W}}_{\text{s}}{{\displaystyle T}}_{\text{col}}\text{)}},$     (4)

where T_col is the column temperature (in K), and W_s is the mass of the stationary phase in the column.  The V_g value is a characteristic for a particular solute on a particular stationary phase and is instrument independent.  This is a quantity that may be compared from instrument to instrument and laboratory to laboratory with a high level of confidence provided the stationary phase used is a single, pure compound or a well characterized mixture.  If the mass of the stationary phase is not known, or is not meaningful, one may use the net retention volume directly, or one may correct the net retention volume to a column temperature of 0 °C (represented by V_N⁰) by simply not including the term for W_s (that is, setting it equal to unity).

It is also extremely valuable to calculate a relative retention, r_a/b:

${{\displaystyle r}}_{\text{a/b}}=\left(\frac{{{\displaystyle V}}_{\text{g}}^{\text{b}}}{{{\displaystyle V}}_{\text{g}}^{\text{a}}}\right)=\left(\frac{{{\displaystyle V}}_{\text{N}}^{\text{b}}}{{{\displaystyle V}}_{\text{N}}^{\text{a}}}\right)$     (5)

where the numerical superscripts refer to the retention volumes of solutes “a” and “b.”  In this case, solute “a” is a reference compound.  The relative retention is dependent only on the column temperature and the type of stationary phase.  For reasons of operational simplicity, this parameter is usually one of the best to use for qualitative analysis.  It can account for small differences in the column temperature, stationary phase considerations, column history, and minor disturbances in the carrier gas flow rate.  It is possible to account for the column temperature by plotting the logarithm of the retention parameters against 1/T, where T is the thermodynamic temperature.  The column pressure is accounted for by variations in the volume measurement; therefore, there is no pressure dependence to these parameters.

We can go beyond the simple retention parameters discussed earlier to incorporate a logarithmic interpolation on a uniform scale by use of the Kovats retention index (Ref. 4).  The isothermal Kovats retention index is calculated by use of the following defining equation:

$I_{\text{sample}}\left(T\right)=100\left[\frac{\log X_{\text{S}}-\log X_{\text{L}}}{\log X_{\text{H}}-\log X_{\text{L}}}+n_{\text{L}}\right]$     (6)

Here, I_sample is the dimensionless Kovats retention index that is a function of both temperature and the stationary phase employed.  The terms represented by X are retention parameters of the sample and standards.  Following this convention, X_S is the retention parameter of the sample under consideration.  Any retention parameter, such as the adjusted retention time, t′, the net retention volume, V_N, the adjusted net retention volume, V_N⁰, and the relative retentions, r_a/b, can be used.  X_S is the retention parameter of the sample under consideration, X_L is the retention parameter of a normal alkane (that is, straight chain or unbranched) of carbon number n_L that elutes earlier than the sample, and X_H is the retention parameter of a normal alkane having a carbon number greater than n_L+1 that elutes after the sample. The retention index of a sample is, therefore, 100 multiplied by the carbon number of a hypothetical normal alkane that shows the same retention parameter on the stationary phase at that temperature.  Thus, a sample that has a retention index of 785, for example, would co-elute with a hypothetical normal alkane that has 7.85 carbon atoms.  By definition, the retention indices of the normal alkanes (on any stationary phase) are equal to 100 multiplied by the carbon number.  Thus, for n-hexane, I = 600, and similarly for the other n-alkanes in this homologous series.  The zero point in the scale is defined for hydrogen, for which I = 0.  Kovats retention indices have been determined for selected compounds on the most common stationary phases (Refs. 5-7).

The temperature dependence of I for a given sample is known to follow a hyperbolic form similar to the familiar Antoine equation used to represent vapor pressure:

$I_{\text{sample}}\left(T\right)=A+\frac{B}{\left(T+C\right)}$     (7)

In this equation, A, B, and C are empirically determined constants, and T is the temperature (in °C).  A nonlinear fitting routine should be used to determine these constants.  When retention indices are available for at least three temperatures, initial values for A, B, and C can be determined by use of the following equations:

$C=\frac{\left(T_2-T_1\right)\left(I_3{T}_3-I_1{T}_1\right)+\left(T_3-T_1\right)\left(I_1{T}_1-I_2{T}_2\right)}{\left(T_3-T_1\right)\left(I_2-I_1\right)-\left(T_2-T_1\right)\left(I_3-I_1\right)}$     (8)

$A=\frac{I_2{T}_2-I_{1}T{}_1+C\left(I_2-I_1\right)}{T_2-T_1}$     (9)

B = (I₂ − A)(T₂ + C)   (10)

In these equations, I₁, I₂, and I₃ are retention indices of the sample measured at temperatures T₁, T₂, and T₃. When additional retention indices are available at other temperatures, we advocate the use of minimum deviation estimates from these three equations to furnish the starting values for the nonlinear fit. When retention indices at four temperatures are available, the best starting values are obtained from the I₁, I₂, and I₃ triplet that minimizes the deviation with the experimental value with that produced by Eq. 7. This approach provides the fastest convergence, and also helps avoid converging to local minima.  Predictions made by use of Eq. 7 can be used for retention indices within the measured temperature range as well as extrapolation somewhat beyond that range on a case-by-case basis.

It is also of value to report and use the temperature dependence as a slope coefficient, δI_sample/10 °C, the variation of I_sample for a particular stationary phase over a particular temperature range.  While not as reliable as the Antoine-type fit, this coefficient is useful for predictions within the range of the measured results.

In all of the above discussion, one must understand that the column temperature is fixed.  The Kovats retention indices can be made applicable to temperature programmed analysis by use of

$I=100\times \left[n+(N-n)\frac{t_{\text{r(unknown)}}-t_{\text{r(n)}}}{t_{\text{r(N)}}-t_{\text{r(n)}}}\right]$     (11)

where I is the retention index, n is the number of carbon atoms in the smaller n-alkane, N is the number of carbon atoms in the larger n-alkane, and t_r is the retention time of the indicated peak.

A useful alternative to the Kovats system is the Lee retention index (isothermal and temperature dependent), based on the polynuclear aromatic hydrocarbon (PAH) standard compounds: naphthalene (I = 200), phenanthrene (I = 300), chrysene (I = 400), and picene or benzo(g,h,i)perylene (I = 500).  Isothermal (and temperature-dependent) Kovats and Lee retention indices for many compounds are tabulated in the NIST Chemistry Web Book (Ref. 8).

References

1. NIST/EPA/NIH (NIST 11) Mass Spectral Database, NIST Standard Reference Database 1A, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg, MD, 20899, 2011.
2. Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Fundamental Spectroscopic Correlation Charts, CRC Press, Boca Raton, FL, 2006. [https://doi.org/10.1201/9781420037685]
3. Bruno, T. J., and Svoronos, P. D. N., CRC Handbook of Basic Tables for Chemical Analysis – Data-Driven Methods and Interpretation, Fourth Edition, CRC Press/Taylor & Francis, Boca Raton, FL, 2021. [https://doi.org/10.1201/b22281]
4. Kovats, E., Helv. Chim. Acta 41, 1915, 1958. [https://doi.org/10.1002/hlca.19580410703]
5. Lubek, A. J., and Sutton, D. L., Chromatogr. and Chromatogr. Commun. 6, 328, 1983. [https://doi.org/10.1002/jhrc.1240060612]
6. Bruno, T. J., Wertz, K. H., and Cacairi, M., Anal. Chem. 68, 1347, 1996. [https://doi.org/10.1021/ac9510191]
7. Miller, K. E., and Bruno, T. J., J. Chromatogr. A 1007, 117, 2003. [https://doi.org/10.1016/S0021-9673(03)00958-0]
8. NIST Chemistry Web Book, NIST Standard Reference Database Number 69, Standard Reference Data Program, National Institute of Standards and Technology, Gaithersburg, MD 20899, 2012

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