J. Calvin Giddings University of Utah Salt Lake City, utah
Elementary Theory of Programed Temperature Gas Chromatography
In the short hut torrid history of gas chromatography, numerous procedures have evolved which have served to extend this powerful technique beyond its original objectives. Large scale columns to prepare adequate samples, special solvents for inorganic analysis, capillary columns for high theoretical plate requirements, and columns in series for structurally diversified molecules are among the examples of procedures evolved and the particular requirements which made them necessary. Perhaps the most widespread of all these variations is the technique of programed temperature gas chromatography (PTGC). While several years ago only a few instrument companies were marketing a PTGC apparatus, today nearly every major manufacture of gas chromatographic equipment produces such an instrument. Its use has been described more frequently in the literature over the past few years as it has been applied to analytical problems which have no other convenient solution. PTGC differs from the usual isothermal technique in that the temperature, T, is methodically increased as the chromatographic run is made. The temperature rise is usually linear in time, t , i.e., where To is the beginning temperature and P is the heating rate in "C per unit time, 1. The iustmment setting for B can commonly be made anywhere from a fraction of a degree to 40 or 50 degrees per minute. Needless to say, the intermediate values are most commonly useful. The initial temperature of a program may be as low as or lower than room temperature. From there the temperature may he increased up to from 300°C to 500°C, depending on the instrument. The value of PTGC lies in its ability to separate mixtures with a wide boiling point range. This is nearly impossible in the usual isothermal technique since any temperature chosen will be either too high or too low for most of the components. Each closely spaced pair of components which requires separation has a rather narrow temperature range within which an isothermal separation is both efficient and reasonably fast. This temperature is found as a compromise between separation efficiency, best obtained at low temperatures, and speed and convenience, a characteristic of high temperatures. The temperature generally chosen as giving the best compromise is usually in the neighborhood of the boiling point. In the programed temperature analysis of a wide boiling mixture one obtains, in succession, a satisfactory temperature range for the separation of each narrow hoiling point fractiou. If care is taken in selecting the heating rate, each group of substances automatically selects its own ideal temperature at which to migrate and separate withm the
column. Prior to reaching the workable temperature range, each substance is essentially dormant, or frozen, a t the head of the column, waiting its turn while the lower boiling compounds are in the process of achieving separation. PTGC obtains its success, then, by automatically selecting an appropriate temperature range for each boiling point group. There are no strange or unusual effects originating in the temperature change itself. "The heating process may he considered merely as a mechanism for obtaining a range of temperatures in proper sequence with no direct effect upon the separability obtainable" (1). Each component peak emerging from a programed column reflects, therefore, the average characteristics of its non-uniform temperature history. Despite the relative simplicity of the working concepts of PTGC, the theory of this technique always leads to rather formidable integrals whose solutions are not easily ohtained (2-8). The approximate theory presented here avoids these mathematical difficulties. Although quantitative results are obtained, the most important role of this theory may be that of showing in simple terms how a solute peak migrates down the chromatographic column. This fundamental process is, unfortunately, obscured by the mathematics of the rigorous theories. Temperature Effects in Gas Chromatography
Because of the crucial role played by temperature in PTGC, it is essential to have a clear understanding of how temperature influences peak migration and resolution. The most important result of a temperature increase is that it forces a larger fraction of the solute into the vapor state. (Other less important results are changes in diffusion coefficients, viscosity, etc.) Of all the compromises in gas chromatography, the balance between solute vapor and solute in solution is the most significant. Peak migration depends entirely upon the solute vapor (since this is the part swept along by the carrier gas), while peak separability depends entirely upon solute in the liquid solution (where selective molecular forces are at work). In order to characterize the balance between the two, we will define m as the fraction of a given solute found in the vapor state. The fraction (1 - a) of the total solute will then be found in the stationary phase. At very low m values (and thus low temperatures), there is so little solute in the form of moving vapor that the solute peak nearly ceases to migrate. In addition to the very long waiting times required for the emergeuce of the compound, the peak under these circumstances is dilute and difficultto detect. At very high m values (near unity), occurring a t high temperatures, essentially all of the solute is "boiled" out of the liquid phase. Volume 39, Number 1 1 , November 1962
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Since interaction with this phase is responsible for separation, very little separation occurs. Thus the temperature for a given solute or several similar solutes should be adjusted so that the values are neither too high nor too low. The best point of compromise is determined after studying some of the quantitative relationships developed in the followillg paragraphs. First of all, an expression for (R must be obtained. If we take a small volume element of the column which contains the desired solute, the number of moles of solute found in the gas phase is the vapor concentration, c., times the volume of the gas contained within this element, Vg. The moles of solute in the liquid phase is similarly c, V,. The fraction of total solute found in the form of vapor is, then, the number of vapor moles divided by the number of total moles vapor plus liquid moles) i.e.,
The ratio (c /%) is, assuming equilibrium, simply the partition coefficient,K, an equilibrium constant relating a component's concentration in different phases (9, 10). It is K which is responsible for the rather rapid increase of m with temperature. Like all other equilibrium constants, K changes exponentially with absolute temperature in the following approximate way where AH, is the heat of vaporization of the solute in the stationary phase and g is a constant related to the entropy of vaporization. The gas constant is R. For reasons to be discussed shortly, there is nearly always more solute in the liquid than in the gas phase (i.e., fR is much less than unity) and it is thus usually a good approximation to ignore V. in the denominator of equation (3). Consequently
where a is essentially constant with respect to temperature changes. This expression shows directly the strong dependence of on absolute temperature, T. (The increase with T shown in this expression is of the same form as the increase of vapor pressure with temperature. This is no coincidence since it is vaporliquid equilibrium which concerns us. The well known, rapid increase of vapor pressure with temperature is, therefore, matched by a similar rapid increase in m with T.) I t is important for the developments of the next section to determine the average increase in temperature needed to just double the W value. If cR is doubled by increasing the temperature from TI to TI, then from equation (5) the ratio of (R values is
which gives AT
=
,693 R T Y A H ,
(7)
where T is the geometric mean of the two temperatures and AT = Tz - TI. Although the chromatographic 570
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Journal o f Chemical Education
process involves vaporization from a solution rather than from pure solute, Trouton's rule is still approximately valid, and AH,/Tb L? 23. Since the process is operated near the solute boiling point, T Tb, the ratio AH/T in equations (6) and (7) may also be approximated by 23. Substituting (R = 1.99 cal deg-' mole-' and the typical operating temperature T = 500°K (227"C), we find the temperature increase that typically doubles the 61 value to be 30 degrees. This value will shift up and down somewhat (usually within the range from 2040 degrees) depending on the operating temperature and its departure from the boiling point, but as a rough rule 30 degrees may be regarded as adequate to double a. Due to the approximation made in obtaining equation (5), this rule applies only when is below or so. As a second matter in establishing a background for PTGC, it is necessary to reflect on the significance of the W value. It may be recalled that for a given solute was defined as the fraction of that solute found in the vapor state. Now obviously the more solute one finds in the vapor state, the faster the peak migrates. The actual velocity of peak migration is a weighted average of the downstream velocity of the vapor (this is the same as the velocity, v, of the carrier gas) and the zero velocity of solute in the stationary liquid phase. Thus, if m = '/a, this means that '/8 of the solute molecules exist as vapor and possess a mean velocity v, and of the solute molecules are stationary within the liquid. The average velocity of the solute aggregate is obviously v/3. In general the velocity of the zone is, then, simply mu, and 6 i may be considered as the migration rate relative to the inert carrier gas. The time required for a chromatographic run is inversely proportional to this velocity, and a rapid analysis cannot there be made if 6i is too low. Gas chromatography is fortunately so rapid anyway that cR values as low as 0.01 can often be used without great inconvenience. Thus in striking a balance between the disadvantage of extremely high and extremely low CR values, the lower limit is rather flexible. This is not true of the higher limit, as we shall see, and m values thus tend to be moderately low, often in the vicinity of 0.1. The upper limit on m values exists, as mentioned before, because of the resolution loss suffered when solute is mostly in the form of vapor. This matter was fimt emphasized by Purnell (11). The detailed theory (11,12) will not be dealt with here. Suffice it to say that the increase in the number of theoretical plates, N, or in the length, L, necessary to offset this loss is indicated by the relationships N = No/(l W)2 or L = Lo/(l - W)*, where No and Lo are the number of plates and column length, respectively, necessary to effect the separation a t low CR values (12). Thus the number of theoretical plates or the column length must be increased in the ratio 1:1.2: 1.8:4: m as C? increases through the series 0, 0.1, 0.25, 0.5 and 1. If resolution is barely marginal a t low m values (low temperatures), then a t high values and temperatures, m = is detrimental and m = is a ceiling above which resolution deteriorates with iutolerable rapidity. The same resolution loss might clearly be expected to occur in PTGC if the temperature were increased too rapidly.
Peak Migration in PTGC
While exact (but complicated) equations are available for the description of peak migration in PTGC, a simple scheme, using the results of the last section, best demonstrates the nature of the migration phenomenon. Let us make the assumption that the migration velocity of the solute zone in the column has a step-like increase, i.e., that it remains constant throughout a 30 degree interval and then jumps suddenly to a value twice as large which it maintains for the next 30 degree interval. This stepfunction increase will be used to approximate the continuous velocity increase found in practice as shown in Figure 1. This diagram applies to a program strated a t 85°C and in which peak elution occurs a t 265°C. The doubling of peak migration velocity every 30 degrees means that the velocity just a t elution is 64 times as large as when the program first starts. This shows that the hulk of peak movement occurs near the latter part of the program. In fact, with such a low velocity at the beginning, we may envision the solute zone as being essentially frozen a t the origin with little displacement occurring until the temperature is a t least within a hundred degrees of elution. This so-called "initial freezing" condition greatly simplifies the quantitative theory of PTGC (1). The quantitative aspects of peak migration in PTGC can easily be formulated using the step-function approximation. Suppose that in the final 30 degrees interval (the 30 degrees just preceding elution) the solute zone migrates some distance, 1, through the column. In the next to last 30' interval the distance covered is half this, 1/2, because the migration velocity is only half as great (as long as the program is linear, the same time interval is involved for each of these successive displacements, and the distance covered is therefore proportional to the peak velocity). In the
I/
1 ACTUAL INCREASE
," t 0.10 rn z
i
,
STEP FUNCTION APPROYIIMTION
C
?i
0 I w
2.
t
-
_I
+ + + + + + +
-
.
z W
005-
& ~m
ixl
160 190 TEMPERATURE. .C
+
+
+
.
Fiaure 2. The distance miorated bv m ~ e a kin mccesive 30 deoree temperature increments.
mi =
0
+
peak velocity, R,v. Thus the interval migration time = L / 2 m t ~= t0/2Rt,where tO = L/v is the passage time of inert (air) peaks through the column. This time must equal the time needed to increase the temperature by 30 degrees since the intervals are defined in terms of 30 degree units of change. This time is equal to the 30 degrees divided by the programed rate of increase of temperature with respect to time, p, i.e., interval migration time = 30°/p. Equating the two times we find that the mean (and presumably uniform) values of the last segment, mi, is
-
0
30" interval previous to this, the distance is again onehalf, or 1/4, and so on down through each l&er and lower 30 degree temperature interval. The total distance migrated is the sum of the migration distance occurring in each interval, 1 1/2 1/4 1/8. . . = . . .). The series, 1 1 (1 '/2 '14 . . ., approaches 2 as a limit as can he '/4 seen by simply adding up a few of the terms. (" ince we wish to count only those intervals going back to the starting temperature, the series is not infinite and the sum is slightly, but usually not much, less than 2. For the example given in Figure 1 the sum is 1.97.) Thus the total distance moved by the zone from beginning to end is approximately 21. This distance must clearly equal the column length, L . Hence we find that in the last 30 degree interval the distance migrated, 1, is one half the column length, L / 2 . In the next to last interval the migration distance of a peak is half of that remaining, or L / 4 , and so on. These facts are illustrated in Figure 2 where a full length column is divided into the distances traveled by the peak in each successive 30" interval. This demonstrates once again that nearly all migration occurs a t the higher temperatures, near the elution temperature T,, and for a long initial period the peaks are essentially frozen at the origin. Although the foregoing study is most useful in showing the nature of peak migration and in choosing optimum conditions for separation, it can he used to make a rough estimate of the retention temperature or time. This is done as follows. Consider the final (and largest) segment of the chromatographic column shown in Figure 2. The peak presumably migrates through this segment with a uniform relative migration rate, &, characteristic of the mean temperature of the interval. The time required to migrate through this interval is the interval length, L/2, divided by the
zo
approximotion I., the ,.te Figure 1. The cso function of temperature.
250
of
265
migral~on
(8)
/3@/60
We know that elution itself occurs right a t the end of the final segment rather than in the middle where mean values are applicable. We also know that a doubles as it passes from the beginning to the end of the 8%meut. If fR, denotes the cR values right a t elution, Volume 39, Number 11, November 1962
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571
then R,/2 would characterize the beginning of the segment, and the "mean" value would be a, = (a, R,/2)/2 = (a/4)(R,. Using this and equation 4 we find the terminal value as
+
This is very close to the value (Bt0/50) deduced as an approximation from a more exacting theory (1). In order to calculate the actual temperature, T,, a t which elution occurs, recall that the relationship between CR and T is given by equation (5). At the point of elution T = T, and = a,, and therefore . a,= u ~ - A H v / R T ~ (10) When this is equated with (9) and T, solved for
This equation will ordinarily yield the absolute retention temperature within 5-10% accuracy. As mentioned previously, however, the important role of the foregoing theory is in establishing a physical picture of the processes occurring in PTGC rather than in obtaining quantitative results. The Significant Temperature
The migration of a peak in PTGC occurs over a wide range of temperatures (usually several hundred degrees), and its final properties, especially peak spreading (plate height), reflect to a certain degree the influence of the entire temperature range. Since most of the migration occurs a t the higher temperatures, however, we would expect the higher temperatures to have a greater influence on the peak's characteristics than the lower temperatures of the program. Thus if there were one single temperature which somehow characterized the entire temperature range (i.e., a temperature which, when used isothermally, would lead to the same amount of peak spreading and the same degree of separability as obtained with the programed run), this temperature would undoubtedly be much closer to the elution temperature than to the starting temperature. The detailed theory shows that a temperature does indeed exist which can be used to characterize the programed operation (1). This temperature is called the significant temperature, T'. Its value is about 40 degrees below the elution temperature. If we accept the thesis that the influence of a given temperature range is proportional to the distance migrated by the peak when within that temperature range, the theory of the last section can be used to establish a weighted average temperature which should closely correspond to the significant temperature, T'. The weighted average involves the following terms. I n the last migration interval of figure 2 the "mean" temperature is T, - 15', and this is weighted (multiplied) by the fraction of the column's length taken up by the internal, i.e., I/*. Thus the first term is (T, - 15") The preceding interval involves a mean temperature of (T, - 45") and a migration distance equal to of the column's length, giving the term (T, - 45') ('/4). The summation of all such terms is T' = (T, - 15')1/2 + (T,- 45")1/4 + =
T, - 15'
- 30°(1/4 +
(T, - 75")1/8 2/8 3/16 4/32
+
572 / lournol of Chemical Education
+
+ ... (12) + .. . )
The series in parenthesis approaches a value of unity, and the significant temperature is thus approximately equal to T' = T. - 45O (13) i.e., the significant temperature is approximately 45 degrees below the elution temperature. The practical use of the significant temperature concept will be briefly explained in the next section. The discussion of the last two sections has been based on the assumption that peak migration velocity is doubled by increasing the temperature 30 degrees. As mentioned, the AT value may differ up to 10 degrees from this figure. This dierence will reflect itself in some of the foregoing equations. Thus if AT does not equal 30 degrees, equations (9), ( l l ) , and (13) are remedied by replacing the number 45 with 3 AT/2. This result can be readily derived by using AT in place of 30 degrees in the derivations. Choice of Practical Parameters
While this paper has been directed more to the concepts than to the practice of PTGC, it would be incomplete without showing how a bridge is formed between the two. First, the practical role of the significant temperature requires explanation. Let us suppose that in a wide-boiling sample mixture there are two solute peaks which are difficult t o resolve completely, and which thus require careful attention. If one wished to separate these two solutes isothermally, a few simple experiments plus consultation of the literature would suggest the best choice for a solid support material, a liquid stationary phase, a carrier gas and its flow velocity, and finally, an operating temperature. I n the programed temperature analysis, the same choice would be made except for the operating temperature which does not exist as a constant value. In place of the operating temperature, one would choose a heating rate, p, which would lead to an elution temperature 40 or 45 degrees above the best isothermal temperature, i.e., the significant temperature, T', should be adjusted as near as possible to equal the best operating temperature of the isothermal method. One would then obtain essentially the same degree of resolution as found in the isothermal run. We see, then, that the significant temperature provides the common link between isothermal and programed temperature methods. When it is necessary to manipulate the parameters of PTGC in order to improve some aspects of the analysis, reference t o equation (11) is useful. For instance, if one desires to adjust fl such that T, equals some p r e assigned temperature, it is helpful to note that T, varies only with the logarithm of 8, and thus large changes in p are needed to effect even small changes in T,. In the usual case, a 1% variation in T, requires a variation in p of about 10%. The same rule concerning the effect on T, applies to variations in tO (which, since to = L/u, may arise through variations in either column length or carrier flow velocity) and in a (which, roughly is inversely proportional to liquid loading percentage). I n each case rather large changes are needed to significantly effect T,. In summary, the parameters and materials used for PTGC are the same as used in the isothermal method, with the exception that one controls the heating rate, 8, in lieu of fixing the operating temperature, T. The
heating rate, however, has much the same role as the operating temperature; in fact a given change in In fi has nearly identical consequences with corresponding changes in T. If fi (PTGC) and T (isothermal) are large, the separation occurs rapidly hut the resolution is less than ideal. If they are too large such that approaches unity in the column, resolution becomes virtually nil. If fi and T are kept reasonably low, excellent resolution will result, and as long as extremely low values are avoided the analysis time will still remain within reason. Within these rather broad limits, a great deal of choice is available. Acknowledgment
This work was supported by the Atomic Energy Commission nnder Contract AT-(11-1)-748. Parts of this article have been obtained from Facts and Methods for Scientific Research, Val. 3, No. 2, 19% published by Fand M Scientific Corporation, Avondale, Pa.
Literature Cited ( 1 ) GIDDINGS,J. C., P ~ e p l i n tof ~ Third International Symposium on Gas Chvornatography, East Lansing, June, 1961.". 41. ~( 2 ) DALNOGARE, S., AND LANGLOIB, W. E.. Anal. Chem., 32, 767 (1960). H. W., AND HARRIS,W. E., Anal. Chem., 32, ( 3 ) HABGOOD, 4--x-11-1 - -9- ,~. i ( 4 ) HABGOOD, H. W . , AND HARRIS,W. E., Anal. Chem., 32, 1206 (1960). . . ( 5 ) FRYER,J. F . , HABGOOD, H. W., A m HARRIS,W. E., Anal. Chem.,33,1515(1961). J . C., J. Chromatog.,4 , 11 (1960). ( 6 ) GIDDINGB, (7) GIDDINGS. J. C.. Anal. Chem.. 34.722 11962). ' (8j ROWAN, R., A&. Chem., 33; 510 (1961). ( 9 ) GIDDINGS, J. C . , in "Chromatography," edited by E. HEFTMANN, Reinhold, New York, 1961. Chapter 3. (10) GIDDING~, J. C . AND KELLER, R. A,, in "~hrorn&to~ra~hy," edited by E. HEFTMANN, Reinhold, New York, 1961, \
-L-.. " "a,,.
". c
( 1 1 ) PORNELL, J. H., J . Chern. SOC., 1268 (1960). ( 1 2 ) GIDDINGS, J. C., Anal. Chem., 32,1707 (1960).
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