Investigation of the Effects of Temperature Gradients and Fluctuations on Gas Chromatographic Retention Data Michel Goedert' and Georges Guiochon Laboratoire de Chimie Analytique Physique, €cole Polytechnique, 17 Rue Descartes, Paris Veme, France
Small temperature gradients along a chromatographic column result in a relative variation of the capacity factor, k ' , proportional to the enthalpy of adsorption or dissolution, A H , and to the temperature gradient, but independent of the isothermal value of k ' . This variation is proportional to column length for straight columns. For coiled columns, it increases as the square of the number of turns at constant length and decreases with increasing column length at constant coil radius. The effects of periodic variations of the column temperature are a systematic variation of k' and a random error resulting from the lack of correlation between the sampling time and the phase of the temperature variation. Both errors are proportional to k ' , AH, and to the fluctuation amplitude when less than about 1 "C. The random error on k' is about proportional to the fluctuation frequency when large and reaches a calculable limit at low values.
In the course of a comprehensive study of the sources of systematic errors in gas chromatography ( l ) ,it occurred to us that column temperature gradients and periodic variations could contribute to such errors. Even if the space, time temperature average is measured, the retention time which is observed does not correspond to that average temperature, as the dependence between retention and temperature is exponential. In other words, the averaging process which determines the retention time is not linear and the corresponding temperature is different from the one measured with a thermometer which usually averages linearly the temperature fluctuations or gradient. The resulting error depends on the magnitude of the temperature gradient or of the periodic variations, on the shape of the column and its orientation with respect to the temperature gradient (usually determined by the oven used) and on the frequency of the temperature fluctuations. These relationships have been investigated here. We shall consider an oven in which the temperature gradient is linear and we shall successively discuss the effect of this gradient on a straight column and on a Ushaped column, both parallel to the direction of the gradient which represents the situation of chromatographic columns in long ovens, and its effect on coiled columns with various coil radii, which is the most conventional situation now. Then we shall study the effect of periodic temperature variations of different frequency. As there is no reason that the sample injection be related to the origin of the temperature fluctuations by a constant phase shift, the statistical effect on a series of measurements has also been calculated. This study is merely theoretical, because temperature effects on retention times are well known, beyond any dis1Present address, c/o Professor R. S. Juvet, D e p a r t m e n t of Chemistry, U n i v e r s i t y of Arizona, Tempe, A r i z . 85281. (1) M Goedert and G Guiochon, Ana/ Chem , 45, in press
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ANALYTICAL CHEMISTRY, VOL. 45, NO. 7 , JUNE 1973
cussion (2, 3j, and can be derived from the classical equations which relate the partition coefficient K to the absolute temperature T:
K
=
RT
(1)
dLog K AH d(l/T) - R where AH and AG are the molar enthalpy and free enthalpy of vaporization of the solute (or of desorption). The temperature effects have been calculated on the column capacity factor h' = ( t R - t,)/t, (3) and on the retention time of an inert peak. A general solution has been derived, whenever possible, and when the equations could not be solved analytically, a computer simulation has been made for some model compounds. The effect of temperature variations on carrier gas flowrate have been studied by Harris and Habgood ( 4 ) in the course of their work on temperature programming. They were interested in large temperature drift, not in temperature gradients or in small periodic variations as we are, so their results are of limited use here. We shall consider the effect of temperature gradient, first on the retention of an inert compound, then on the partition coefficient, and last on the effect of periodic temperature variations.
RETENTION TIME OF AN INERT COMPOUND This time is related to the experimental parameters (5) by the following equation: t, =
47L2
-x-
3hpo
PJ-1 (P>-l)*
(4)
where 7 is the viscosity of the carrier gas, L the column length, k the column permeability, p o the outlet pressure, and P the inlet to outlet pressure ratio. The simplest and most efficient way to control the carrier gas flow-rate is to control the inlet and outlet pressures. Then, if the column temperature changes, the effect on column length and permeability can be neglected, as they are related to dimensional changes of the column and packing, which are very small (-10-5 "C-I). So the only temperature dependent parameter in the right hand side of Equation 4 is the gas viscosity. This dependence may usually, and especially in a small temperature range, be approximated by the equation: R. S. Juvet and S. Dal Nogare, "Gas Chromatography," Interscience. New York, N.Y., 1962. A. 6 . Littlewood, "Gas Chromatography," Academic Press, New York, N.Y.. 1962. W . E, Harris and H. W . Habgood, "Programmed Temperature Gas Chromatography." Wiley. New York, N.Y., 1966, Chap. 2. G. Guiochon, Chromatogr. Rev., 8, 1 (1966).
d77 =5 pdT T
(5)
tl and: dt, -- 65 X dTT
(6)
tm
Alternatively, the flow-rate can be controlled. This solution gives identical results in isothermal work, since the flgw-rate is then determined only by the inlet and outlet pressures, but is less satisfactory as the time constant of flow controllers is larger than that of pressure controllers. A flow-rate controller keeps constant the mass flow-rate of carrier gas. So if the column temperature changes, the carrier gas velocity will change (since p u / T = = constant) as well as the pressure profile since the inlet pressure increases with increasing temperature as a response of the system to keep the mass flow-rate constant in spite of the increase in gas velocity and viscosity. T o calculate the variation oft, with T i n such conditions, let: pouo -T
-'
(15)
(POUO)/P
The carrier gas velocity is also related to Darcy's law ( 5 ) by:
Combination of Equations 14, 15, and 16 gives:
k (Pi' -Po?)
(8)
2tlLPO
we derive the inlet pressure: Pi' = PO*
where u is the local velocity. Assuming steady-state conditions, the carrier gas mass flow-rate is constant, so: u =
(7)
where is a constant determined by the setting of the flow-rate controller. From the conventional equation ( 5 )relating the carrier gas velocity to the experimental parameters: uo =
The above calculations give the effect of temperature variations from one analysis to another. They are not valid, however, if the temperature changes during an analysis. We shall calculate this effect now, in the case of pressure control only. Similar calculations show that when the flow-rate is controlled the effect is of opposite direction but of the same order of magnitude. Furthermore, it will be shown that except for very small k ' , the temperature effect on the retention is much more important than on the carrier gas viscosity, so it did not seem necessary to discuss this second calculation in details. The retention time is given by:
and:
+
27LXT k
(9)
dt=
--
k
(POUO>'
as a function of temperature. Differentiation of Equation 4 by respect to temperature gives:
p2dp 77
(18)
The column length and retention time are thus:
L
=
k pouo
JP::
y
( 10)
Differentiation of Equation 9, taking Equation 5 into account, givesp,dp,/dT, hence:
(111 Combination of Equations 7 and 8 gives: (12) Combination of Equations 5 , 11, and 12 finally results in:
3.3 po2
tm
dT
6T
pi2
+ pip0 +
Po2
The retention time of the inert peak decreases always with increasing temperature, the effect being larger when the column pressure drop is small (p, p , ) . Comparison of Equation 6 and Equation 13 shows that for the two practical cases discussed (pressure and flowrate control), the effects of temperature variations are opposite and of the same order of magnitude. In both cases, the effect decreases slowly with increasing temperature. But since almost all experimental gas chromatography is carried out between ambient temperature and 300 "C, which means a twofold increase of absolute temperature only, the order of magnitude of the effect is not changed.
-
Equation 19 defines the relationship between the pressure profile and the viscosity profile, the latter of which is a function of the temperature profile along the column. Unfortunately, Equations 19 and 20 are not analytical expressions that we could solve or integrate numerically. This is so because we know of no analytical expression of the pressure profile to introduce in Equations 19 and 20, to express the two functions in the integrand, p and 7 , with a common variable, x. In fact the conventional pressure profile in isothermal columns is obtained by integration of Equation 19, when 7 is constant ( 5 ) . Therefore we solved the problem numerically in the following manner. Assuming a constant column temperature, Equations 19 and 20 can be solved, resulting in Equations 8 and 4, respectively. Using the corresponding value of u,, Equation 19 is then solved numerically using the local values of 7 corresponding to the temperature profile. The result is different from the actual column length. The calculation is then repeated with a different value of u,. A suitable choice of the adjustment procedure of u, allows reaching the actual L value within less than after a limited number of such calculations. Then, we know the mass flow-rate of carrier gas through the column in the given experimental conditions (pt, p,, temperature profile). Using this value of u,, t , is calculated by numerical integration of Equation 20. This iteration procedure might ANALYTICAL CHEMISTRY, VOL. 45, N O . 7, J U N E 1973
1181
Table I. Experimental Conditions for the Study of the Effects of Temperature Gradients Column length ( c m ) Column permeability ( c m 2 ) a
102
103 10-6
10-5
104 10-7
Outlet flow velocity ( c m / s e c ) b 3 10 30 Carrier gas: H2 ( 7 = 0.9 X c g s ) . He (7= 2.2 X cgs) a Typical permeabilities of packed columns are (crn2) to lo-', of capillary columns around ( 4 ) . *The inlet pressure is calculated accordingly. p o = 1 atm.
L
c
I
0
,
T0ti-l
seem lengthy but it is straight forward, accurate, easy to program, and uses only short computer time. This calculation has been done with the following conditions (cf. Figure 1): straight column, linear temperature gradient:
T
=
T=T,+ax T = To + a ( L - x)
'
I
Figure 1. S c h e m e of the columns studied and their temperature
(22)
T
=
T o + T I sin ( x / R )
L
=
2nnR
112
9(p,L - Po?)?
The range of variation of f l and f 2 is rather small. When p J p 0 increases from 1 t o ” , f i decreases from 2.5 to 2 and f i from 11.66 to 8, so we can take in first approximation their average values, respectively, 2.25 and 9.8 which corresponds tOPLlP0 = 2.
Thus Equation 34 shows that for small temperature gradients the relative variation of k':
Akf k,' ~
= -0.45
LAH a RTO2
(37)
~
is proportional to the column length, to the vaporization enthalpy, and to the temperature gradient. Then, with the parameters chosen here (k' = 1, L = 1 m, T o = 300 "K cf. Table 11), Equation 34 becomes:
* k,,'
= - 1.26 a
+ 1.35 a2
(38)
As we shall see, this result is in excellent agreement with the calculations. For a coiled column it is easy to derive an equation for k ' , similar to Equation 31:
R This equation shows that again, for small temperature gradients, the relative variation of k ' will be proportional to AH and to the temperature gradient; it is however not possible to integrate sin x t / a x b, although it can be shown that the relative variation of k' will be smaller than in the previous case (straight column). As shown above, however, t , increases with increasing temperature and this will result in another negative contribution to the variation of k ' . It may be expected that this contribution is not a major one, as the retention is an exponential function of temperature. As, anyhow, a numerical solution is necessary for coiled columns, the calculation of the velocity profile and its integration as discussed above have been included in the computer program and calculations run for straight columns as well. Straight Column. Figure 2 shows the relative variation of the retention times with the temperature gradient. It is almost proportional to a and much larger than the variation of t,, 3 times for the first compound (k' = l), 7.5 and 14 times for the other two. For this last compound, a temperature gradient of 4 x 10-3 "C/cm results in a 1% error on the retention time. This shows that the effect of temperature variations is much larger on the retention than on the gas viscosity so under controlled carrier gas flow-rate, the relative variation of k' will be about the same. Figure 4 shows the relative variation of k' for the three compounds studied. l k ' is negative-i. e., values obtained are lower than the isothermal ones. If the measured values are reported to the average temperature ( i . e . , temperature a t half column length), the error is much smaller, about 7 times, but still not zero as the temperature averaging process is exponential and not linear. From the data on Figure 4 (curve 2) the equation for the variations of l k ' l k , ' with a can be derived for the first compound ( k ' = 1):
+
This is in excellent agreement with Equation 37, for a < 0.12. As the Ak' error is proportional to AH when a is small, part of the data for k' = 3 and k' = 10 has been omitted from Figure 4. The effect of the carrier gas velocity ( u ) and viscosity ( v ) and of the column permeability ( k ) are extremely
small, the relative changes in a k ' / k , ' when u, 7 , and k are varied in the range studied (cf. Table I), being less than 1% for any value of the temperature gradient. These effects can thus be neglected. U-Shaped Column. The results are shown also in Figure 4. For a given value of the temperature gradient, the error is 1.6 to 1.7 times smaller than for a straight column. As for straight columns, the variation in k' is proportional to the column length, for small gradients ( c f . Equation 37). This is to be expected as the maximum temperature difference between two points in the column increases with column length for a given value of the gradient. Coiled Columns. Figures 5 and 6 show the effect of the temperature gradient in such columns. As shown above the relative variation of k' is proportional to AH.This is in agreement with the results of the calculations made for various values of T I lower than 1, so only the results corresponding to the compound for which k' = 10 are given. At low values of the gradient, the systematic error on k' is proportional to the gradient and inversely proportional to the square of the number of turns for a given gradient ( T I I R ) .Comparison of Figures 3 and 5 shows that the contribution of the variation of the carrier gas velocity is much smaller than the one of the partition coefficient. At larger values of the temperature gradient, the systematic error on the column capacity factor changes its sign. It is negative a t low gradients; it becomes positive for the larger gradients and increases very fast. The gradient a t which the error becomes zero and changes its sign is the same for all columns: 0.85 "C/cm for columns with a full number of turns, 2.15 "C/cm for a full number of half turns. For a given column and different compounds, this effect occurs for values of the temperature gradient which are inversely proportional to the vaporization enthalpy as shown by the calculations. This effect arises from the nonlinear dependence of k' on the temperature. Numerical instabilities of the calculation are ruled out by the fact that this phenomenon is unaffected by changes in column permeability, carrier gas viscosity, or velocity. If the column length is increased from 1 to 2 m while keeping the same coil radius ( i . e . , doubling the number of turns), the relative variation in k' a t low temperature gradients (TI < 0.5 "C) is divided by 1.6 which is less than proportional to L . EFFECT OF PERIODIC TEMPERATURE VARIATIONS In this section, we shall discuss the other type of temperature fluctuations that a compound may experience during its elution through a column: temporal changes. We shall consider only periodic sinusoidal variations as temperature fluctuations around the average value can be expanded in Fourier's series. Their effects depend on three parameters: the amplitude of the fluctuation, the period 7 of the fluctuation, and the phase shift between its origin and the injection. We can expect that temperature fluctuations with periods much shorter than the retention time will be averaged out, with a systematic error originating only from the nonlinear dependence between the partition coefficient and the temperature as illustrated above. Temperature fluctuations with periods very long compared to the retention time (temperature drift) will act like a temperature gradient on a straight column. Temperature fluctuations with periods which are of the same order of magnitude as the reA N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 7 , J U N E 1973
*
1185
e/
QM
0.1
1
T1
Figure 7. Variation of k ’ with the amplitude of temperature fluctuations No phase shift. isothermal k’ = 1. t R = 34.86 sec. Periods 1. 80 sec; 2, 200 sec; 3 , 50 and 316 sec; 4, 20 sec; 5 , 12.6 sec; 6 , 31.6 sec; 7 , 2 sec, 8, 3.2 sec
tention times do not give results similar to those obtained with coiled columns placed in a temperature gradient, because in coiled columns the temperature variation is determined by the position of the zone, and the apparent period is a fraction of the retention time equal to the inverse of the number of turns, while the period of temperature fluctuations is the same for all compounds. The effect of temperature fluctuations on the viscosity of the carrier gas and the retention time of the inert peak have not been taken into account here as they would be too difficult to calculate. Furthermore, it has been shown in the previous section that the effect of temperature on the retention is the major one. Figure 7 shows the variation with the fluctuation amplitude of the error on k’ for different periods of the temperature fluctuations. For the sake of simplicity, all these fluctuations have their origin at injection time (Le., T = To + 2’1 sin 2 A t / ~ In ) . most cases, we have observed that the error is proportional to the importance of the fluctuation, except when this amplitude is large, but we shall consider here only well controlled equipment for which T I is much less than 1 “C, especially if we take into account the fact that several sinusoidal fluctuations with different periods are usually responsible for the observed temperature variations. At first glance, the effect of the period seems to be quite complex. Figure 8 shows the variation of the error on k‘ with the period of sinusoidal fluctuations of amplitude 1 “C, when the origin of the fluctuations is a t injection time. It is remarkable that the effect is extremely small when the period is equal to the retention time of the inert peak or of the compound or is a simple fraction of this retention time (y2, y3, l/4, l/g, lho, . . .). This corresponds to situations which are similar to that of the coiled column. Also the limit for large period is the one which corresponds to the effect of a temperature gradient (as the effect of temperature fluctuations on the carrier gas viscosity are not accounted for in the model, only the temperature effect on t R ’ = t R - t,, as derived from Figure 2 has to be taken into account to calculate this limit). 1186
ANALYTICAL CHEMISTRY, VOL. 45, NO. 7, JUNE 1973
Figure 8. Variation of k ‘ with the period of temperature fluctuations No phase shift. Isothermal k’ = 1. Amplitude:, T , = 1 ‘C,t~ = 34.86 sec, tm = 17.43 sec. Dotted lines: 1, derived from Figure 2 for a temperature gradient 2?r/i: 2, maximum deviations at low periods
[AT k ’ 7 %k’ lo-’
Figure 9. Variation of k‘ with the period of temperature fluctuations. Results on series of 20 experiments with random phase shift Amplitude = 1OC. Isothermal values: 1, 2, 3 , k’ = 1 , t ~ . = , 34.86 sec; 4, 5 , 6 , k’ = 10; t ~ , 2= 191.73 sec; 2, 5. relative variation of the average value of k‘: 1, 4, relative standard deviation of the results: 3 , 6 , maximum value ?f the standard deviation at low periods
The origin, the importance, and the details of the variation of Ak’/k’ with T have not been studied in detail, as there is no reason to assume that the fluctuation always starts just a t the injection time. So the calculations have been repeated 20 times, introducing each time a random shift between the origin of the sinusoidal fluctuation and the injection:
T
=
T o + T , sin27r(t/z
+
0.1 6)
(41)
where 6 is a random integer selected between 0 and 10 with uniform probability distribution. The average value and the standard deviation of the results are plotted in Figure 9 as a function of 7 . The average value is very small, but the standard deviation is much larger: this effect is mainly a source of random error, more than of systematic error. When the period becomes very large, the limit of the standard deviation is somewhat smaller but near the value which can be calculated by differentiation of Equation 1:
dk' k' -
A H dT RT2
It is observed (cf. Figure 9) that the relative standard deviation u k . / k ' is proportional to AH (cf. Table 11). The calculation of the effect of temperature fluctuations would need to calculate the Fourier transform of the record of the temperature of the column oven. This is much too complicated in most cases and a good approximation can be obtained by calculating the effects corresponding to the main temperature fluctuations (oscillations a t the controller period and drift).
CONCLUSION The results discussed above show that in the high precision chromatograph we have built ( I ) , when the tempera"C/cm and the maximum temture gradient is 5 x perature fluctuations kO.01 "C a t 180 "C with a main period of 2.5 minutes and a maximum drift of about 0.005 "C/h occurring during a maximum of about 2 hours, the errors introduced on the determination of k' are about 10-5 for a coiled column 2 m long, 8 turns (4 8 cm) and a compound with AH = 10 kcal/mole and k' 3. The contribution of temperature fluctuations to the random error is about 1.1 x The main contribution comes from the temperature drift and can be partly corrected. It should be pointed out that most commercial equipment is not suitable for the determination of thermodynamic data by gas chromatography as temperature gradients are often around 0.1 "C/cm and the amplitude of temperature fluctuations around 0.5 to 1 "C. These values lead to rather large errors, typically in the 0.5 to 2% range, and lack of reproducibility. This is probably the origin of the well known lack of reliability of retention data found in the literature.
--
Received for review October 31, 1972. Accepted January 26, 1973.
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