decomposition occurred with several sugars above this temperature. Glycoside formation is known to occur very easily in solutions containing 2-deoxy sugars (8). Stepped chromatograms were recorded for 2-deoxy sugars at 100" C. To avoid any detectable formation of ethyl glycosides of 2-deoxy sugars the temperature had to be decreased to 75" C. A typical application of partition chromatography on a cation exchange resin in its lithium form is given in Figure 2. The seven monosaccharides applied in this run were separated in 5 hours, including the retention time in the analyzing system. Among these sugars rhamnose and 2-deoxyglucose cannot be separated o n anion exchange resins in the sulfate form. The separation of xylose and tagatose is also extremely difficult in the latter system, whereas the other sugars can be separated as well on this type of resin as on the cation exchange resin. Another example of failure to achieve satisfactory separation on an anion exchanger in its sulfate form is that of tagatose and fructose. As can be seen in Figure 3, these two ketoses can be easily separated on a cation exchanger in the potassium form. Sorbose was also included in this run. With the short column used in this work sorbose overlapped with tagatose but fairly accurate evaluation of the chromatogram can nevertheless be made. If greater accuracy in the quantitative analysis of the three ketoses is desired, the separation can be made on a longer column. The application of the resin in its potassium and lithium forms is compared in Figure 4. In the separation of 2-deoxy (8) W. Pigman, "The Carbohydrates," Academic Press, New York, 1957.
sugars the lithium form is superior to the potassium form. The separation is more effective and the time of elution is shorter. With other sugars the other ionic forms are sometimes superior to the lithium form. The distribution coefficients reported in Table I can serve as a guide in the choice of resin form. One disadvantage of the lithium form in comparison with the sodium and potassium forms is that, compared at constant flow rate, the pressure drop in the column was much higher. For this reason all runs with the lithium resin were carried out a t moderate flow rate t o avoid breakage of the column. Some simple separations-for instance, that of xylose and glucose-could be made a t very high flow rates by using the potassium form of the resin. When very rapid separations are desired, a high temperature should be chosen whenever possible. This permits lower peak elution volumes and a decreased pressure drop in the column. A practical example is the separation of xylose and glucose, which could be made in 45 minutes at 100" C. Compared with anion exchange resins in the sulfate form, cation exchange resins in the forms studied in this work have certain disadvantages. Most striking is the fact that mannose cannot be separated from glucose on the cation exchanger. Cation exchange resins in lithium, sodium, and potassium forms can, therefore, not replace the anion exchange resin in all determinations of monosaccharides. The main advantages of the cation exchangers are in analyses of solutions containing deoxy sugars and ketoses. RECEIVED for review January 16, 1967. Accepted April 26, 1967. Work supported by the Swedish Technical Research Council.
Calculation of Essential Parameters in Programmed Temperature Gas Chromatography from Programmed Temperature Data Alone Robert Rowan, Jr. New Mexico State Uniuersity, Uniuersity Park, N. M .
METHODSHAVE BEEN PRESENTED (I) for the calculation, o r prediction, of retention temperatures in programmed temperature gas chromatography (PTGC). The methods most pertinent to the present work utilize the experimental variables along with two characteristic constants, here designated B and B (or In B). These are the slope and intercept, respectively, of the well known linear equation: In
);(
-
-; --f1nB
where V , = net retention volume and T = absolute temperand is dependent ature. Theta is essentially AH(vanorilation)/R only o n the solute-solvent system. B is also a function of the solute-solvent system but in addition depends linearly o n the amount of solvent present. The reverse calculation-i.e., finding B and B from retention temperatures-is not a straightforward task because of the complexity of the equation relating the quantities involved. ( 1 ) P. Chovin, Ann. Chim. (Paris),7,727 (1962) (review containing
pertinent references). 1 158
0
ANALYTICAL CHEMISTRY
This paper describes a method for computing B and B from PTGC data alone and presents results confirming the validity of the method. Because these quantities are not entirely independent of temperature, the results obtained will be to some small degree a function of the temperature interval of the run and will be average values for this interval. THEORY
Significant variables and characteristic constants are related by the following equation ( 2 , 3 ) : rB e-#? 42 - = - _ -
FB
$2
$1
where r
=
temperature rise rate,
F
=
flow rate, ml/minute SjTi eIT2 air peak temperature, " K retention temperature
$1
=
$2
=
TI = Tz =
C/minute
(2) R. Rowan, Jr., ANAL.CHEM., 33, 510 (1961). (3) Zbid., 34, 1042 (1962).
Although Equation 2 is an approximate relation, because of the use of net retention volume in place of retention volume in its derivation, the error involved is negligible in the case of packed column operation under any except the most extreme conditions ( 4 ) . The equation can therefore be used with confidence in the present instance as though it were exact. Taking T1 to be air peak temperature rather than injection temperature reduces the error, and this has been done here. Since both 0 and B are unknown, it is necessary to have two independent equations jn order to compute them from the other quantities. Because of the complexity of the relationships, however, this cannot be done directly, and we must resort to a process of successive approximation. The essence of the approach used in this paper is that equations are transformed so that they can be solved iteratively with rapid convergence. Two suitable equations are the following, representing two PTGC runs at different temperature rise rates. Flow rates and starting temperatures for the two runs can be the same or different. (3)
Table I. First Test of Calculation Procedure (Theta, degrees) PTGC Untreated Internal Normal paraffin Isothermal data standard 3164 3613 4150 4626 5055 5336 5692
c 6
G c 8
CQ c10
CI1 ClZ
=
In Y
8(C2 - 1) X
-
T’
Equation 8 is a quadratic where a=(Cz-l)
where g and h refer to the low and high temperature rise rates, respectively. Then
b
It has been shown (3)that function @ can be represented very accurately by the relation: @
=
exp [(C, - 1) $2
+ C I $ ~ ’+/ ~CO]exp [(CZ- 1) +
where CZ, C1, and COare empirical constants determined in a manner described elsewhere (3). In the present instance the values CZ= 0.15177, C1 = -2.1834, and Co = 0.55290, which provide good results in the range @ = 6 to 4 = 20, were used, but other sets of constants to cover more limited ranges with greater accuracy may be chosen. The quantities + g and & can therefore be stated similarly by adding subscripts g and h to the respective +l’s and &’s. For convenience, let Qih
= (CZ- 1) + l o = the 4 i h term
+ C1
$1g1/2
+ CO
=
the
$lo
term
etc. Then:
Replacing agwith 6:
a. + eQ10
= ~ Q w
(5)
ah + eQlh
=
(6)
W a h
eQ2h
=
CI
--(io
7 9
[(k)“’ ($)“’] -
c = -1nY
and
+ COI (4)
$1
Qlg
3053 3518 4148 4867 5056 5189 5447
e
Since 6 =
1796 2492 3190 3886 4087 4208 4437
in Equation 5 and dividing by Equation
(4) J. F. Fryer, H. W. Habgood, and W. E. Harris, ANAL.CHEM., 34, 882 (1962).
(9) L
Once 8 is determined, B can be found from either of Equations 3. The left-hand side of Equation 7, which is Y in Equation 8, contains 8. This makes it necessary to assume some value of 8 so that Y can be calculated to put into Equation 8. Since both the terms containing 0 which appear in Y pertain to the start of a run-Le., they are terms-they have only a small effect, in nearly all cases, on the value of 0 calculated by Equation 9. The procedure is to assume an initial value of 0, compute Y, use Y in Equation 9 to compute 0, then compute a new value of Y , compute 0 again, etc., until two successive values of 0 differ by some predetermined amount (say 3 units). In the calculation of Y using some value of 0, it is of course necessary to evaluate a h (see Equation 7). This could be done by means of Equation 4 but has been done in this study by the approximations of Hastings, Hayward, and Wong (5).
The initial assumption is made that Y = W-Le., that 41 terms are negligible. This is equivalent to assuming that at the starting temperatures the solutes are substantially frozen in the column, and in most cases this is fairly close to the truth. The degree of departure from this condition dictates the number of successive approximations necessary to reach the limit set for convergence. The following diagram indicates the sequence of operations in the iterative procedure: ( 5 ) C. Hastings, J.
T. Hayward, and J. P. Wong, “Approximations for Digital Computers,” Princeton University Press, Princeton,
N. J., 1955. VOL. 39,
NO. 10, AUGUST 1967
e
1159
Table 11. Actual and Calculated Retention Temperatures Rate, deg/min Measured. Degrees K
Run Amyl bromide
1
Ethylbenzene
2 3 1 2 3
Butyl ether
Predicted
336.2 363.1 381.5 340.1 368.8 388.2 344.3 373.6 392.9
1
2 3
335.7 362.5 378.6 339.4 368.1 384.9 343.9 372.7 388.8
Measured 0.42 1.89 3.71 0.42 1.89 3.71 0.42 1.89 3.71
Calcd. (see text) 0.44 1.94 4.15 0.44 1.95 4.20 0.43 1.97 4.37
Other data T(injeetian) T(air) Initial pressures, torr Degrees K In out 315.9 316.2 995 666 2 316.0 317.3 994 665 3 316.5 319.1 992 665 Rates used in calculation of 0: 0.44, 1.95, 4.20 degrees per minute. 3500 Flow rate. F = __ + 23.16, where T = column temp., ' K ; F = gas-expansion-corrected flow rate, ml/rnin. Run 1
Initial flow rate, ml/minb 44.9 44.9 44.9
T
a
(Net retention time) X (measured temp. rise rate). Measured at 28" C with bubble-type flowmeter. Uncorrected.
Assume Yo = W -+ Bo
M and N are constants, F is net flow rate, and T i s absolute temperature. The treatment is exactly the same as that employed previously (3) for calculations on PTGC a t constant pressure. In principle, flow rates for the two runs ( g and h) could be different-Le., two sets of constants M and Nbut the present program does not provide for this.
b/
Y , -+
e,
RESULTS AND CONCLUSIONS CALCULATIONS
Calculations were made by means of a digital computer. The arithmetic is so extensive and tedious that hand methods would not be feasible. The computer program is written in Fortran 11. IBM Model 1620 o r CDC Model 3300 computers were employed. A refinement of the program having a beneficial effect involves the treatment of flow rate as a variable. As has been explained (3), pressure drop-corrected flow rate can be expressed as a function of reciprocal absolute temperature if helium is the carrier, even when a flow controller is used. Of the several mechanical controllers in the author's experience, all allowed some variation of net flow rate with temperature. In these cases, it is helpful to incorporate the following equation in the program: F = M / T N , where
+
Table 111. Second Test of Calculation Procedure Theta, degrees Programmed temperature Material Isothermal Runs 1 and 2 Runs 1 and 3 Adjusted rates, internal std. (See text) Amyl bromide 4657 4696 4703 Ethylbenzene 4726 4803 4757 Butyl ether 5215 5130 5057 Unadjusted rates Amyl bromide 4657 4718 4382 Ethylbenzene 4726 4824 4456 Butyl ether 5215 5150 4762 ~
1160
ANALYTICAL CHEMISTRY
No data specifically designed to test this method have so far been available. However, an initial test was made o n some data originally obtained for another purpose, and the agreement between values of 0 obtained from isothermal runs and by this procedure shows clearly that the method is valid. All pertinent information such as flow rates, heating rates, and injection temperatures, is given by Rowan (2) and is not repeated here. Data treated were from runs o n C 6 to CI2 n-paraffins with a n injection temperature of 363' K. F o r reasons not known, perhaps nonlinearity of r or biases in flow-rate measurements, there were biases in the calculated results. It was therefore decided to use a n internal standard technique. In this procedure, the temperature rise rate necessary to make B for n-octane found by this method correspond with that found from isothermal runs, was calculated. This value of r was then used in all other calculations on PTGC data. (This changed rB from 3.2 to 3.4 degrees per minute and rh from 8.3 to 10.6 degrees per minute.) From a practical standpoint this procedure is perfectly feasible. The results obtained are shown in Table I. Although the agreement is poor in some cases even for the internal standard case, the results in Table I show that the calculation method is valid. A second test of the procedure utilized data which, w a i n , were obtained for another purpose. I n this case the instrument was a n Aerograph Model 1520 with an %foot, experimental, 6-millimeter O.D. glass column bent in the shape of a double hairpin projecting some 18 inches above the oven. This required a n oven extension, and temperature uniformity
during the program was poor. The packing was 10% DC200 silicone o n C-22 firebrick (actually, 0.0116 gram of substrate per cm of column). Runs were made a t different temperature rise rates o n a mixture of three compounds which are listed in Table 11. Here the measured temperature rise rate is compared with the value calculated to be necessary to make the predicted (by Equation 2) retention temperature equal the actual retention temperature. Table I1 shows that discrepancies between the calculated and actual retention temperatures
are small, and the calculated rates are therefore close to the real ones. F o r the calculation of theta for all three compounds, the “calculated” temperature rise rate values for ethylbenzene were used. Two sets of theta values were computed: One was obtained by pairing the 0.42 and the 3.71 degrees per minute runs and the other by pairing the 0.42 and 1.89 degrees per minute runs. The results are shown in Table 111.
RECEIVED for review December 12,1966. Accepted May 17, 1967.
Current Transients at a Rotating Disk Electrode Produced by a Potential Step Stanley Bruckenstein and Stephen Prager Department of Chemistry, Unicersity of Minnesota, Minneapolis, Minn.
MANYPARTIAL DIFFERENTIAL EQUATIONS which occur in electrochemical problems are not amendable to closed-form solution. An example of such a problem is the calculation of the current transient accompanying a potential step which results o n a rotating disk electrode a t hydrodynamic equilibrium. The electrochemical problem is, in itself, of fundamental interest for a t least two reasons. First, the “response time” of the disk electrode current to a potential change, and hence a t a n electrode surface concentration change, must be known in order t o evaluate the analytical utility of the electrode under certain conditions-e.g., in following the kinetics of an extremely rapid reaction. Second, film formation o r adsorption often interferes with the interpretation of steady-state limiting currents, and the desired information can be obtained from the current transient before such complications interfere. The approximate mathematical solution t o this problem and to many others of electrochemical interest may be obtained by using the well-known “method of moments” ( I ) o r the “Galerkin method” (2). A reviewer of this paper has pointed out that this latter method has been applied by Gelb (3)recently t o other electrochemical problems.
55455
On applying a constant potential t o a disk electrode, the surface concentration of the electroactive material becomes constant c(0, r)
=
cs
(3)
t>O
A convenient, approximate, solution t o the above problem may be obtained by using the “method of moments” ( I ) or the “Galerkin” method (2). If Equation 1 is integrated with respect to y from 0 to 03, a gross mass balance is obtained for the entire diffusion layer
where
K
=
0.51 [
~ ~ / v ] l ’ ~ ,
Assuming a simple, time-dependent, Nernst diffusion layer of thickness 6(t) adjacent to the disk surface, we write our concentration distribution in the form
Therefore, from Equation 4
THEORY
The convective diffusion equation for a rotating disk electrode which has attained hydrodynamic steady-state is and I n Equation 1 , D is the diffusion coefficient of the electroactive species whose bulk concentration is cb. The normal distance from the electrode surface is y , cy ( = 0.51 [ ~ ~ / v ] l ’ 2 y 2 ) is the velocity of the supporting electrolyte at a distance y , w is the angular velocity, and v is the kinematic viscosity of the supporting electrolyte. Initially there is a uniform concentration distribution throughout the solution c(y, 0 ) =
Cb
O
