sitivity to be useful with UV detectors based on the absorption of light from the mercury resonance line at 254 nm. Although some detectors are available with a choice of wavelengths, their function in the low wavelength range of the UV can be limited by the end absorption of the solvents employed. Solvents such as chloroform, ether, and acetonitrile; which have end absorption in this region, would create problems at precisely those wavelengths (210-220 nm) which would be most useful for compounds with unconjugated chromophores-witness the spectrum
of DHA. Double beam monochromatic instruments with variable wavelength capability could probably alleviate the end absorption problem, but cost would enter as a factor. Derivatization will present itself as an attractive alternative for some time to come, especially when it can aid in improving chromatographic resolution as well as sensitivity. Received for review December 4, 1972. Accepted July 5, 1973.
Mathematical Analysis of Velocity Programmed Chromatography Dwight W. Underhill and James A. Reeds Harvard University, School of Public Health, 665 Huntington Avenue, Boston, Mass. 021 15
Richard Bogen Massachusetts Institute of Technology, Project MAC, 545 Technology Square, Cambridge, Mass. 021 39
A general iterative procedure is developed for the calculation of the spatial moments of sorbate in a velocity programmed chromatography column. This procedure is applied to a column in which mass transfer is a function of film limited diffusion between the mobile and stationary phases, molecular diffusion in the mobile phase, and a mobile phase velocity which increases linearly with time. The zeroth and first three spatial moments were calculated for this case using a computer program (MAXSY MA) designed to manipulate symbolic expressions. The results are simple algebraic expressions which clearly show how the parameters cited above affect the number of theoretical plates and skewness of the sorbate as it passes across the column.
Quite often well defined Gaussian curves are obtained in programmed velocity chromatography, but there are factors which work against such a result. Severe deviations from Gaussian behavior may be expected if the peak has travelled only a short distance, as well a t the high carrier gas velocities encountered later in the programming. The analysis presented here permits the spatial moments of the concentration of sorbate to be calculated directly from the partial differential equations describing mass transfer. From' the zeroth and first four spatial moments can be determined, in order, the average distance from the point of injection, and the standard deviation, the skewness, and the kurtosis of the distribution of sorbate across the column. This gives the means to characterize the mass transfer by the best fitting Gaussian curve, as well as to determine the extent of the deviations from such a curve. Because of the length of the calculations, in this paper we present and discuss only the zeroth and first three moments of the linearly programmed column. But the method for calculating the higher moments, as well as for calculating moments for nonlinear velocity programmed chromatography, is established. 2314
MATHEMATICAL ANALYSIS Definitions a n d Boundary Conditions. For the case of linear velocity programmed chromatography, the interparticle velocity is the following function of time,
V = a + b t (1) The partial differential equation for mass transfer resulting from a combination of interparticle diffusion, convection, and interphase mass transfer resistance is:
where U = interparticle concentration of sorbate, moles/ cm3; D = interparticle diffusion coefficient, cmz/sec; V = interparticle mobile phase velocity, cm/sec; g = interphase mass transfer function, sec-1; and * denotes convolution, Le.,
In this paper, the interphase mass transfer function, g, is defined as the rate of mass transfer to the volume of stationary phase contained within a unit volume of mobile phase, following exposure to a constant unit concentration of sorbate in the mobile phase starting at time, t = 0. In making this calculation, it is assumed that a t time, t = 0, the stationary phase is sorbate free. The boundary conditions for the column are based on a unit 6 function input a t time, t = 0, at the origin, x = 0, into a column whose cross sectional interparticle area is equal to 1 cm2. The choice of 1 cm2 for the interparticle area simplifies the calculations that follow by letting the factor U also represent the quantity of interparticle sorbate per unit column length. To simplify further the calculations which follow, it is assumed that mass transfer occurs in an infinite column filled with an incompressible fluid. These restrictions are not too severe for many practical applications. The assumption of an incompressible fluid limits the results to
A N A L Y T I C A L C H E M I S T R Y , VOL. 45, NO. 1 4 , D E C E M B E R 1973
either a column filled with liquid or to a gas chromatograph in which the compression effects are not great. The choice of a n infinite, rather than a semi-infinite, column for our analysis means a loss of any non-Gaussian behavior resulting from interparticle diffusion at low carrier velocities. In a real chromatograph, this effect is generally not important. The external boundary conditions are then
The kth ordinary spatial moment, as calculated from the interparticle concentration, is
The concentration variable, U, is a function of both distance and time. Equation 5 integrates out the distance variable and leaves the moments as calculated as only a function of the time variable. Integration. The following results are useful in the calculations which follow.
(6)
To calculate the zeroth moment, integrate Equation 2 with respect to x, obtaining
$IX0(t)+ so
+
where t = fractional interparticle void volume, dimensionless; W = the time constant for mass transfer, sec-I; and k = partition coefficient for sorbate between equal volumes of stationary and mobile phases, dimensionless. Therefore (1 - t)kW ZS)= t(W s) Then
+
(JTo*g)(t)l
+ (R,*g)(t)
ET&)
We already knoy g o ( s ) an$ Gl(s),and can use this formula to obtain R ~ ( S then ) , M ~ ( Sand ) , so on. This method can also be extended to nonlinear velocity programmed chromatography. For example if V = a + bt c t 2 , the same general met_hod would hold, but we would nEed to calculate (d2/ds2)(Mb-1(s)) in addition to (d/ds)(Mk-1(s)).This merely adds to the algebraic difficulty, but it does not alter the iterative procedure developed here. Mathematical Calculations. The calculations given below were carried out at Project MAC, an interdepartmental laboratory at the Massachusetts Institute of Technology, using MACSYMA (Project MAC,'s Symbol MAnipulation System), a large program written in the language LISP. This computer program was designed to manipulate algebraic expressions and it has a capacity to differentiate, integrate, take limits, solve equations, factor polynomials, expand functions in a power series, and take or invert the Laplace transform ( I ) . The ability to perform these operations by machine can save much time. The calculations which follow took about forty minutes of machine time. In making these calculations, it is necessary to select a mass transfer function, g ( s ) . If mass transfer between the moving and the stationary phases is assumed to be controlled by a thin film, then ( 2 )
Mo(s) =
=
..(")""" -
or
w + s
where C is an as yet undefined constant. But since U(x,t = 0 ) = fi(x),
(23) where
R,(O> = 1 and because g is smooth,
H
(Go+g)(O) = 0
=
(-)k
t
+1
(25)
+
Ro(t) (No*g)(t) = 1
(15) Taking the Laplace transform and solving for if?o(s) gives (16)
a,,multiply Equation 2 by x and then integrate VJTO(t)
=
&ut)
+
(M1+gXt))
Taking the Laplace transform and solving for
(24 1
By inversion
Hence C = 1 and
To find
(22)
(17)
gives
This procedure can be developed in a sequential procedure for calculating momemts. In general
The zeroth moment shows an exponential decay from an initial value of unity to a steady state value of 1/H, with a time constant of HW. Physically what has happened is that a t time t = 0, a unit pulse of sorbate was introduced into the mobile phase of the column at x = 0. As time passes, the sorbate redistributes between the mobile and stationary phases, and as the spatial moments calculated here are calculated from the mobile phase concentrations, these moments should also be expected to contain functions having an exponential decay. The zeroth moment, the simplest moment, shows this effect most clearly. Oxtoby ( 3 ) has discussed these exponential factors in constant flow (unprogrammed) chromatography. (1) MACSYMA Users' Manual, Richard Bogen, Ed., Project MAC, Massachusetts Institute of Technology, Cambridge, Mass., Publisher,
1973. (2) 0. Grubner and D. W. Underhill, J. Chrornatogr., 73,1-9 (1972). (3) J. C. Oxtoby. J. Chem. Phys., 51,3886-90 (1969). A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 14, D E C E M B E R 1973
2315
If the exponential terms are important, e.g., t < 3 / W H , then the distribution of sorbate is so poorly established that from the standpoint of chromatographic separation, the columns are uninteresting. For this reason and to make our equations more readily understandable, we omit from our calculations the exponentially decaying terms which represent the initial redistribution of sorbate between the mobile and stationary phases. Equations 26-28 give, respectively, the first ordinary spatial moment and the second and third central spatial moments. Each moment has been normalized with respect to Mo(t), which for the long term case examined here is equivalent to multiplying these moments by the factor, H. Specifically these moments were calculated by Equations 29-31.
M,(t)
+
+ + +
varies as t - 1 I 2 for large t. But in the absence of interparticle mass transfer resistance, the skewness becomes equal to zero, because in the infinite column assumed in these calculations, the distribution of sorbate would then be purely Gaussian. In this paper we have developed a method for the theoretical analysis of programmed chromatography and shown how a computer program could be used to obtain results rapidly. It is hoped that the application of these procedures will lead to improved design of chromatographic separations.
NOMENCLATURE a,b = Coefficients of the linear velocity equation; cm/sec and cm/sec2, respectively D = interparticle diffusion coefficient, cm2/sec
+ +
+ +
{ 2 ( H 4- H3)W3b2t3 ( 3 ( H 4 - 5H” 4 H 2 ) W 2 b 2 6 ( H 4 - H3)W3ab)? 4H2)W2ab 6(H4 - H3)W3a2)f 5H)Wb’ 6(H4 - 5H3 ( 6 H 5 W 4 D - 6(2H3 - 7 H 2 6(H4 - 5 H 3 4 H 2 ) W 2 ~ 2 } / 3 H 6 W(42 7 ) 12(H5 - H 4 ) W 3 D 3 0 ( H 2 - 3H 2)b2
=
+
+
+ +
+
+ +
+
+
+ +
(3(H5 - 3H4 2H3)W3b3t4 ( 4 ( H i - 1 2 H 4 27H3 - 16HZ)W2b3 (12(H6 - H 5 ) W 4 b D - 6 ( 6 H 4 - 47H3 91H2 - 5 0 H ) W b 3 E ( H 5 - 3H4 2H3)W’ab2)t3 12(H5 - 12H4 27H3 - 1 6 H 2 ) W 2 a b 2 1 8 ( H 5 - 3 H 4 + 2 H 3 ) W 3 a 2 b ) t 2 ([12(H6 - 5 H 5 4H4)W3b 24(H6 - P ) W 4 a ] D 60(2H3 - 13H2 23H - 12)b3 - 12(6H4 - 4 7 H 3 91H2 - 50H)Wab’ 12(H5 - 12H4 27H3 - 1 6 H 2 ) W 2 a 2 b 12(H5 - 3 H 4 2 H 3 ) W 3 a 3 ) t 24(@ - 5 H 5 4H4)W3aD 120(2H3 - 13H2 23H - 12)ab2 8 ( H 5 - 12 H 4 27H3 - 1 6 H 2 ) W 2 a 3 } / 2 H B W(j 2 8 )
M,(t)
=
+
+ +
+
+
+
+
+
M,(t)/M,(t) (29) M2(t) = M , ( t ) / R , ( t ) - M,(tI2 (30) M3(t)/Mo(t) - 3 R d t ) R 1 (t ) / R ~ (+ t )2) ~ ~ ~ ( t M,(t)
~ , ( t )=
+
=
(31) The number of theoretical plates a t time t is
N
=
R,(tY/M2(t)
(32)
For very large t, in the presence of interphase mass transfer resistance ( W < m ) , the number of theoretical plates is proportional to t, and therefore the HETP, the height equivalent to one theoretical plate, varies inversely with the square root of the distance moved by the peak. On the other hand, in the absence of appreciable interphase mass transfer resistance, the behavior of the column will be controlled by the interparticle diffusion coefficient, D. In this case the number of theoretical plates is proportional to t 3 , which is in accord with the idea that this diffusional effect is less important a t higher flow velocities. The analytical procedure developed here allows other characteristics of the breakthrough curve to be examined in detail. For the general case, which includes interparticle mass transfer, the skewness, defined as
2316
+
+ + + +
+
+
+
+
+ +
g = interphase mass transfer function, sec-I H = defined by Equation 24, dimensionless k = partition coefficient for sorbate between equal vol) umes ~ of stationary and mobile phases, dimensionless M l ( t ) ,M z ( t ) ,M s ( t ) = defined by Equations 29-31 i&Tk(t) = kth ordinary spatial moment, cmk N = number of theoretical plates, dimensionless s = coordinate of Laplace transform, sec-I t = time, sec U = interparticle concentration of sorbate, moles/cm3 V = interparticle mobile phase velocity, cm/sec W = time constant for mass transfer, sec-I x = axial distance, cm = fractional interparticle void volume, dimensionless y = skewness, dimensionless T = dummy variable for integration
‘Received for review April 6, 1973. Accepted July 2 , 1973. This work was supported in part by Grant No. ES00002 between Harvard University and the National Institute of Environmental Health Sciences, U S . Department of Health, Education, and Welfare, and from contract N00014-70-A-0362-0001 between MIT and the Advanced Research Projects Agency, the Office of Naval Research.
A N A L Y T I C A L CHEMISTRY, VOL. 45, NO. 14, D E C E M B E R 1973
