The Evaluation of the Parameters in the van Deemter Equation Harvey W. Moody United States Air Force Academy. USAFA, CO 80840 Several analytical texts describe experiments measuring the efficiencv of chromatoera~hic - . columns (1-3). This experiment expands on these principles and allows one to calculate the ~ a r a m e t e rin s the van Deemter equation. The resolution or kfficiency of chromatographic peaks in gas chromatography is determined by hoth column and stationary phase characteristics. One measure of the efficiency of a gas chromatographic column is the height equivalent to a theoretical plate (HETP). A relationship has been developed by van Deemter, Zuiderweg, and Klinkenherg ( 4 3 ) ,commonly called the van Deemter equation:
where u is the velocity of the carrier gas and A, B, and C are constants for a particular column. The eddy diffusion term, A, represents the distance a flowing stream of the vapor moves before its direction is changed by the column packing. Because of the multitude of different routes molecules can travel through the column, different molecules will arrive at the outlet of the column at different times. A is independent of carrier gas velocity, since this pathlength is independent of carrier gas velocity. Eddy diffusion is proportional only to the average particle diameter (d,) of the solid support and a constant (A) related to the geometry of the support particles and how uniformly they are packed. A = Ad,
TiME
Figure 1.
A
typical gas chromatographic peak
(2)
The molecular diffuiiunal term. H , is a function of the dif'fuand the time sion cwfticient of the solute in the gas uhnse (Do! spent in the column. Since B is dkp&dent on the time the solute resides in the column, it must he dependent on carrier gas flow rate. Therefore,
where J is a correction factor for interpnrticle spaces. 'l'hr resistance to mass transfer, C , is a function of physical pnlcesses of crossing the gas-liquid phase boundary and also within the liquid phase to the gas phase, where
and d r = the liquid film thickness Dl = the diffusioncoefficient of solution in the liquid phase K = a constant relating column geometric factors and column capacity The complete van Deemter equation is
H~LIUMFLOW R A ~ E (mllminl Figure 2. Graph of height equivalent to a theoretical plate versus carrier gas velocity.
rium between the gas and liquid phases. The numher of theoretical plates for a particular solute is given by (7)
The value of the H E T P can he obtained by dividing the column length of the chromatographic column ( L ) by the numher of theoretical plates in the column, ( N ) :
where t , is the rentention time of the solute and t b is the width of the chromatographic peak at its base (Fig. 1).Substituting eqn. (7) into eqn. (6)gives L tb2 HETP = lfi
A theoretical plate can he considered as one stage of equilih-
A plot of height equivalent to a theoretical plate versus carrier gas velocity is shown in Figure 2. Where the curve exhibits a minimum value of HETP, an optimum carrier gas
290
Journal of Chemical Education
);(
velocity (u,,t) is ohtained. The theoretical optimum velocity is
Table 1. Multiple Linear Regression Analysis. Helium Flow Rate (mllmin)
Retention Time
Peak Width
HETP
(mm)
(mm)
(mmlplate)
One can determine values of A, B, C, and uOpt for a given column, temperature, and species. The actual u,,t can be obtained from a plot of HETP versus carrier gas flow rateuODtis the point where HETP is a minimum. The constants A, B, and C may he determined from the solution of three simultaneous equations of the form (9):
B HETP3=A+-+CUQ u3
(12)
Where H E T P I , ~and , ~ ul,z,3 are the values for the variables a t three different flow rates.
Table 2. Results of the Evaluation of the Parameters in the van Deemter Equation. Method of Three SimultaneousEouations
Experimental All gas chromatographic studies were carried out using a Carle GC 8700 Basic gas chromatograph equipped with a thermal conductivity detector. A five-foot (1524mm) by &-in.stainlesssteel column packed and a column temoerature of 82°C was used with dinonvl . ~hthalate . in !he analysis. T ~ ~nrtrumrntal P ourput was rprorded un n Ilw~ton recorder nt 1 m V full rralr. The rat rhroIn*rrumenr Omnis~rih~ rnatugraphiu analysis were made with 1.0 rrl sample
