# liquid chromatography column - Scott R.P.W.

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Consequently, since (n) will always be greater than 100,

nloge

\ 1 111 1

'¦ft = n ^ 2n 3^ 4n2

Ã 1 1 _L

* n - 2~ ç^ï "" 4n

(7)

Now, substituting for, n loge

\

, from equation (7) in equation (6)

Now ,

ç/ï

Consequently, the term

and above, can be Ignored with respect to \-

Then,

'°9eS = - j-

and,

: = e2= 0.6065 h

(8)

It is seen from equation (8) that the points of inflection occur at 0.6065 of the peak height and the peak width, at that height, is equivalent to two standard deviations (2c) of the Gaussian curve.

Many peaks that are measured will be only a few millimeters wide and, as the calculation of the column efficiency requires the width to be squared, the distance (x) must be determined very accurately. The width should be

50

measured with a comparitor reading to an accuracy of t 01 mrn The measurements are taken from the inside of one line to the outside of the

Figure 6 Measurement of Peak Width

adjacent line, in order to eliminate errors resulting from the finite width of the ink line drawn on the chart This procedure is shown in figure (6) The measurement should be repeated using the alternate edges of the line and an average taken of the two readings to avoid errors arising from any variation in line thickness. At least three replicate runs should be made and the three replicate values of efficiency should not differ by more than 3% if the data acquisition system has software to measure efficiency then this can be used providing its accuracy is carefully checked manually. Noise on tne detector can often introduce inaccuracies that are less likely to occur with manual measurement.

The Gaussian Form of the Elution Equation

In order to change the Poisson form of the elution equation into the Gaussian form it is necessary to effect a change of origin. Consider the elution curve shown in figure (7) The origin for the Poisson form of the equation is at the point of injection whereas, the origin for the Gaussian equation will be at the peak maximum, (n) plate volumes from the Injection point. Thus, a point A, (v) plate volumes from the injection point, will be (v-n) = w plate volumes from the peak maximum The necessary change in origin is demonstrated in Figure (7).

Now the Poisson form of the elution equation is given by,

Figure 7

The Difference in Axes Between The Poisson Function and the

Gaussian function

Substituting (w+n) for (v)

Xne +n\w + n)n

n!

Now, from Stirlings Theorem, n! = e nnn^2nn.

Therefore,

Xm(n) ~

X e (w + n\w + n)n

_ ëî'

e'nnnV2jtn

It follows,

Fvnandinn as the series

fen

l°9e Xm(n) = loge

- J^e-W l + -

X0 ... _____I. w

- W + nlOQp I + —

&é\ I n;

Xn

2 3

w w w°

52

. X0 w2 w3

=þäå -r== - w + w - — + —

V2nn 2n 3nZ

Now (n) is large and the whole elution curve is practically contained between w = -2^n and w = 2^n (i.e. contained within four standard

deviations of the Gaussian Curve) thus — will always be very much

2n

W3 u/®

greater than —ã and thus, —ã and all higher terms can be Ignored with 3nz 3n2

w2

respect to —.

2n

w2

X e

Therefore, logeXm(n) = loge-?=

y2 nn

x e2n

Xfp(n) I .................................................... (9)

V2nn

Equation (9) is the well known Gaussian form of the elution curve equation and can be used as an alternative to the Poisson form in all applications of the Plate Theory.

References

1/ " Contemporary Liquid Chromatography", (R. P. W. Scott), John Wiley and Sons, New York, (1976)31

2/ " Gas Chromatography-Second Edition", (AI.M.Keulemans), Reinhold Publishing Corporation, Amsterdam, (1959)122

3/ ” Contemporary Liquid Chromatography", (R. P. W. Scott), John Wiley and Sons, New York, (1976)35

Chapter 5

Applications of the Plate Theory

The elution curve equation, derived from the Plate Theory, can be used in a number of ways to describe the chromatographic performance of an LC column operated under a variety of conditions (1,2). It can also be used to explain other physical chemical phenomena that occur in a column, such as the change in temperature of the contents of a theoretical plate during the passage of solute through it (3) and to examine pressure changes that can occur in a theoretical plate in GC (4) . In this chapter the elution curve equation, or its derivatives, will be applied to a number of practical situations in LC to, either expand our knowledge of the physical chemical processes involved, or to utilize the process to an analytical advantage.

The Maximum Sample Volume

Any sample placed on to an LC column will have a finite volume, and the variance of the injected sample will contribute directly to the final peak variance that results from the dispersion processes that take place in the column, it follows that the maximum volume of sample that can be placed on the column must be limited, or the column efficiency will be seriously reduced. Consider a volume Vi, injected onto a column, which will form a rectangular distribution on the front of the column. The variance of the peak eluted from the column will be the sum of the variances of the injected sample plus the normal variance of the eluted peak. The principal of the Summation of Variances will be discussed more extensively in a later chapter, at this time it can be stated that,

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