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

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Inserting the full expression for f(v) in equation (28), v

0V = 9e-»v|e»v î

/ 1 < ¦ -< < . \

c > > 1 <L> — |eCa fe Ca c > > 1 <L> dv

x0-- n!

0 n I \ iCa j j 0 >

dv

(29)

Equation (29) is the explicit equation for the temperature of a detector and can be used to synthesis the different shaped curves that the detector can produce. Employing a computer in the manner of Smuts et al (21) Scott (22) calculated the relative values of (0) for (v= 74 to 160) for a column of 100

theoretical plates, and for (Ca) ranging from 0.25 to 4 and (ô) ranging from 0.01 to 1.25. The curves obtained are shown in figure (8).

Figure 8

Theoretical Temperature Curves and the Integral Temperature Curves Obtained from the Heat of Absorption Detector

A Integra) Curves

The twenty curves shown are graphs of (0) versus (v) and the integral of (e) versus (v) for different values of (Ca) and (ô) and are all normalized to the same peak height. The curves cover the practical range of heat loss factors that might be expected from an heat of adsorption detector cell. They demonstrate the effect on the shape of the (0) versus (v) curves of changes In 'detector cell c’apacity'/'column plate capacity' ratios that'would result from different cell designs but when detecting a peak of constant width. The curves for different values of (C0) would also represent the effect on peak shape of solute of different retention, and consequently, having different peak widths passing through a cell of fixed dimensions. It is seen that the major effect on peak shape is the 'detector cell capacityV'column plate capacity' ratio, (Ca). when tne capacity of the detector cell is less than the plate capacity of the column, (Ca<l), the negative part of the signal dominates and when the detector cell capacity exceed the column plate capacity, (Ca>l), the positive part of the signal

dominates. For this reason when (Ca>1), the integral curve rises to a maximum but does not return to the baseline. Conversely., when (Ca<D, the integral curve first rises and and then falls below the baseline and does not return. Only when (Ca= 1) does the detector signal simulate the differential form of the elution curve and consequently the integral curves describes a true Gaussian peak.

The effect of the heat loss factor (ô), on peak shape is small but the magnitude of the signal varies inversely as (ô) although, this is not apparent in figure (8), as all curves are normalized. It should also be noted that for low values of (p), where the maximum sensitivity is realized the peak maximum is displaced. However, for large values of (p), the maximum of the integral curves for (Ca=1), is almost coincident with the maximum of the elution curve.

It follows, that if the detector was to be effective and produce the true 6aussian form of the eluted peak then, (Ca), must at all times be unity and consequently the detector must have the same plate capacity as that of the column. This means that the detector must employ the same absorbant, have the same geometry and be packed to give the same plate height as the column. It is obvious to accomplish this, the column must also be the detecting cell and the temperature sensing element must be placed in the column packing itself.

Restating the expression for f(v),

f(v) = X0

<T> 1 < < 3

n! cj

-v

,Ñß

Lc.fc^'

I n!

dv

Now, if the sensor is in the column packing, Ca= 1 Consequently,

o-Vv/Ï Ã ,.fe-vvn

dv

e-vvn f(v) = X0-—~—X0e"

r\l

j.-È

' V J

86

Thus,

f(v)= X0

e vvn v e~v “7T"X° n!

,n+1

n+1

= X,

e-vvn

n!

= Xn _X(n+i)

0—v y(n+l)

’ (n+1)!

Recalling the basic differential equation for the elution curve (9) given on page 18 in chapter (2)

dX(n+l)

f(v) -

dv

(30)

Now, as there is no (n+1) plate that constitutes the detector, the sensing element can be considered to be placed in the (n)th plate.

Thus,

f(v) = ^

dv

(31)

If the (n)th plate of the column acts as the detecting cell, there can be no heat exchanger between the (n-l)th plate and the (n)th plate of the column. Thus, there will be a further convective term in the differential equation that will take into account the heat brought into the (n)th plate from the (n-1 )th plate by the flow of mobile phase (dv).

Thus the heat convected from the (n-1 )th plate to plate (n) by dv will be

dm5m6(n-1 )Cadv ....................................................... (32)

Substituting for f(v) from equation (31) ion equation (26) and inserting the extra convection term from (32)

e(n-1)

den_ dXp_ Ppca + C CD dmSmca

dv 1HP) dv lHP HP

where the subscript (p) accounts for the change from the already defined physical characteristics of the detecting cell to those of the last plate of the column.

á?

Thus,

den dxn ïë - = «—-âåï+óåÛ)

(33)

where.

a_^p Ppca _ dmSmce

Hr

Hr

Hr

A solution to the differential equation (33) is given by:-

r-n

0n =

=^2

r=0

(y-1)

(B-l)

d^(n-r)

dv

(34)

The validity of this solution can be confirmed by differentiation as follows:-

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