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The Optimum Column Length
Substituting the optimum value for the particle diameter from equation (18) in equation (13) the optimum column length can be obtained,
I (opt) —
f ( / î \ \a5l ( 9 .0.5 T
4(1+k')^2 2ß.+ ( 1+6k‘+l Ik*2) 8(l+k) 2A. 3y (l+k)
k(a-l)j Y 3(l+k')2 / k'(a-l) ôÐ (l+6k + l Ik'2) +Y
( I 2
(l+6k+! Ik j
' . î V
Çó()+ê'Ã (i+6k'+i Ik'2)
Equation (19) shows that the optimum column length is inversely proportional to the third power of the function (a-l) and thus, will increase very rapidly with the difficulty of the separation It is also seen that the length is inversely proportional to the square root of the available inlet pressure and, consequently, has a similar sensitivity to pressure as the optimum column radius.
A graph relating optimum column length to the separation ratio of the critical pair for inlet pressures of 2000, 4000, and 6000 p.s.i. is shown in figure (2).
Graph of Log. Optimum Column Length against Separation Ratio
d 2000 p.s.i ¦ 4000 p.s.i. î 6000 p.s.i.
It is seen from figure (2) that the length of the optimum column ranges from nearly 30 meters for the separation of solute pairs having a separation ratio of 1.01 at an inlet pressure of 2000 p.s.i., to less than I cm for a solute pair having a separation ratio of 1 12 at an inlet pressure of 6000 p.s.i. It is also seen that rlesnite the lonarithrnir scale for column lennth channinn the inlet pressure from 2000 p.s.i. to 6000 p.s.i. does not have a profound effect on the optimum column length. This of course is because the optimum length is inversely proportional to the square root of the inlet pressure.
The Minimum Analysis Time
The minimum analysis time is that achieved by employing the column of optimum length, packed with particles of optimum diameter and operated at the optimum velocity. Thus, the minimum analysis time, (t(min)), is given
tmin = (' + K2)^ z' U0pt
where k'2 is the capacity ratio of the last eluted peak
Substituting for (uopt) from equation (9) and the column length (1) from equation (13) then analysis time is seen to be given by,
t - (l+k2)-
4(1+k')\2 2ß.+ (l+6k'+l lk'2) ¦ 051
dp (l+6k'+l lk'2)
|[4(l+k')] 2 21 + (l+6k'+l 1 k'2) 0.51 1+6 ê'+1 lk'2 0.5 9 d2 p
4k'(a-l)J V î 3(1+ ê) 3 Y(l + ê')2 4 Dm
Substituting for dp(0pt) from equation (18),
(l + 6k'
3Y(l + kf
f 8(1 + k'))
3Y(l + kf
Simplifying and rearranging,
tmin — (' +^2
y(l + 6k +1 lk'2j
3(l + k')"
It is interesting to note from equation (21) that the analysis time does not depend on the magnitude of the diffusivity of the solute in the mobile phase but only on the viscosity of the mobile phase. It does, however, increase
Graph of Log. Minimum Analysis Time against Separation Ratio
î 2000 p si.
¦ 4000 p 5 i
* 6000 p si.
dramatically as the inverse of the fourth power of the expression (a-1) The analysis time is now inversely related to the inlet pressure and not the square root of the inlet pressure and so pressure has a more significant effect on the analysis time than on the column length. Employing equation (21) curves were constructed relating analysis time to the separation ratio
r\f tKo ë-r-i 11 rv'iir f r\r- ir\1 f rrccci »r-csc r\f ÎÏÏË n ñ i
VI fJUII I 11 I ! «. «I VI i-vw -
p.s.i. and are shown in figure (3).
/qoO p s I 3nd 60CC
It is seen from figure (3) that there is a very wide range of analysis times extending from just over ten days, which was required to analyze a mixture
where the critical pair had a separation ratio of 1.01 and with an inlet pressure of 2000 p.s.i to about 15 seconds where the separation ratio was
1 12 and the inlet pressure 6000 p.s.i.. It is also seen that, again bearing in mind the logarithmic scale for the analysis time in figure (3), the effect of pressure on analysis time is not as great as might be expected. Increasing the inlet pressure from 4000 psi. to 6000 p.s.i only reduces the analysis time by about one third.
The Optimum Column Radius
The optimum column radius has been discussed in chapter (1) but the expression obtained was for any column and not specifically an optimized column. Consequently a slightly cifferent derivation will be employed here.
Starting with equation (9) from chapter (10) The maximum value the extra column dispersion, (oe), that can take place is given by,
where (oe) >s the total extra column dispersion, and (oc) is the column dispersion.
The limitation of (oe) to (0.32oc ), allows the variance of the peak eluted from the column to be increased by a maximum of 10% as a result of the extra column dispersion and the width of the peak by a maximum of 5%.