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The Optimum Length of an Open Tubular Column
The column length is given by equation (2), viz.,
Substituting for (n) and (Hmjn) from equations (I) and (8) respectively,
Equation (12) shows that the optimum column length is inversely proportion to the third power of (a-l). This is an analogous relationship to the length of a packed column. In a similar manner the column length is inversely proportional to the square root of the pressure drop across the column. Using equation (12) the optimum column length was calculated that was necessary to separate a series of mixtures having (a) values ranging from
1.01 to 1.12 for inlet pressures of 1,10,100,1000 p.s.i. respectively. The results are shown as graphs relating column length (I) to separation ratio
(a) in figure 2.
It is seen from figure 2 that, operating at an inlet pressure of only I p.s.i, the optimum column length varies from about 15 meters for the separation of solute pairs having a separation ratio of 1.01 to only 2cm for solutes having a separation ratio of 1.12 At the other extreme of pressure, 1000 p.s.i., the column length varies from about 65 cm for a separation ratio of
1.01 to only 0.3 of a millimeter for a separation ratio of 1.12 it is obvious that the optimum conditions for open tubular columns are not practical for the separation of simple mixtures at high pressures as both
the column radius would be too small for the extra column dispersion of the equipment and the column would be far too short to construct. At low pressures, and for difficult separations, open tubular columns might be practical from the point of view of column diameter and column length.
Graph of Log. Column Length against Separation Ratio
E) 1 p.s.i.
¦ 10 p.s.i.
¦ 100 p.s.i.
î 1000 p.s.i.
1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14
Minimum Analysis Time
The analysis time is given by the following equation,
tmin = <1 + ê'2>^ uopt
Substituting for (l0pt) from equation (2), for (uopt) from equation (7) and for (Hmin) from equation (8),
t_,n = (l + k'oJn v mm ã ||
tmin - (1 +k'2) Ï 2C
Substituting Ãîã (C) from equation (6),
tmin - d+k 2 ^n
and for (n) and Ñr0pt) from equations (1) and(11), respectively,
4(l + k‘)'|2| + 6k' + l Ik'2 256(l + k')2Dmn
Equation (13) gives the minimum analysis time that can be obtained from an open tubular column, when separating a mixture of defined difficulty, under given chromatographic conditions. It is seen that, in a similar manner to the packed column, the analysis time is inversely proportional to the fourth power of the function (a-l) and inversely proportional to the inlet pressure. The contribution of the function of (k'), to the analysis time is not clear and can be best seen by calculation. It is also seen (perhaps a little surprisingly) that the analysis time is compiete’y independent of the diffusivity of the solute in the mobile phase but is directly proportional to the viscosity of the mobile phase
Using equation (13) the minimum analysis time was calculated for the separation of a series of mixtures having (a) values ranging from 1.01 to 1.12 and for inlet pressures of 1,10,100,1000 ps.i. respectively. The results are shown as graphs relating analysis time (1) to separation ratio (a) in figure 3.
Figure 3 shows that, there might indeed, be a limited practical range of column dimensions and operating conditions mat would make tne open tubular column a possible alternative to the packed column in LC. To separate a solute mixture with the separation ratio for the critical pair of
1.01 and an inlet pressure of I p.s.i would require an analysis time of 6 5
days which even for this difficult separation is tediously long. However, the analysis times would be commensurate to those obtained from an optimized packed column, separating a mixture of the same difficulty and would have a similar length. Furthermore, it would be far easier to fabricate and coat a 15 meter capillary column than it would to efficiently prepare a packed column of comparable length.
Graph of Log. Analysis Time against Separtion Ratio
At 10 p.s.i the diameter of the column is down to 26 micron which is a little small for easy practical use, but the analysis time would be reduced to about 15 hours. However, it is now necessary to calculate the mobile phase flow rate and, in particular, the maximum extra column dispersion that can be tolerated to determine the ultimate practicality of open tubular columns in LC.
The Optimum Flow-Rate
The optimum flow-rate (Gopt) will be given by..
Substituting for (Â) and (Ñ),
Qopt ~ 4nropt
1+ 6k'+ ilk"
Finally, substituting for (ropO, from equation (11)
It is seen that (00pt), varies inversely as (a-1) and the square root of the inlet pressure but directly as the diffusivity of the solute to the cower of 3/2 Curves relating the optimum flow-rate to the separation ratio of the critical pair are shown in figure 4 for inlet pressures of 1,10,100,and 1000 p.s.i