Purnell equation

Column design involves the application of a number of specific equations (most of which have been previously derived and/or discussed) to determine the column parameters and operating conditions that will provide the analytical specifications necessary to achieve a specific separation. The characteristics of the separation will be defined by the reduced chromatogram of the particular sample of interest. First, it is necessary to calculate the efficiency required to separate the critical pair of the reduced chromatogram of the sample. This requires a knowledge of the capacity ratio of the first eluted peak of the critical pair and their separation ratio. Employing the Purnell equation (chapter 6, equation (16)). 

Now, the column length (L) can be defined as the product of the minimum plate height and the number of theoretical plates required to complete the separation as specified by the Purnell equation. 

Rearranging equation (36), chapter 12, and substituting for (n) from the Purnell equation, an expression for the column radius can be obtained. 

Now, if the column is to effect a particular separation, where the pair of solutes that are eluted closest together have a separation ratio of (a) and the first of the pair, ( solute A), is eluted at a capacity ratio value of (k A), then the value of (n) is given by the Purnell Equation, which was discussed in the chapter on The Applications of the Plate Theory, viz... 

In practical GC, it is important that the number of theoretical plates required (Nreq) in order to provide a desired resolution (Rs) for a given pair of components (defined by a and k) can be calculated. This can be achieved by using the Purnell equation (eqn [1]) ... 

Berezkin and Retunsky [80, 81] proposed a modified equation that approximates the Purnell equation (eq. (3-23)) with good accuracy (the relative error being less than 2% for both capillary and packed columns) ... 

Separation factor according to Purnell, Equation 2.82 surface area ... 

Equation (16) was first developed by Purnell [3] in 1959 and is extremely important. It can be used to calculate the efficiency required to separate a given pair of solutes from the capacity factor of the first eluted peak and their separation ratio. It is particularly important in the theory and practice of column design. In the particular derivation given here, the resolution is referenced to (Ra) the capacity ratio of the first... 

It should be noted that Purnell s equation utilizes the thermodynamic capacity ratio calculated using the thermodynamic dead volume. 

Starting with the same basic equation of Purnell (chapter 6) which is applicable to all forms of chromatography, and allows the number of theoretical plates required to separate the critical pair of solutes to be calculated. 

Equation (14) was first developed by Purnell in 1959 (7) and has proved to be one of the most important equations in column design and one that is the greatest use as an aid in column selection for the... 

Fig. 3. Arrhenius plots for the decomposition of dimethyl mercury. All rate coefficients are at or near the high-pressure limit. If a radical scavenger has been used it is shown in brackets following the authors names. 1, Krech and Price (benzene) 2, Kallend and Purnell (propene) 3, Russell and Bernstein (cyclopentane) 4, Russell and Bernstein 5, Laurie and Long 6, Kominar and Price (toluene) O, Weston and Seltzer (cyclopentane) , point calculated from the steady-state equation of Kallend and Purnell.

The mechanism proposed by Kallend and Purnell explains many features of the dimethyl mercury pyrolysis but two difficulties arise. Their explanations are valid only if addition of NO does, in fact, increase the methyl radical concentration. The process by which this occurs has not been specified and none comes readily to mind. In fact, the equilibrium CH3+NO CH3NO might reasonably be expected to lower the methyl radical concentration. The second difficulty arises when high pressure limiting values of calculated from Kallend and Purnell s steady-state equation... 

Equation (3), however, was developed for a gas chromatographic column and in the case of a liquid chromatographic column, the resistance to mass transfer in the mobile phase should be taken into account. Van Deemter et al did not derive an expression for fi(k ) for the mobile phase and it was left to Purnell (3) to suggest that the function of (k ), employed by Golay (4) for the resistance to mass transfer in the mobile phase in his rate equation for capillary columns, would also be appropriate for a packed column in LC. The form of f (k ) derived by Golay was as follows,... 

EFFECTIVE NUMBER OF PLATES. Desty et al. (31) introduced the term effective number of plates, N, to characterize open tubular columns. In this relationship adjusted retention volume, VR, in lieu of total retention volume Vp, is used to determine the number of plates. This equation is identical to Purnell s separation factor discussed below. 

Martire and Riedl and coworkers (8,9) developed a method of determination of complex constants which is less time consuming than the Cadogen-Purnell method, although it makes additional assumptions. They have demonstrated that the specific retention volume of A is related to the association constant by the following equation ... 

What are the best ways to optimize separations One way to answer this question is to consider a popular form of the resolution equation that is originally attributed to Purnell.1... 

The gas chromatographic method is based on the relation between the differential enthalpy of adsorption at zero coverage and the temperature dependence of the Henry s law constant, kn, as expressed in the form of Equation (4.3). In the low-pressure region, where Henry s law applies, the specific retention volume, Vj, is a linear function of kH (Purnell, 1962 Littlewood, 1970). This relationship makes it possible to make use of elution chromatography since... 

Purnell [7] suggested to center attention on the second peak of the pair, thus using the peak width of the second component as a base width (meaning that the width of the first peak is equal to the width of the second peak). This assumption leads to the following equation ... 

Since typical biological mixtures are exceedingly complex, adequate chromatographic resolution is imperative for both identification and quantitation purposes. Improved resolution is feasible through either increasing the column efficiency (number of theoretical plates), or phase selectivity. Alternatively, a combination of both can be practiced. The number of theoretical plates, required for adequate resolution of two adjacent peaks (98% separation of the peak areas) is related to the column selectivity (relative retention, a) and to the capacity ratio, k, according to the well-known equation derived by Purnell [70] ... 

Resolution between peaks is dependent on the retention characteristics of each component (k), the ability of the stationary phase to selectively retain the components (a) and the overall efficiency of the column (N). The equation derived below was first developed by Purnell in 1959 to accurately relate the degree of separation attainable in a GC column in terms of the separating capability and efficiency [13-15],... 

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