# Retention uniformity

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Retention uniformity

Retention uniformity, or "RU", is a concept in thin layer chromatography, designed for quantitative measurement of "equal-spreading" of the spots on the chromatographic plate and one of the Chromatographic response functions. It is calculated from the following formula:

$R_\left\{U\right\} = 1 - sqrt\left\{frac\left\{6\left(n+1\right)\right\}\left\{n\left(2n+1\right)\right\}sum_\left\{i=1\right\}^\left\{n\right\}\left\{left\left(R_\left\{Fi\right\}-frac\left\{i\right\}\left\{n+1\right\} ight\right)^2$

where "n" is the number of compounds separated, "Rf (1...n)" are the Retention factor of the compounds sorted in non-descending order.

Theoretical considerations

The coefficient lies always in range <0,1> and 0 indicates worst case of separation (all Rf values equal to 0 or 1), value 1 indicates ideal equal-spreading of the spots, for example (0.25,0.5,0.75) for three solutes, or (0.2,0.4,0.6,0.8) for four solutes.

This coefficient was proposed as an alternative to earlier approaches, such as "D" (separation response), "Ip" (performance index) or "Sm" (informational entropy). Besides its stable range, the advantage is a stable distribution as a random variable, regardless of compounds investigated.

In contrast to the similar concept called Retention distance, "Ru" is insensitive to "Rf" values close to 0 or 1, or close to themselves. If two values are not separated, it still indicates some "uniformity" of chromatographic system. For example the "Rf" values (0,0.2,0.2,0.3) (two compounds not separated at 0.2 and one at the start ) result in "RU Calculation

The calculation of the "RU" requires some operations and is quite difficult to perform in spreadsheets. The following implementations may help. They take the vector of "Rf" values, returning the single "RU" value.

The R (programming language)/S-PLUS implementation:

ru <- function (x) { x <- sort(x) n <- length(x) i <- (1:n)/(n+1) s <- sum((x-i)^2) ru <- 1-sqrt( (6*(n+1)) / (n*(2*n+1)) * s ); return(ru);

}

The GNU Octave/Matlab implementation:

function res = ru(x) x = sort(x); n = length(x); i = (1:n)./(n+1); s = sum((x-i).^2); res = 1-sqrt((6.*(n+1))./(n.*(2.*n+1)).*s);endfunction

The Scilab implementation:

function res = ru(x) x = gsort(x,"g","i"); n = length(x); i = (1:n)./(n+1); s = sum((x-i).^2); res = 1-sqrt((6.*(n+1))./(n.*(2.*n+1)).*s);endfunction

* Chromatographic response function

References

* Komsta Ł., Markowski W., Misztal G., A proposal for new RF equal-spread criteria with stable distribution parameters as a random variable. J. Planar Chromatogr. 2007 (20) 27-37.

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