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On a chromatogram, the ideal elution peak would have the same form as the graphical representation of the normal distribution of random errors (Gaussian curve). In keeping with the classic notation, μ corresponds to the retention time of the eluting peak and σ to the standard deviation of the peak (σ² represents the variance). y represents the signal as a function of time x from the detector located at the outlet of the column (Figure 1.4).

This is why ideal elution peak signals of a compound are usually described by the probability density function (Eq. (1.2)).

Equation (1.1) is a mathematical relationship describing a Gaussian function, whatever the x variable. In this expression, σ represents the width unit to describe the peak and μ corresponds to the horizontal axis of the Gaussian curve (in this case, retention time t_R). If we make the peak symmetry axis correspond with the new time origin (μ or t_R = 0), we obtain Eq. (1.2)).

(1.1)

Figure 1.4 Characteristics of an ideal chromatographic peak. Meaning of the three classic parameters and summary of characteristics of a Gaussian curve.

(1.2)

This function is characterized by a symmetrical curve (maximum at x = 0, y = 0.399) possessing two inflection points at x = ±1 (Figure 1.4), whose y‐value is 0.242 (i.e. 60.6% of the maximum value). The width of the curve at the inflection points is equal to 2σ (σ = 1).

In chromatography, δ represents the full width at half‐maximum (FWHM, δ = 2.35σ) and σ² the variance of the peak. The width of the peak ‘at the base’ is labelled ω and corresponds to the base of the triangle formed from the tangents to the inflection point I of the Gaussian curve. It is measured at 13.5% of the peak height. At this position, for a Gaussian curve, ω = 4σ by definition.

Real chromatographic peaks often deviate significantly from the ideal Gaussian form. There are several reasons for this. In particular, the peak’s half‐width at the inflection point is not only due to elution in the column but also to injection and detection, which we summarize with the following expression:

(1.3)

where σ ² _tot, σ ² _inj, σ ² _col, σ ² _det are, respectively, the total variance (as observed experimentally), the variance due to injection (injection time, time for the sample to penetrate into the column), the variance due to the column (elution), and the variance due to detection (dead volume between column outlet and detector, detector response time, etc.).