**2. Basic theory and properties of derivative spectrophotometry**

Derivative spectrophotometry is a technique which is based on derivative spectra of a basic, zero-order spectrum. The results of derivatisation of function described a run of absorbance curve is called the derivative spectrum and can be expressed as:

$$\mathbf{^nD\_{x,\lambda} = d^n A / d\lambda^n = f(\lambda) \quad \text{or} \quad \mathbf{^nD\_{x,v} = d^n A / d\mathbf{v^n} = f(v)}$$

where: n – derivative order, nDx,λ or nDx,ν represents value of n-order derivative of an analyte (x) at analytical wavelength (λ ) or at wavelength number (ν), A- absorbance.

Derivative spectrophotometry keeps all features of classical spectrophotometry: Lambert-Beer law and law of additivity.

Lambert-Beer low in its differential form is expressed as:

$$\prescript{}{}{}{}{D} = \frac{d\prescript{}{}{}{}{A}}{}{} = \frac{d\prescript{}{}{}{\mathcal{E}}}{}{}{}{\mathcal{E}} \prescript{}{}{\cdot} \cdot \prescript{}{\cdot} \cdot \prescript{}{\cdot}$$

Where ε-molar absorption coefficient (cm-1mol-1l) , c – concentration of analyte (mol l-1), lthickness of solution layer (cm).

Derivative spectrum of **n**-component mixture is a sum of derivative spectra of individual components:

$$\mathbf{^nD\_{mi\dot{\alpha}}=nD\_1+nD\_2+...+nD\_n}$$

A new feature of derivative spectrophotometry is a dependence of derivatisation results on geometrical characteristic of starting, zero-order spectrum. A shape and an intensity of the resulted derivative spectrum depend on half- heights width of peak in basic spectrum:

$$\mathbf{^nD} \mathbf{=} \mathbf{P}^n \mathbf{A}\_{\text{max}} \mathbf{L} \mathbf{^1}$$

where Pn- polynomial described run of n-derivative curve, n- derivative order, L- width of half- heights of peak of zero-order spectrum.

Due to this property broad zero-order spectra are quenched with generation of higher orders of derivatives while narrow undergo amplification. If the zero-order spectrum possess two bands A and B which differ from their half- heights width (LB>LA), after a generation of n-order derivative a ratio of derivatives intensity can be expressed as:

$$\mathrm{"{n}D\_{A}/{n}D\_{B} = (L\_{B}/L\_{A})^{n}}$$

This dependence leads to increase in selectivity and/or sensitivity of assay. It allows to use for analytical properties a narrow band, overlapped or completely hooded by a broad ones.

The shape of derivative spectrum is more complicated than its parent one (Fig. 1). New maxima and minima appeared as results of derivatisation. The generation of **n-th** order derivative spectrum produces (n+1) new signals: an intense main signal and weaker bands, so called satellite or wings signals. Position of maxima or minima depend on order of derivative. The main extreme of derivative spectra of even order is situated at the same wavelength as maximum in zero-order spectrum, but for 2, 6 and 10-th order it becomes minimum in

Derivative spectrophotometry is a technique which is based on derivative spectra of a basic, zero-order spectrum. The results of derivatisation of function described a run of absorbance

nDx,λ=dnA/dλn=f(λ) or nDx,ν=dnA/dνn=f(ν) where: n – derivative order, nDx,λ or nDx,ν represents value of n-order derivative of an

Derivative spectrophotometry keeps all features of classical spectrophotometry: Lambert-

*n n dA d <sup>D</sup> c l d d*

Where ε-molar absorption coefficient (cm-1mol-1l) , c – concentration of analyte (mol l-1), l-

Derivative spectrum of **n**-component mixture is a sum of derivative spectra of individual

nDmix=nD1+nD2+...+nDn A new feature of derivative spectrophotometry is a dependence of derivatisation results on geometrical characteristic of starting, zero-order spectrum. A shape and an intensity of the resulted derivative spectrum depend on half- heights width of peak in basic spectrum:

nD=PnAmaxL-1 where Pn- polynomial described run of n-derivative curve, n- derivative order, L- width of

Due to this property broad zero-order spectra are quenched with generation of higher orders of derivatives while narrow undergo amplification. If the zero-order spectrum possess two bands A and B which differ from their half- heights width (LB>LA), after a

nDA/nDB=(LB/LA)n This dependence leads to increase in selectivity and/or sensitivity of assay. It allows to use for analytical properties a narrow band, overlapped or completely hooded by a broad ones. The shape of derivative spectrum is more complicated than its parent one (Fig. 1). New maxima and minima appeared as results of derivatisation. The generation of **n-th** order derivative spectrum produces (n+1) new signals: an intense main signal and weaker bands, so called satellite or wings signals. Position of maxima or minima depend on order of derivative. The main extreme of derivative spectra of even order is situated at the same wavelength as maximum in zero-order spectrum, but for 2, 6 and 10-th order it becomes minimum in

generation of n-order derivative a ratio of derivatives intensity can be expressed as:

 λ= = ⋅⋅

ε

analyte (x) at analytical wavelength (λ ) or at wavelength number (ν), A- absorbance.

*n n <sup>n</sup>*

λ

**2. Basic theory and properties of derivative spectrophotometry** 

curve is called the derivative spectrum and can be expressed as:

Lambert-Beer low in its differential form is expressed as:

Beer law and law of additivity.

thickness of solution layer (cm).

half- heights of peak of zero-order spectrum.

components:

derivative spectrum and for 4, 8 and 12-th order it remains as a maximum (Fig. 1). The point of initial maximum converts into the point of inflection in derivative spectra of odd order. A narrowing of new signals is observed during generation of consecutive derivative spectra. This feature leads to narrowing bands and as a consequence to separation of overlapped peaks.
