Standard addition method applied to solid-state stripping voltammetry: determination of zirconium in minerals and ceramic materials

Abstract

An application of the standard addition method to stripping voltammetry of solid materials immobilized in inert electrodes is described. The method allows the determination of the mass fraction of a depositable metal M in a material on addition of known amounts of a standard material containing M to a mixture of that material and a reference compound of a second depositable metal, R. After a reductive deposition step, voltammograms recorded for those modified electrodes immersed in a suitable electrolyte produce stripping peaks for the oxidation of the deposits of M and R. If no intermetallic effects appear the quotients between the peak areas and the peak currents for the stripping oxidation of M and R vary linearly with the mass ratio of the added standard and the reference compound, thus providing an electrochemical method for determining the amount of M in the sample. The method has been applied to the determination of Zr in minerals, ceramic frits, and pigments, using ZnO as reference material and ZrO₂ as the standard.

Appendix

Analysis of errors

To evaluate the relative uncertainty associated with the mass fraction of M in the sample, ɛ_MA, we shall use the conventional theory of error propagation. Because the uncertainty of r_MS, r_RX, M_M, and M_R is negligible, ɛ_MA depends on the uncertainties of m_MA, m_MS, m_RX, and those associated with peak current or peak area measurements.

Taking the first, the relative uncertainty in peak current (or peak area) measurements can be expressed as:

$$ \varepsilon _q = \Delta q\left( {\frac{1} {{q_{{\text{Zr}}} }} + \frac{1} {{q_{{\text{Zn}}} }}} \right) $$

(13)

where it is assumed that an identical uncertainty Δq(= Δq(M)=Δq(R)) is obtained for the peak areas or peak currents of both stripping processes. Assuming that an identical total charge q(=q(M)+q(R)) passes in the electrochemical experiments, one can obtain:

$$ q_{\text{M}} = \frac{{H\left( {\Gamma _{\text{M}} /\Gamma _{\text{R}} } \right)}} {{1 + H\left( {\Gamma _{\text{M}} + \Gamma _{\text{R}} } \right)}} $$

(14)

$$ q_{\text{R}} = \frac{q} {{1 + H\left( {\Gamma _{\text{M}} + \Gamma _{\text{R}} } \right)}} $$

(15)

where H (=H_R/H_M) represents the coefficient of response associated with the electrochemical measurements. Combining these equations, the relative uncertainty associated to the quantity q(M)/q(R) (or Δi_p(M)/Δi_p(R)) is:

$$ \varepsilon _{\text{q}} = \frac{{\Delta \left( {q_{\text{M}} /q_{\text{R}} } \right)}} {{q_{\text{M}} /q_{\text{R}} }} = \frac{{\Delta q\left[ {1 + H\left( {\Gamma _{{\text{Zr}}} /\Gamma _{{\text{Zn}}} } \right)} \right]^2 }} {{qH\left( {\Gamma _{{\text{Ze}}} /\Gamma _{{\text{Zn}}} } \right)}} $$

(16)

The values of ɛ _q depend on the values of H and the quotient Γ_M/Γ_R in an identical way. The position of the minimum of uncertainty depends on the value of these parameters. Because H (=H_R/H_M) increases on decreasing the Γ_M/Γ_R ratio, a compromise between both parameters must be found. Thus, for Γ_M/Γ_R=1, the minimum of ɛ _q is obtained for H=1.

For the second, one can take a typical uncertainty Δm (=Δm_MA=Δm_MB=Δm_RX) equal to 0.01 mg. Then, the relative uncertainties of the quotients m_MA/m_RX and m_MS/m_RX are:

$$ \begin{aligned} \frac{{\Delta \left( {m_{{\text{MA}}} /m_{{\text{RX}}} } \right)}} {{m_{{\text{MA}}} /m_{{\text{RX}}} }} & = \Delta m\left( {\frac{1} {{m_{{\text{MA}}} }} - \frac{1} {{m_{{\text{RX}}} }}} \right) \\ \frac{{\Delta \left( {m_{{\text{MS}}} /m_{{\text{RX}}} } \right)}} {{m_{{\text{MS}}} /m_{{\text{RX}}} }} & = \Delta m\left( {\frac{1} {{m_{{\text{MS}}} }} - \frac{1} {{m_{{\text{RX}}} }}} \right) \\ \end{aligned} $$

For fixed values of Δm and the sum of the masses, numerical calculations indicate that the value of such terms is minimal for m_MA/m_RX and m_MS/m_RX equal to unity.