Catalysts are chemical compounds, which decrease the activation energy of a reaction. This way they decrease the time necessary to approach equilibrium. Proteins acting as catalysts are called enzymes (or ferments). They are biocatalysts, to which also belong ribozymes and catalytic RNA molecules. All other catalytically active molecules, ions, atoms, metals or solids are simply called catalysts. Enzymatic analysis has a twofold meaning: it is either (i) the determination of enzyme activity (concentration) based on the conversion of a substrate or (ii) the determination of the concentration of a substrate that is converted by an enzyme. In this lecture text, it is not necessary to repeat the basics of enzymatic analysis, as this is easily available in modern textbooks (e.g. [1]) and it is a classic part of teaching chemistry and biochemistry.

Students acquainted with enzymatic analysis may wonder, whether there are methods using catalysts not belonging to enzymes for analytical purposes. The simple answer is, yes there are plenty of such methods. However, these methods were mostly pushed aside by modern instrumental methods, like Atomic Absorption and Emission Spectroscopy, Isotope Dilution Analysis, etc. At present, papers describing catalytic methods of analysis relating to inorganic catalysts are only published from time to time. Hence it could be argued that these catalytic methods are outdated and it is not anymore justified to teach this subject. The author of the present lecture text is convinced that basic knowledge of catalytic methods of analysis for inorganic elements should still be mentioned in lectures at University level, since students can never know, what problems they will have to solve during their career, and there is a good chance that simple catalytic methods may provide an optimal solution for their problem. In the last decades, the absence of catalytic methods in the university curricula, may well have contributed to their little use in applications. However, the simplicity of catalytic methods of analysis, together with their inexpensive instrumentation and their possible application for on-site analysis (esp. for environmental analysis) are clear advantages, which should be exploited much more often.

The principles of catalymetry

Catalymetry means the determination of concentration of inorganic catalysts. Another term is catalytic methods of analysis. Like enzymatic methods of analysis they are variants of kinetic methods of analysis, as they are all based on measuring reaction rates in order to quantify compounds or elements.

Basic reaction kinetics

Imagine a reaction \(} \to }\), which can proceed in the presence of a catalyst C according the following scheme:

$$}\mathop \rightleftarrows \limits_}} }}^}} }} }$$

(I)

$$}\xrightarrow}} }} }$$

(II)

where S is the substrate, C the catalyst, P the product, \(k_}}\) the rate constant of forward reaction forming the intermediate catalyst-substrate complex SC, \(k_}}\) is the rate constant of the backward reaction, and \(k_}}\) is the rate constant of decomposition of the intermediate complex SC. This intermediate complex forms with a rate

$$\frac}c_}}} }}}t}} = k_}} c_}} c_}} - \left( }} + k_}} } \right)c_}}}$$

(1)

Here t denotes time. In the case of a stationary concentration of SC (\(\frac}c_}}} }}}t}} \approx 0\)) follows:

$$\left( }} + k_}} } \right)c_}}} = k_}} c_}} c_}}$$

(2)

When the reaction rate of the catalysed reaction is taken as \(\frac}c_}} }}}t}}\) follows:

$$\frac}c_}} }}}t}} = k_}} c_}}} = \frac}} k_}} c_}} c_}} }}}} + k_}} }} = k_}}} c_}} c_}}$$

(3)

Here, \(k_}}}\) is called the rate constant of the catalytic reaction. Hence, this rate constant is:

$$k_}}} = \frac}} k_}} }}}} + k_}} }}$$

(4)

In case of catalytic reaction the rate constant \(k_}}}\) is larger than the rate constant for the non-catalytic reaction \(} \to }\). When the catalysed reaction \(} \to }\) is used to determine the concentration of the catalyst, this reaction is called indicator reaction. To determine \(c_}}\) it is necessary to evaluate the rate of the catalysed reaction.

Evaluation strategies

When the indicator reaction can be written as

$$}\xrightarrow}} }$$

(III)

where A and B are the reactants and X and Y are the products and C is the catalyst, the reaction rate \(v\) is:

$$v = \frac}c_}} }}}t}} = (k + k_}}} c_}} )c_}} c_}}$$

(5)

Here, k is the rate constant of the reaction in the absence of the catalyst.

It is possible to follow the course of reaction III, by measuring the changes of the concentration of A, B, X and Y in time. Defining the variable \(\xi\) (extent of reaction, symbol: Greek letter xi) as follows:

$$\xi = c_}t = 0}} - c_}t = t_ }} = c_}t = 0}} - c_}t = t_ }} = c_}t = 0}} - c_}t = t_ }} = c_}t = 0}} - c_}t = t_ }}$$

(6)

and introduction of \(\xi\) in Eq. 5 yields:

$$\frac}\xi }}}t}} = (k + k_}}} c_}} )\left( }t = 0}} - \xi } \right)\left( }t = 0}} - \xi } \right)$$

(7)

When the reaction is studied at its very beginning, i.e., when \(\xi\) is negligible small, it is possible to write

$$\frac}\xi }}}t}} \approx \frac\xi }}t}} = (k + k_}}} c_}} )c_}t = 0}} c_}t = 0}} = k^ + k^_} c_}}$$

(8)

Measurements of the reaction rate \(\xi } \mathord\xi } t}}} \right. \kern-0pt} t}}\) based on Eq. 8 are called differential evaluations.

When the extend of reaction \(\xi\) is not negligible, it is necessary to use the integral evaluation: In case of an excess of B (compared to A), follows:

$$\frac}\xi }}}t}} = (k + k_}}} c_}} )c_}t = 0}} \left( }t = 0}} - \xi } \right) = (k^} + k_}}}^} c_}} )\left( }t = 0}} - \xi } \right)$$

(9)

Equation 9 is a first-order differential equation [2] and its integration in the limits \(\xi = 0\) to \(\xi = \xi\) and \(t = 0\) to \(t = t\) yields:

$$\ln \frac}t = 0}} }}}t = 0}} - \xi }} = (k^} + k_}}}^} c_}} )t$$

(10)

The experimental evaluation are either based on application of Eq. 8 (differential evaluation) or of Eq. 10 (integral evaluation). For this, the so-called tangents method can be used by plotting \(\xi\) versus time to get \(\tan \alpha\) (slope) of these plots. When the \(\tan \alpha\) data are then plotted versus \(c_}}\) of calibration solutions, the concentration of an \(c_},}}}\) in an unknown solution can be read. In case of the integral variant (B in excess of A) \(\ln c_}} } \mathord\ln c_}} } t}}} \right. \kern-0pt} t}}\) is plotted versus time for several \(c_}}\) of the calibration solutions. Finally, the \(\tan \alpha\) data of these dependencies, are plotted versus \(c_}}\) of the calibration solutions (Fig. 1).

Fig. 1

Left side: plot of \(\xi\) (extent of reaction) versus reaction time. Right side: plot of tanα of the left side lines versus catalyst concentration

It is also possible to simplify the evaluations by fixing the time and plotting \(\xi\) versus \(c_}}\) (differential variant) or \(\ln c_}}\) versus \(c_}}\) (integral variant). Another possibility is to fix the concentration (\(\xi\) for differential variant or \(\ln c_}}\) for integral variant) and plotting \(t}}} \right. \kern-0pt} t}}\) versus \(c_}}\) (Fig. 2).

Fig. 2

Left side: plot of \(\xi\) (extent of reaction) versus reaction time. For a fixed \(\xi\) the corresponding times \(t_}}\) are read. Right side: plot of \(}} }}} \right. \kern-0pt} }} }}\) versus catalyst concentration

The preceding presentation gives just the simplest cases of reaction kinetics and evaluation. There are many more complicated cases for which we refer to detailed monographs [3,4,5,6,7,8]. There are also reactions, which exhibit an induction period, i.e., during that period, no reaction is observed and after that period the reaction starts. In such cases, the concentration of the catalyst is inversely proportional to the time of the induction period. This makes the calibration very easy, as it is sufficient to measure the time until the reaction starts.

Catalymetry using homogeneous chemical reactions

As mentioned before, the catalysed homogeneous chemical reaction used for assessing the concentration of a catalyst is called the indicator reaction. Such reaction has to meet two requirements: (i) The equilibrium constant has to strongly favour the reaction products, i.e. the free energy has to be very negative. (ii) The reaction has to be kinetically hindered, i.e., has to have a large activation energy, that can be lowered by a catalyst. It is highly desirable that the catalyst has a high selectivity, i.e., has to be very specific. These criteria are fulfilled in case of many redox reactions, esp. when they have no simple stoichiometry and multi-step reaction pathways. However, the selectivity is usually not very high in the presence of chemically similar ions. In that case, the selectivity can be improved by modifying the indicator reaction, by masking agents or by preliminary separations [9].

Since this is a lecture text and no review, only a few examples of indicator reactions are given in Table 1. Extensive tables of catalysed reactions and their application for analysis are given in the following references: [3,4,5,6].

Table 1 Homogeneous indicator reactions and their catalysts (complete reaction equations are only given, when clearly established)

The Sandell-Kolthoff reaction \(} \to }\) is still a frequently used standard method for analysis of iodine in urine [12]. The reaction can be followed by spectrophotometry (Ce(IV) solutions are yellow), or also by potentiometry using a platinum electrode [17]. This reaction is a good example to illustrate the complex mechanisms of catalytic reactions: Rodriguez and Pardue [18] have proposed the following reaction pathways:

$$}^ + })\xrightarrow }}}) + }^$$

(IV)

$$}^ }\mathop \rightleftarrows \limits_ }}^ }} }$$

(V)

$$}\xrightarrow }} }^$$

(VI)

$$}\xrightarrow }} }^$$

(VII)

$$}^ }\mathop \rightleftarrows \limits_ }}^ }} }^$$

(VIII)

These authors came to the conclusion that Reactions IV–VI, and Reactions I, VII, and VIII represent two different catalytic cycles. Neither pathway VI, V, VI, nor IV, VII, VIII taken alone adequately describe the experimental data. However, a branched mechanism involving both cycles is very probable.

Instrumentation for catalymetric determinations

Two kinds of instruments are necessary to perform catalymetric analyses: First, it needs instruments to mix three solutions: the two solutions of the substrates A and B, and the solution of the catalyst. This mixing has to be as quickly as possible and always highly reproducible. For this a simple possibility is to use a glass vessel with three connected tubes (see Fig. 3). Since such vessels are not commercially available they need to be made by a glassblower. For this an Erlenmeyer flask and three glass tubes can be used. When the three solutions A, B, C are transferred with a pipette to the three tubes, the flask is closed by a stopper and by turning the flask around, a clock for time measurements is started. The second kind of instrumentation needs to perform measurements to follow the catalytic reaction: for this, all kinds of spectrometry, thermometric and electrochemical methods (potentiometry or amperometry) are useful. Performing the catalytic reaction using the mixing flask shown in Fig. 3 is the simplest way. It needs reactions, which are not too fast, i.e., the time of mixing should be negligible short compared to the overall reaction time. Using this mixing flask is, however, excellently suited for student laboratory experiments. A more sophisticated and much more reproducible experimental approach is to use flow-through devices, especially flow-injection [19,20,21], which need a flow-through mixing chamber.

Fig. 3Catalymetry using heterogeneous chemical reactions – Electrochemical catalymetryDissolved catalysts

Almost ideal conditions for catalymetric determinations are provided by electrochemistry, esp. by voltammetry and amperometry. Current measurements lend themselves for kinetic methods of analysis because current is charge per time, i.e., itself a reaction rate. In contrast to the homogeneous chemical reactions discussed above, in which the reaction rate relates to a volume, the electrochemical reaction rates relates to the electrode surface. For the purpose of catalymetry, it is an advantage (even a prerequisite!) that most electrode reactions are irreversible, i.e., they proceed with a more or less high overvoltage, which –in many cases– can be lowered by catalysts. For understanding the meaning of reversible and irreversible electrochemical reactions, the papers [22,23,24] can be consulted. When the catalyst is dissolved in the solution, it can often act as a mediator for the irreversible electrode reaction of a substrate. A typical example is the Mo(VI)-catalysed reduction of chlorate ions on a mercury electrode (see the detailed explanation in [24]). Figure 4 shows a scheme of the catalytic reduction of chlorate ions by the electrochemically generated Mo5+ ions in the presence of tungsten [25]. This is a very good example for the high selectivity which can be achieved with catalytic methods, especially bearing in mind the similarity between molybdenum and tungsten chemistry. In this paper [25], the term ‘polarographic catalymetry’ has been introduced. The homogeneous chemical reaction proceeds in the near-electrode layer. Of course, Mo6+ and Mo5+ only indicate the oxidation state. These ions are present as complexes because such highly charged ions cannot exist as aqua complexes in aqueous solutions [26]. The back-oxidation of Mo5+ to Mo6+ causes a much steeper concentration gradient of Mo6+ at the electrode surface, leading to higher currents of Mo6+ reduction. This current enhancement seriously decreases the detection limit of molybdenum. Catalytic currents have been used especially for the voltammetric (polarographic) determination of elements possessing several oxidation states, like Mo, W, Fe, V, Ti, U, Cr, Co, Ni, Pd, Pt, because they are highly active catalysts of redox reactions. The detection limits are down to \(10^\) and even \(10^\) mol L−1.

Fig. 4

Left side: Scheme of catalytic cycle of electrochemical reduction of chlorate ions in the presence of Mo(VI) ions. Right side: Voltammograms of increasing concentrations of of 0, 20, 40, 50, 80, 100, and 120 nmol L−1 Mo(VI) in the presence of 0.2 mmol L−1 W(VI)

Sinyakova [27] and Milyavskiy [28, 29] have proposed to label voltammetric systems similar to the molybdenum cycle in Fig. 2 as ‘current catalysis”, indicating that the catalyst enhances the current response. In the next paragraph, the other type of catalysis, i.e., ‘potential catalysis’ will be discussed.

Immobilized catalysts

When the electrode itself is the catalyst, this is labelled as electrocatalysis. This term has been used for the first time by Dmitriy Viktorovich Alekseyev (Дмитpий Bиктopoвич Aлeкceeв) [30]. This kind of catalysis of electrochemical reactions is most important for construction of biosensors, where the catalysts are enzymes. The archetype of such biosensors is the glucose sensor [31]. Here the enzyme glucose oxidase and a mediator, usually a ferrocene derivative [32], are immobilized on the electrode surface. The ferrocene derivative is the electron-transfer catalyst, which transfers electrons between the metal electrode and the glucose oxidase, and the latter transfers the electrons to the glucose. Electrocatalysis [33] is most important for fuel cells and many technical electrolyses, e.g., the electrolysis of brine solutions to produce chlorine. The purpose of the catalysts is to decrease the overpotential of electrode reactions in order to make the electrode reactions more economical.

For the purpose of analytical determination of catalysts, methods have been designed, in which the catalyst decreases the large overpotential of proton reduction on mercury electrodes. An excellent example is the determination of platinum traces (down to 4 × 10–14 mol L−1) by using the adsorption of a formazone-Pt(II) complex that decreases the hydrogen overpotential on mercury and produces a hydrogen reduction signal [34]. The formazone-Pt(II) complex forms in the bulk solution, and the catalytic hydrogen wave caused by the complex adsorbed on the electrode is a function of the platinum concentration in solution. Catalytic hydrogen waves have been extensively studied by Mayranovskiy [35].

Scientists, who have contributed to the formation of catalymetry

The use of catalysed homogeneous chemical reaction for quantitative analysis started and flourished in the in the twentieth century. Most important and very early contributions have been made in the USA by Izaak Maurits Kolthoff (Feb. 11, 1894 – March 4, 1993) [10, 11] and Ernest Birger Sandell (Feb. 20, 1906 – March 17, 1984) [10, 11]. Much later, Gerry Arthur Rechnitz (*Jan. 1, 1936) [4] and Harry B. Mark, Jr. (Feb. 28, 1934 – March 3rd, 2003) have propagated kinetic methods of analysis by their very influential monograph [4]. In Europe it were two Hungarians, László Szebellédy [36,37,38] (April 20, 1901 – January 23, 1944) and his student Miklós Ajtai (chemist and later politician, May 19, 1914 – Feb. 14, 1982) [36, 37], who published catalymetric methods already in the 1930-ies. A very important scientist for developing catalymetry was Konstantin Borisovich Yatsimirskiy (Кoнcтaнтиин Бopииcoвич Яцимииpcкий) [5, 39] (Fig. 5). Most of the Eastern European scientists working in catalymetry are linked to him, e.g. Helmut Müller (Feb. 27, 1939 – Feb. 27, 2013) [6] in GDR (East Germany), Panajot Rankov Bontschev (Bontchev) (Пaнaйoт Paнкoв Бoнчeв, Dec. 31, 1933 – April 11, 2015) [40] in Bulgaria, Iraida Ivanovna Alekseyeva (Иpaидa Ивaнoвнa Aлeкceeвa, 1930-?) in Moscow and many more. Another notable scientist was Samuil Usherovich Kreyngol’d (Caмyил Ушepoвич Кpeйнгoльд, Oct. 3, 1936 – Oct. 16, 2003). His monograph [41] provides many examples for applying catalymetry for the analysis of high purity materials. When these materials are catalytically inactive and the minute impurities are catalysts, catalymetry can give splendid analytical results. Scientists, who have made fundamental contributions to electrochemical catalymetry are: Yakow Yosifovich Tur’yan (Якoв Иocифoвич Typьян, July 25, 1922 – Nov. 12, 2023) [42,43,44], Pëtr Mikhailovich Zaytsev (Пeтp Mиxaйлoвич Зaйцeв, *Feb. 2, 1932) [45], Ovsey Evelevich Ruvinskiy (Oвceй Eвeлeвич Pyвинcкий, Oct. 23, 1936 – 2016 (?)) [42], Sof’ya Il’inichna Sinyakova (Coфья Ильиничнa Cинякoвa, March 8, 1901 – Nov. 1, 1970) [27], Elena Grigor’yevna Chikryzova (Eлeнa Гpигopьeвнa Чикpызoвa, Feb. 26, 1921 – May 1, 2008) [46], Mark Borisovich Bardin-Shtejn (Mapк Бopиcoвич Бapдин-Штeйн, 1919–1987) [46], Stal’ Grigorievich Mairanovskiy (Cтaль Гpигopьeвич Maйpaнoвcкий, Feb. 12, 1926 – Sept. 28, 1991) [47, 48], and Herbert Weisz (April 25, 1922 –March 15, 2018) [8, 49].

Fig. 5

Konstantin Borisovich Yatsimirskiy was born April 4, 1916 in the Ukrainian village Pologi (Gaysins’kiy rayon). He graduate from a Forestry College in 1934 and was at the Uzbek Experimental Forest Station until 1936, when he entered the Chemical Faculty of the Central Asian State University in Tashkent. There he worked under the supervision of Isaak Platonovich Tsukervanik (Иcaaк Плaтoнoвич Цyкepвaник) (Organic Chemistry) and Michail Il’ich Usanovich (Mиxaил Ильич Уcaнoвич) (Physical Chemistry). From 1945 to 1962 he was at the Ivanovo State University of Chemistry and Technology, and until 1981 he was professor at Taras Shevchenko State University in Kiyv. From 1969 to 1981 he served as director of the Pisarzhevskiy Institute of Physical Chemistry of the Ukrainian Academy of Science. His scientific work concerned the thermochemistry of complex compounds, the application of catalytic reactions in chemical analysis (i.e., catalymetry) and bioinorganic chemistry. He passed away on June 21, 2005 in Kiyv