Thermochemical Data from Gas-Phase Adsorption and Methods of Their Estimation

Abstract

Nowadays, gas-phase chromatography represents one of the fastest and most efficient techniques for the investigation of chemical properties of superheavy elements. The classical gas-adsorption chromatography experiment with transactinide species performed in an isothermal regime or in a temperature gradient, at ambient gas pressures, or under vacuum conditions delivers observables for single atomic species that are dependent on the parameters of the experiment. In Part I of this chapter we present several methods to derive thermodynamic data of the investigated species from these observables, which are not dependent on the experimental parameters and which are therefore intercomparable. A reversal of these methods suggests the behavior of transactinide species based on predicted thermochemical data, which is instrumental for the design of experiments. In Part II of this chapter we demonstrate several methods for the prediction of thermochemical properties of the heaviest elements based on chemical trends established in the Periodic Table. This provides data limits complementary to results of modern relativistic calculations. Finally, only the direct comparison of predicted data to experimental results allows for conclusions to be made on trends of chemical properties among transactinides and their corresponding homologs and for further improvement of the mostly empirical prediction models suggested here.

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Part I: Basic Principles of the Determination of Adsorption Properties Using Gas-Phase Adsorption Chromatographic Methods

1.1 Introduction

Volatilization, combined with gas-adsorption chromatographic investigations, is a well established method in nuclear chemistry. Fast reactions and high transport and separation velocities are crucial advantages of gas chromatography methods over alternative chemistry methods. Additionally, the fast sample preparation for α-spectroscopy and spontaneous fission (SF) measurements directly after the gas-phase separation is a very advantageous feature. Formation probabilities of defined chemical compounds and their volatility can be investigated on the basis of both experimentally determined and theoretically predicted thermochemical data: the latter are discussed in Part II of this chapter.

Volatile elements, as well as a large variety of volatile chemical compounds can be investigated by using a broad assortment of reactive carrier gases. Moreover, different stationary phase materials, available for gas-adsorption chromatography, are further broadening the areas of application; see Table 1. Apart from that, the separation quality can be influenced by the size of the chromatographic surface, e.g., introducing fills into the chromatography column. Experimental investigations were carried out in the temperature range from 85 K [1] to 2400 K [2]. Suitable chemical states and stationary phases exist for almost all elements to allow for their investigation via adsorption chromatography.

Table 1 Chemical states and the reactive carrier gases and stationary phases applicable for each

As the quantities of elements investigated in nuclear chemistry experiments are often small, down to one-atom-at-a-time, deposition and volatilization are predominately related to adsorption and desorption phenomena, respectively. Essentially, pure condensed phases do not occur. The entire gas-phase transport along the chromatography column is called gas-adsorption chromatography. The gas-phase transport through a chromatography column depends on several factors:

• Experimental parameters: e.g., flow rate and the type of carrier gas.

• The design of the chromatography columns like shape, material, and filling.

• The temperature, applied to the stationary phase.

• The interaction mechanism between the gaseous atoms and molecules with the stationary phase.

This chapter describes basic physico-chemical relationships between the gas-phase transport of atoms and molecules and their thermochemical properties, which are related to the adsorption–desorption equilibrium. These methods can either be used to predict the behavior of the adsorbates in the chromatographic processes, in order to design experiments, or to characterize the adsorbate from its experimentally observed behavior in a process. While Part I of this chapter is devoted to basic principles of the gas-phase transport and of the gas-adsorption process, the derivation of thermochemical data is discussed in Part II. Symbols used in the following sections of Part I are described in Sect. 1.5. Chapters “Experimental Techniques” and “Gas-Phase Chemistry of Superheavy Elements” discuss technical aspects and experimental results, respectively, which are related to the evaluation methods in gas-adsorption chromatography outlined in this chapter.

1.2 Thermochemical Description of the Transport of Substances in Gas-Adsorption Chromatography

1.2.1 The Adsorption Equilibrium

Here we consider the simple adsorption–desorption reaction equilibrium of a reversible mobile adsorption process:

$$ {\text{A}}_{\text{gas}} \Leftrightarrow {\text{A}}_{\text{ads}} $$

(1)

The description of the chromatographic transport of a substance in an ideal linear gas chromatography is based on the differential equation for the transport velocity [3]:

$$ \frac{{\rm d}y}{{\rm d}t} = \frac{u}{{1 + k_{i} }} $$

(2)

where k _i represents the partition coefficient corrected by the ratio of both phases, the solid and the gas phase:

$$ k_{i} = \frac{s}{v} \cdot K_{\text{ads}} $$

(3)

For the formulation of the dimensionless adsorption equilibrium constant the definition of a standard state is crucial. The standard state is freely selectable, regardless of the possibility of its physical realization. It is defined according to convenience. The standard state of adsorption is assumed to be the ratio of a standard molar volume to the standard molar surface [4].

$$ \frac{V}{A} = \frac{{c_{\text{ads}}^{o} }}{{c_{\text{gas}}^{o} }} = 1 cm\;\left( {\text{cgs}} \right)\; \to \;\frac{V}{ 1 0 0\cdot A} = \frac{{c_{\text{ads}}^{o} }}{{c_{\text{gas}}^{o} }} = 1 m\;({\text{SI}}) $$

(4)

$$ K = K_{\text{ads}} \cdot \frac{ 1 0 0\cdot A}{V} = \frac{{c_{\text{ads}} }}{{c_{\text{gas}} }} \cdot \frac{{c_{\text{gas}}^{o} }}{{c_{\text{ads}}^{o} }} $$

(5)

The dimensionless equilibrium constant of a simple reversible adsorption reaction is related to thermodynamic standard quantities—the standard adsorption enthalpy and the standard adsorption entropy, which are assumed to be temperature independent:

$$ \Updelta G_{\text{ads}}^{0} \; = \; - \,R \cdot \,T\, \cdot \,ln\,(K)\; = \;\Updelta H_{\text{ads}}^{0} \; - \;T\, \cdot \,\Updelta S_{\text{ads}}^{0} $$

(6)

For a localized adsorption the concentration of the adsorption sites has to be taken into account [4]. In other cases a reversible change of the chemical state of the adsorbate in the chromatography process has to be considered, e.g., dissociative adsorption and substitutive adsorption as described in Part II of this chapter (Sect. 1.3, Eqs. 55, 56). The reaction enthalpy and entropy have to be introduced into the calculations [5–8] as well as the equilibrium constant for the chemical reaction including the standard states of the occurring chemical states as:

$$ K = K_{r} \cdot K_{\text{ads}} \cdot \frac{ 1 0 0\cdot A}{V} $$

(7)

$$ \Updelta G_{{r,{\text{ads}}}}^{o} = - R \cdot T \cdot ln\left( K \right) = (\Updelta H_{r}^{o} + \Updelta H_{\text{ads}}^{o} ) - T \cdot (\Updelta S_{r}^{o} + \Updelta S_{\text{ads}}^{o} ) $$

(8)

The entropy of the mobile adsorption process can be determined from the model given in [4]. It is based on the assumption that during the adsorption process a species in the gas phase, which has three degrees of freedom (translation), is transferred into the adsorbed state with two translational degrees of freedom parallel to the surface and one vibrational degree of freedom vertical to the surface. From statistical thermodynamics the following equation for the calculation of the adsorption entropy is derived:

$$ \Updelta S_{\text{ads}}^{o} = R \cdot \left( {ln(\frac{ 1 0 0\cdot A}{{V \cdot \nu_{b} }}) \cdot \sqrt {\frac{R \cdot T}{{ 2\cdot \pi \cdot M_{a} }}} + \frac{ 1}{ 2}} \right) $$

(9)

As a good approximation it is assumed that the adsorbed species are vibrating in resonance with the lattice phonon vibrations of the solid stationary phase. The phonon frequency can be evaluated from phonon spectra, from the standard entropy of solid metals, from the Debye temperatures, or from the Lindemann equation [9].

1.2.2 Isothermal Gas Chromatography

Combining Eqs. (2–6), including the linear temperature dependency of the carrier gas flow rate, and integrating over the column length the following relation for the retention time in isothermal gas chromatography is obtained [10]:

$$ t_{r} = \;\frac{{L \cdot T_{o} \cdot {\varnothing}}}{{\overline{V}_{o} \cdot T_{\text{iso}} }}\; \cdot \;\left[ { 1+\frac{s}{v} \cdot \frac{V}{ 1 0 0\cdot A} \cdot { \exp }\left( { -\frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{iso}} }}}\right) \cdot { \exp }\left( {\frac{{\Updelta S_{\text{ads}}^{o}}}{R}} \right)} \right]$$

(10)

The first addend in Eq. (10) can be neglected. Thus the retention volume is calculated as:

$$ ln\;\left( {t_{r} \cdot \overline{V}_{o} \cdot \frac{{T_{\text{iso}} }}{{T_{o} }}} \right)\; = \;ln\;\left( {\frac{L \cdot {\varnothing} \cdot s \cdot V}{v \cdot 1 0 0\cdot A}} \right)\; + \;\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{iso}} }}} \right)\; + \;\left( {\frac{{\Updelta S_{\text{ads}}^{o} }}{R}} \right) $$

(11)

In experiments with long-lived nuclides species, with T _1/2 much longer than the experimental duration, the retention time equals the time of the experimental duration. However, for short-lived nuclides, at the temperature where 50% of the species containing a specific nuclide pass the isothermal chromatography column, the retention time equals the half-life of the nuclide:

$$ t_{r} = T_{1/2} $$

(12)

Using the mobile adsorption entropy (Eq. 9) $\Updelta H_{\text{ads}}$ can be evaluated from at least a pair of experiments at two different isothermal temperatures under otherwise identical experimental conditions.

1.2.3 Thermochromatography

1.2.3.1 Gas Thermochromatography

The given description of the gas-phase transport in a tube with a temperature gradient is only valid for the adsorption equilibrium of reversible mobile adsorptions without any superimposed chemical reaction. The temperature profile along the chromatography column is approximated to be linear by:

$$ T = T_{s} - a \cdot y $$

(13)

The substitution of the corrected partition coefficient and the introduction of the standard state (Eqs. 3–5) lead to:

$$ t_{r} = - \frac{ 1}{a} \cdot \int\limits_{{T_{s} }}^{{T_{\text{dep}} }} {\frac{{\left( { 1+ \frac{s}{v} \cdot \frac{V}{ 1 0 0\cdot A} \cdot K(T)} \right)}}{u(T)}} \cdot dT $$

(14)

Introducing

$$ u(T) = u_{o} \cdot \frac{T}{{T_{o} }} $$

(15)

and

$$ u_{o} = \frac{{\bar{V}_{o} }}{{\varnothing} } $$

(16)

the integration yields

$$ \begin{gathered} t_{r} = - \frac{{T_{o} }}{{a \cdot u_{o} }} \cdot { \ln }\;\left( {\frac{{T_{\text{dep}} }}{{T_{s} }}} \right) + \frac{{s \cdot {\varnothing} \cdot T_{o} \cdot \frac{V}{ 1 0 0\cdot A} \cdot { \exp }\left( {\frac{{\Updelta S_{\text{ads}}^{o} }}{R}} \right)}}{{\overline{V}_{o} \cdot a \cdot v}} \cdot \hfill \\ \cdot \left[ {Ei^{*} \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right) - Ei^{*} \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{s} }}} \right)} \right] \hfill \\ \end{gathered} $$

(17)

where Ei*(x) is the exponential integral function of x. This function can be estimated by:

$$ Ei^{*} (x) = \frac{exp(x)}{x}\quad \left( {{\text{for}}\;x\gg 1} \right) $$

(18)

The first addend in Eq. (17) is negligible for T _dep « T _s , which is usually the case and

$$ Ei^{*} \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right)\;\gg \;Ei^{*} \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{s} }}} \right) $$

(19)

Hence, the following simplification [11] is conceivable:

$$ \frac{{a \cdot t_{r} \cdot \overline{V}_{o} }}{{s \cdot T_{\text{dep}} \cdot \frac{V}{ 1 0 0\cdot A}}} = \left( {\frac{{R \cdot T_{o} }}{{ - \Updelta H_{\text{ads}}^{o} }}} \right) \cdot { \exp }\;\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right) \cdot { \exp }\;\left( {\frac{{\Updelta S_{\text{ads}}^{o} }}{R}} \right) $$

(20)

Thus, the deposition temperature and the thermodynamic state function of the adsorption are combined and they can easily be determined from each other. The retention time for a short-lived radioactive species is calculated as the radioactive lifetime of the nuclide:

$$ t_{r} = \frac{{T_{1/2} }}{{{ \ln }\left( 2\right)}} $$

(21)

or equals the duration of the experiment for a long-lived species, having a T _1/2 much longer than the experimental duration. Note, that Eqs. (17) and (20) are only valid if the chemical state of the investigated species does not change along the entire temperature range. A simplification of Eq. (17) is not possible if very short-lived nuclides (T _1/2 < 1 s) are investigated. Numerical integration methods have to be used instead. Using the mobile adsorption entropy (Eq. 9) $\Updelta H_{\text{ads}}$ can be evaluated from one experiment. If several experiments are performed at varied experimental parameters over large ranges, e.g., varying the experimental duration or the gas flow rate, $\Updelta H_{\text{ads}}$ and $\Updelta S_{\text{ads}}$ can be evaluated from an Arrhenius-type plot.

1.2.3.2 Vacuum Thermochromatography

The model of ideal linear gas chromatography (Eq. 2) also describes the transport of a chemical species along the temperature gradient of a vacuum tube [12]. At molecular flow conditions, the linear velocity of the carrier gas, which is identical to the transport velocity of the adsorbate in the gas phase, has to be substituted by the fraction of the column length over the average retention time of the species in the column:

$$ u = \frac{y}{{t_{r} }} = \frac{{ 4\cdot d_{i} \cdot \sqrt {\frac{ 2\cdot \pi \cdot k \cdot T}{m}} }}{ 3\cdot y} $$

(22)

The deposition site of a species in a defined temperature region under conditions where the retention time in the adsorbed state exceeds the duration of the experiment is assumed to be similar to a corresponding effective length of the column at which this species would have exited. This length is the distance of the deposition region, T _dep, from the beginning of the column held at T _s . Thus, the average retention time is a function of the column volume and the conductivity (geometric shape) of the column for inert gas-phase species at molecular flow conditions. The probability of residence of the species in the adsorbed state is mainly dependent on the temperature-dependent equilibrium constant and thus, on the thermochemical constants of the adsorption reaction. Increasing the adsorption probability by lowering temperatures decreases the migration velocity of the adsorbate zone. A limited experiment duration or radioactive lifetime of the species leads to a deposition zone of the adsorbate in the column or a zone of increased decay observations, respectively. Applying Eqs. (2) and (22) and substituting the corrected partition coefficient by a gas phase kinetic formulation of the adsorption equilibrium constant:

$$ k_{i} = \frac{ 4}{{d_{i} }} \cdot \frac{h}{k \cdot T} \cdot \sqrt {\frac{k \cdot T}{ 2\cdot \pi \cdot m}} \cdot { \exp }\;\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{R \cdot T}} \right) $$

(23)

gives the following functional dependency between the retention time, the experimental parameters, T _dep and $\Updelta H_{\text{ads}}$:

$$ \begin{aligned} t_{r} & = 2\cdot A \cdot \left( {\frac{ 1}{ 3} \cdot \sqrt {T_{\text{dep}}^{ 3} } - T_{s} \cdot \sqrt {T_{\text{dep}} } + \frac{ 2}{ 3} \cdot \sqrt {T_{s}^{ 3} } } \right)\; + \\ & + B \cdot \left[ {T_{\text{dep}} \cdot { \exp }\;\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right) - T_{s} \cdot \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right)} \right] + \\ & + B \cdot \left( {T_{s} - \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{R}} \right)} \right) \cdot \left[ {{\text{Ei}}^{ *} \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right) - {\text{Ei}}^{ *} \left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{R \cdot Ts}} \right)} \right] \\ \end{aligned} $$

(24)

with

$$ A = \frac{ 3}{{ 4\cdot a^{2} \cdot d_{i} \cdot \sqrt {\frac{ 2\cdot \pi \cdot k}{m}} }} $$

(25)

and

$$ B = \frac{ 3\cdot h}{{ 2\cdot a^{2} \cdot d_{i}^{2} \cdot k}} $$

(26)

For practical applications some simplifications [13] result in:

$$ t_{r} = \frac{{R \cdot T_{s} \cdot T_{\text{dep}} \cdot B}}{{ - \Updelta H_{\text{ads}}^{o} }} \cdot { \exp }\;\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{{R \cdot T_{\text{dep}} }}} \right) $$

(27)

with B taken from Eq. (26).

Hence, from the experimental parameters and from the measured deposition temperature, the adsorption enthalpy $\Updelta H^{o}_{\text{ads}}$ can be determined directly. Again this simplification holds only for t_r » 1 s [14, 15]. For experiments with very short-lived nuclides Eqs. (13, 21–23) have to be substituted into Eq. (2) and the resulting integral has to be solved numerically.

Introducing the mobile adsorption model into this derivation [16] k_i from Eq. (23) changes to:

$$ k_{i} = \frac{4}{{d_{i} }} \cdot \frac{V}{ 1 0 0\cdot A}{ \exp }\left( {\frac{{\Updelta S_{\text{ads}}^{o} }}{R}} \right) \cdot { \exp }\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{R \cdot T}} \right) $$

(28)

Then Eqs. (9, 13, 21, 22, and 28) have to be merged into Eq. (2) and, again, the resulting integral can be solved numerically.

1.3 Microscopic Kinetic Description of the Adsorption Process in Gas Chromatography: Monte-Carlo Methods

This section explains basic principles of Monte-Carlo models, which are also successfully applied for the evaluation of adsorption enthalpies in gas-phase and vacuum adsorption chromatography investigations of the heaviest elements and their lighter homologs (see chapters “Experimental Techniques” and “Gas-Phase Chemistry of Superheavy Elements”). The application of Monte-Carlo simulation methods in gas-phase adsorption chromatography is based on ideas given in [17]. All models use a microscopic description of the chromatographic adsorption–desorption process on the atomic scale. Hence, they are kinetic models of gas-adsorption chromatography. They can be applied to thermochromatography as well as to chromatography in the isothermal regime. To determine $\Updelta H^{o}_{\text{ads}}$ of the adsorbate on the stationary phase the experimental parameters and the physical data of the carrier gases and of the adsorbate are used as an input into these models. For a set of $\Updelta H^{o}_{\text{ads}}$ values, which is the only varied parameter, the simulation yields chromatograms, which have to be compared to the experimental result. The resulting $\Updelta H^{o}_{\text{ads}}$ value is obtained from the best agreement between the model and experiment. The simulation can be either used for an evaluation of the $\Updelta H^{o}_{\text{ads}}$ from gas-adsorption chromatographic results or, vice versa, to predict the behavior of an adsorbate in gas chromatography from its known adsorption data and hypothetical experimental data.

1.3.1 Gas Chromatography with Carrier Gases

A flow chart of the Monte-Carlo model is presented in Fig. 1. The formalism can easily be adapted to a PC-based program. The formulas used in the PC-based simulation are given here. For their derivations see [17–22]. The Monte-Carlo principle uses random numbers which are selected between 0 and 1 and are never equal to 1.

Fig. 1

Flow chart of the PC-based Monte-Carlo simulation of gas chromatography. The indices i and j stand for the number of the atom and of the individual jump, respectively

The randomized values in the Monte-Carlo simulation of a gas-phase adsorption process for radioactive species are:

• 1. The radioactive lifetime.

For every given atom a random lifetime, which is distributed logarithmically according to the radioactive decay law with the half-life, T _1/2, of the nuclide is calculated as:

$$ t_{\lambda } = - \frac{\text{ln(2)}}{{T_{1/2} }} \cdot {\text{ln(1}} - {\textit{random)}} $$

(29)

• 2. The jump length and the time of flight in the gas phase.

The mean jump length of the adsorbate can be approximated by Golay’s approach [18] for cylindrical columns:

$$ Y_{\text{jm}} = \frac{{ 1 1\cdot \overline{V} (p,T)}}{ 4 8\cdot \pi \cdot D(p,T)} $$

(30)

For rectangular channels the derivations given in [19, 20] shall be considered.

The pressure and temperature dependent diffusion coefficient of the adsorbate (a) in the carrier gas (g) is either tabulated or is approximated, e.g., according to Gilliland’s empirical equation [21]:

$$ D = 0. 4 3\cdot 1 0^{ - 6} \cdot \frac{{T^{ 1. 5} }}{p} \cdot \sqrt {\frac{ 1}{{M_{g} }} + \frac{ 1}{{M_{a} }}} \cdot \left( {\root{3} \of {{\frac{{d_{g} }}{{M_{g} }}}} + \root{3} \of {{\frac{{d_{a} }}{{M_{a} }}}}} \right)^{ - 2} \left[ {{\text{m}}^{ 2} \cdot {\text{s}}^{ - 1} } \right] $$

(31)

The gas flow also depends on the temperature and on the pressure. Assuming an ideal gas behavior of the carrier gas it can be calculated from the flow rate under standard conditions:

$$ \overline{V} (p,T) = \overline{V}_{o} \cdot \frac{{T \cdot p_{o} }}{{T_{o} \cdot p}} $$

(32)

Subsequently, the random jump length along the column, which is distributed logarithmically, is calculated as:

$$ y_{j} = - Y_{\text{jm}} \cdot {\text{ln(1}} - {\text{random)}} $$

(33)

Thus, the time the atom remains in the gas phase, i.e., the time of flight can be derived:

$$ t_{j} = \frac{{y_{j} \cdot \pi \cdot d_{i}^{ 2} }}{{ 4\cdot \overline{V} (p,T)}} $$

(34)

• 3. The residence time.

The mean number of wall collisions along Y _jm is:

$$ N_{m} = Y_{\text{jm}} \cdot \frac{{d_{i} }}{{ 2\cdot \overline{V} (p,T)}} \cdot \sqrt {\frac{ 2\cdot \pi \cdot R \cdot T}{{M_{a} }}} $$

(35)

The mean residence time of the atom in the adsorbed state is calculated by a Frenkel-type equation:

$$ t_{\text{am}} = \frac{ 1}{{\nu_{b} }} \cdot { \exp }\left( { - \frac{{\Updelta H_{\text{ads}}^{o} }}{R \cdot T}} \right) $$

(36)

The overall residence time of the atom in the adsorbed state between two jumps is assumed to be distributed logarithmically and can be calculated as:

$$ t_{a} = - N_{m} \cdot t_{\text{am}} \cdot {\text{ln(1}} - {\text{random)}} $$

(37)

1.3.2 Gas Chromatography in Vacuum

A Monte-Carlo simulation of gas chromatography under vacuum conditions was suggested in [16] and follows the same scheme as of Fig. 1.

The lifetime of each atom or molecule is calculated using Eq. (29). The mean residence time of the atom in the adsorbed state at a defined temperature is calculated using Eq. (36). The residence time of the atom in the adsorbed state, which is distributed logarithmically, can be calculated as:

$$ t_{a} = - t_{\text{am}} \cdot { \ln }\left( { 1- {\text{random}}} \right) $$

(38)

The transport of species along the chromatography column is assumed to be dependent only on the solid angle of the desorption direction and on the dimensions of the column.

The solid angles of desorption from a plane surface into the vacuum are calculated according to Knudsen [22] assuming a cosine law:

$$ \theta = { \arcsin }\sqrt {\text{random}} $$

(39)

$$ \varphi = 2\cdot \pi \cdot {\text{random}} $$

(40)

It is well presented in [20] that the corresponding straight flight length to the next surface in a cylindrical tube is calculated as:

$$ Y_{j} = d_{i} \cdot \frac{{{ \cos }\left( \theta \right)}}{{\left( { 1- { \sin }^{ 2} \left( \theta \right) \cdot { \sin }^{ 2} \left( \varphi \right)} \right)}} $$

(41)

The corresponding jump length projected to the column wall of a cylindrical tube is calculated as:

$$ y_{j} = d_{i} \cdot \frac{{{ \cos }\left( \theta \right) \cdot { \sin }\left( \theta \right) \cdot { \sin }\left( \varphi \right)}}{{\left( { 1- { \sin }^{ 2} \left( \theta \right) \cdot { \sin }^{ 2} \left( \varphi \right)} \right)}} $$

(42)

The velocity of the atoms can be set as the arithmetic mean velocity from the Maxwell–Boltzmann distribution. It only depends on the temperature and their molecular weight:

$$ \overline{v} = \sqrt {\frac{ 8\cdot R \cdot T}{{\pi \cdot M_{a} }}} $$

(43)

Hence, the time the atom is in flight can be calculated as:

$$ t_{j} = \frac{{Y_{j} }}{{\overline{v} }} $$

(44)

For simplicity, the column is assumed to be “closed” at the “entrance”, i.e., the atoms cannot escape from the side of the column where they entered.

1.4 Summary

Some advantages and disadvantages of the thermodynamic and the kinetic approach are summarized here to facilitate the decision as to which approach should be selected for the subsequent data evaluation from a gas-phase chromatography experiment.

• 1. Advantages of the thermodynamic model:

  • It is less time consuming.

  • Experiments, where complex stationary phases have been used (e.g., filled columns) can be evaluated.

• 2. Disadvantages of the thermodynamic model:

  • A linear temperature gradient has to be assumed;

  • Possible “long jumps” cannot be described in the model of linear gas chromatography. Hence separation factors are overestimated, especially in open columns.

  • The radioactive decay time is uniformly approximated through an average lifetime.

• 3. Advantages of the kinetic Monte-Carlo model:

  • A description of the microscopic chromatographic process at realistic temperature conditions at the surfaces (real temperature gradient) is possible.

  • The probability distribution of the radioactive decay time of the adsorbate is included.

  • The probability that the adsorbate is transported along the column by “long jumps” is taken into account, which is important to describe the chromatographic separation quality, especially in open columns and at high gas flow rates.

• 4. Disadvantages of the kinetic Monte-Carlo model:

  • The evaluation might be more time consuming, due to the Monte-Carlo programming.

  • The stationary surface geometry has to be simple.

1.5 Symbols

A	…	Inner surface per 1 m column length, m2
A	…	Negative temperature gradient, K/m
$c_{\text{ads}}$	…	Surface concentration of the adsorbate, particles/m2
$c_{\text{ads}}^{0}$	…	Standard surface concentration of the adsorbate, 2.679×1023 particles/m2
$c_{\text{gas}}$	…	Gas concentration of the adsorbate, particles/m3
$c^{o}_{\text{gas}}$	…	Standard gas concentration of the adsorbate, 2.679×1025 particles/m3 (ideal gas at STP)
d a	…	Density of the adsorbate at its melting point, kg/m3
d g	…	Density carrier gas at its melting point, kg/m3
d i	…	Inner diameter of the column, m
D	…	Diffusion coefficient of the adsorbate in the carrier gas, m2/s
∅	…	Free open cross-sectional area of the column, m2
φ, θ	…	Solid angles of desorption
H	…	Planck’s constant, J·s
$\Updelta G^{o}_{\text{ads}}$	…	Free standard adsorption enthalpy of adsorption, J/mol
$\Updelta G^{o}_{{r,{\text{ads}}}}$	…	Free standard adsorption enthalpy with chemical reaction, J/mol
$\Updelta H^{o}_{\text{ads}}$	…	Standard adsorption enthalpy at zero surface coverage, J/mol
$\Updelta H^{o}_{r}$	…	Standard reaction enthalpy, J/mol
K	…	Equilibrium constant of adsorption, dimensionless
K ads	…	Distribution constant, m
k i	…	Partition coefficient corrected by the phases, dimensionless
K r	…	Reaction equilibrium constant, dimensionless
K	…	Boltzmann constant, J/K
L	…	Length of the column, m
M	…	Atomic or molecular mass of the adsorbate, kg
M a	…	Molar weight adsorbate, kg/mol
M g	…	Molar weight carrier gas, kg/mol
N	…	Number of atoms
N m	…	Mean number of wall collisions
ν b	…	Maximum lattice phonon vibration frequency, 1/s
P	…	Pressure, Pa
p o	…	Standard pressure, 101325 Pa
Random	…	Random number
R	…	Gas constant, 8.31441 J/mol·K
S	…	Open surface of the column per 1 m column length, m2
$\Updelta S^{o}_{\text{ads}}$	…	Standard entropy of adsorption at zero surface coverage, J/mol·K
T	…	Time, s
t a	…	Time the adsorbate keeps adsorbed, s
t am	…	Mean time the adsorbate keeps adsorbed, s
t exp	…	Duration of the experiment, s
t j	…	Transport time, s
t λ	…	Lifetime, s
t r	…	Retention time, s
T	…	Temperature, K
T dep	…	Deposition temperature, K
T iso	…	Isothermal temperature, K
T o	…	Standard temperature, 298.15 K
T 1/2	…	Half-life, s
T s	…	Upper (start) temperature of the gradient, K
U	…	Carrier gas velocity, m/s
u o	…	Carrier gas velocity at STP, m/s
V	…	Inner volume of the column, m3
V	…	Open volume of the column per 1 m column length, m3
$\overline{V}$	…	Carrier gas flow, m3/s
$\overline{V}_{o}$	…	Carrier gas flow at STP, m3/s
Y	…	Coordinate longitudinal to the column, m
y j	…	Jump length, m
Y j	…	Path length, m
Y jm	…	Mean jump length, m

2 Part II: Derivation of Thermochemical Data with Respect to the Periodicity of Properties and Their Interrelations

2.1 Introduction

The discovery of new chemical elements—the transactinides or superheavy elements—stimulated the work on theoretical predictions of their chemical properties. Our intention is to present in Part II of this chapter empirical methods from [23–37] and from our partly unpublished work [38–50], which are used to predict chemical properties of elements and compounds relevant to gas-phase chemical studies of transactinides.

Gas-phase chemical methods can be successfully applied to the chemical characterization of transactinides. They provide a fast separation of transactinides from a wide variety of by-products, which are instantly produced in the nuclear formation reactions of transactinides. These methods are mainly based on the measurement of adsorption properties of the atomic or molecular state of transactinides on different stationary phases.

Due to the extremely low production rates of transactinides in nuclear fusion reactions, all chemical characterizations are carried out at the single atom level (see chapter “Fundamental and Experimental Aspects of Single Atom-at-a-Time Chemistry”). The chemical reaction products are characterized on the basis of their behavior in the separation process or, to be exact, in the gas-phase-adsorption chromatographic process (see Part I of this chapter). In this process the formation probability of defined stable chemical states of transactinides and the subsequent interaction of the formed species with a solid state surface are studied.

The stability and the volatility are chemical properties that define the behavior of a transactinide element or its compound in the gas-adsorption chromatographic process. Therefore, the predictions of these properties are instrumental for the design of experiments (see Chapter “Experimental Techniques” for instrumental aspects) and they are also indispensable for the interpretation of experimental results; see chapter “Gas-Phase Chemistry of Superheavy Elements”. “Empirical” and “exact” methods are both of high importance for the prediction of properties of transactinide elements. The “exact” ab initio methods, which are described in chapter “Theoretical Chemistry of the Heaviest Elements”, yield the atomic ground state configuration, ionization potentials, atomic radii, and ionic radii as well as binding energies in isolated single gaseous molecules of defined chemical compounds with high accuracy. However, it still appears impossible to exactly calculate the behavior of a single atom or molecule in a given experimental setup. For the ab initio calculation of the stability of pure solid phases, which is necessary for the evaluation of the volatility¹ of compounds, the accuracy of the available methods is limited. Therefore, up to now, only empirical methods have been used to predict the volatility of compounds. These predictions represent an important part in the multiple step process of the chemical characterization of transactinide elements.

The main steps in this process are:

• I. The evaluation of thermochemical data of chemical compounds in the solid and in the gaseous phase.

• II. Calculations of the volatilities of the elements and of their compounds.

• III. The experimental determination of empirical correlations between the volatility of the pure macroscopic phase and the adsorption behavior of single atoms or molecules on defined surfaces.

• IV. The prediction of the adsorption behavior of transactinides or their compounds at zero surface coverage from the predicted volatility (II.) applying the empirical correlations (III.).

• V. The physico-chemical description of the gas-adsorption chromatographic process.

• VI. Model experiments using short-lived nuclides of lighter homologs for the determination of reaction rates, retention times, and decontamination.

• VII. Design of an experimental setup for experiments with transactinides with respect to the predicted properties and its half-life using III. and V.

• VIII. The interpretation of the results according to the:

  • formation of compounds with similar volatility as known from their homologs,

  • relative volatility of equivalent chemical compounds compared with the homologs,

  • adsorption enthalpies and the volatilities of fictive macroscopic phases,

  • confirmation and/or improvement of prediction methods of the chemical reaction behavior and of the volatility, and

  • periodicity of chemical properties and trends along the groups of the Periodic Table.

This contribution focuses on steps I–V. Steps VI–VIII are partly discussed in chapters “Experimental Techniques” and “Gas-Phase Chemistry of Superheavy Elements”.

2.2 Thermochemical Data

2.2.1 Extrapolative Analysis of Thermochemical Data for Transactinides and Their Compounds

It is assumed that the stability of a chemical state can be expressed in terms of the standard formation enthalpies of gaseous and solid compounds $\Updelta_{f} H^{o}_{ 2 9 8} \left( {{\text{g}},{\text{s}}} \right)$ and of their atomic standard formation enthalpies $\Updelta_{f} H^{*}_{ 2 9 8} \left( {{\text{g}},{\text{s}}} \right)$. Both values differ only by the presumed elemental state: The first value refers to the standard state of the gaseous element and the second one to the gaseous monatomic state of the element at standard conditions. These values represent relative stability measures in combination with a competing chemical state. Thermochemical state functions describe the behavior of a large number of atoms or molecules. For the description of single atoms we use these values as a measure of the formation probability of different chemical states and to quantify the binding energy in molecules. Gas-adsorption chromatography is a multi-step process, during which a single atom or molecule changes its chemical state in thousands of adsorption–desorption steps. Therefore, the resulting information on the adsorption properties of the investigated chemical species is as reliable as the results from experiments with a very large number of atoms or molecules from a one-step evaporation-deposition experiment. For more detailed information on this topic see Chapter “Fundamental and Experimental Aspects of Single Atom-at-a-Time Chemistry”.

Trends of chemical properties of the elements and their compounds exist within the groups of the Periodic Table. Therefore, assuming that all transactinides systematically belong to defined groups, rough estimates about their chemical properties should be feasible. For these estimates, shell effects and relativistic effects, which depend on the atomic number and lead to discontinuous variations of properties along the groups, have to be considered [51–58]. Examples for such effects are: the half occupancy of electron shells, which has a larger effect on the chemical properties of elements with lower atomic numbers, and the sub-shell closure of the p-, d-, and f-electron levels with an increasingly strong spin–orbit coupling for elements at the higher number periods up to the transactinides. A detailed discussion of relativistic effects including the spin–orbit coupling can be found in Chapter “Theoretical Chemistry of the Heaviest Elements”. In general, elements with a shell closure or a half-filled electron shell in their ground state configuration form relatively volatile condensed phases and less stable chemical compounds compared to elements in the same group, which do not exhibit such effects. This creates a dilemma for extrapolative predictions of thermochemical data of transactinides along their groups as these configurations change with atomic or period number. Anyhow, no direct functional dependence exists between thermochemical state functions and electron configurations.

In the following part, several possibilities are outlined, which can be used to master the described dilemma. Hence, more trustworthy extrapolative predictions are obtained by correlating thermochemical state functions mutually. In this case electron shell effects in the homologues are already implemented. To estimate standard formation data of transactinide compounds in their solid and in their gaseous states, the standard enthalpies of monatomic gaseous elements $\Updelta H^{*}_{ 2 9 8} \left( {{\text{E}}_{{({\text{g}})}} } \right)$ are of fundamental importance. They are used as primary input values for extrapolative predictions and they are mostly equal to the standard sublimation enthalpy $\Updelta H^\circ_{\text{subl}}$ of these elements. They have to be estimated empirically for the transactinides, though. Experimentally deduced tabulated thermochemical data of the lighter homologs in the corresponding groups of the transactinides in the Periodic Table include relativistic effects in their electronic structure. Hence, rough estimations, e.g., of $\Updelta H^\circ_{\text{subl}}$ can be obtained by a simple extrapolation by atomic number (Z) along the groups of the Periodic Table [27–30, 34]. These estimations are not to be considered as non-relativistic, because they include the increasing relativistic effects with increasing Z along the lighter elements in the corresponding groups of the Periodic Table. As an example, data included in Table 2 for s- and p-elements are presented in Fig. 2. Note here, the deviation of some of the elements in the first two rows of the Periodic Table.

Table 2 Predicted thermochemical data for transactinides and compounds in kJ/mol

Fig. 2

Extrapolation of the standard enthalpies of monatomic gaseous elements $\Updelta H^{*}_{ 2 9 8} \left( {{\text{E}}_{{({\text{g}})}} } \right)$ along the groups of the Periodic Table based on atomic number Z (adapted from [30])

In an illustrative way, the standard enthalpy of monatomic gaseous elements can be seen as the dissociation enthalpy of a (macromolecular) elemental crystal [59]. This value is a substantial constituent of the binding enthalpy of compounds. Therefore, a coupling between the standard or the atomic standard formation enthalpies of solid and gaseous compounds and the standard enthalpy of monatomic gaseous elements can be expected and is certainly observed, e.g., see Fig. 3.

Fig. 3

Extrapolation of the atomic standard formation enthalpies of gaseous compounds $\Updelta_{f} H^{*}_{ 2 9 8} \left( {{\text{comp}}_{{({\text{g}})}} } \right)$ based on the standard enthalpies of monatomic gaseous elements $\Updelta H^{*}_{ 2 9 8} \left( {{\text{E}}_{{({\text{g}})}} } \right)$ for group 6 compounds with M = Cr, Mo, W, and Sg (adapted from [35])

The enthalpies of formation obtained by this extrapolation are compiled in Table 2. Several results are given for the formation enthalpies, if different values estimated for the standard enthalpies of monatomic gaseous elements were used for the prediction. Correlations between the standard formation enthalpies of the solid state versus the corresponding values of the gaseous state have been used [35–44]. For similar types of compounds of elements along one group with equivalent oxidation states linear correlations can be observed. More generally, this type of correlation is observed for different compounds of transition elements in their highest achievable oxidation state; see Fig. 4.

Fig. 4

Correlation of the standard formation enthalpies of oxychlorides and oxyhydroxides in their gaseous $(\Updelta_{f} H^{o}_{298} \left( {\text{g}} \right))$ and solid state $(\Updelta_{f} H^{o}_{298} \left( {\text{s}} \right))$ (adapted from [36])

Another interesting type of linear relations was suggested by Golutvin [60] correlating the standard formation enthalpies of solid and gaseous compounds of elements ($\Updelta H^{o}_{{{\text{eq}} .}} \left( {{\text{g}},{\text{s}}} \right)$, see Eq. 45), which are normalized to the oxidation state of the metal ion in this compounds (w) and to the number of metal ions (metal equivalents) in the compounds (eq_metal), with the logarithm of the oxidation state of the metal ion in this compounds.

$$ \Updelta_{f} H_{{{\text{eq}}.}}^{o} \left( {g,s} \right) = \frac{{\Updelta_{f} H_{ 2 9 8}^{o} \left( {g,s} \right)}}{{w \cdot {\text{eq}}_{\text{metal}} }} $$

(45)

Thus, just by assuming the oxidation state of any compound in a selected compound class, the formation enthalpy of a compound in its solid or gaseous state can be predicted. As an example the stability of various bohrium oxides in the gas phase and in the solid phase are predicted as a function of oxidation states in Fig. 5 [36, 41, 42].

Fig. 5

Correlations between the normalized standard formation enthalpies $\Updelta H^{o}_{{{\text{eq}} .}} \left( {{\text{g}},{\text{s}}} \right)$ of gaseous and solid compounds of group 7 elements and the logarithm of the oxidation state (w) of the metal ion in these compounds (adapted from [41])

2.2.2 Volatility of Compounds

The temperature dependent vapor pressure is frequently used to define volatility. For the adsorbed state the relevant quantity is the desorption pressure, which depends on the temperature and on the surface coverage. The individual crystal lattices, with their characteristic binding properties and, thus, their standard entropy of the pure solid phase, may largely influence the vapor pressure; see, e.g., [39, 44]. Thus, the standard sublimation enthalpy $\left( {\Updelta H^\circ_{\text{subl}} } \right)$ is preferred as a measure for the volatility of a chemical species. The prediction of the adsorption behavior of a transactinide compound starts with the derivation of the sublimation enthalpy $\left( {\Updelta H^\circ_{\text{subl}} } \right)$ of the pure compound. Depending on the availability of data, the standard sublimation enthalpy can be calculated using the following methods:

• Calculations employing the difference between the standard formation enthalpies of compounds in the gaseous and in the solid state:

  $$ \Updelta H_{ 2 9 8}^{o,(*)} \left( g \right) - \Updelta H_{ 2 9 8}^{o,(*)} \left( s \right) = \Updelta H_{\text{subl}}^{o} $$

  (46)

• Correlations between the standard formation enthalpies of the gaseous and of the solid state with the linear regression coefficients a and b given, e.g., in Fig. 4:

  $$ - \Updelta H_{ 2 9 8}^{o,(*)} \left( s \right) = a \cdot \Updelta H_{ 2 9 8}^{o,(*)} \left( g \right) + b $$

  (47)

The standard sublimation enthalpy is then calculated by:

$$ ( 1- a) \cdot \Updelta H_{ 2 9 8}^{o,(*)} \left( g \right) + b = \Updelta H_{\text{subl}}^{o} $$

(48)

Significant correlations exist between ionic radii in the solid state—or radii of electron orbitals of ions—and the sublimation enthalpies for various compound classes; especially for the halides. It is possible to calculate the standard sublimation enthalpy of transactinide compounds from their calculated ionic radii by making use of a radii-volatility correlation and of the corresponding radii of the homologs [43]. However, in this procedure it is important to use consistent sets of radii for all homologs, see, e.g., Fig. 6. The sublimation enthalpy of RfCl₄ derived from this approach is included in Table 2.

Fig. 6

Radii-volatility correlation for group 4 elements and rutherfordium. The filled squares represent the radii of the outer orbital (r _(orbit.), top axis) the open circles show the ionic radii in solid crystals (r_(cryst.), bottom axis). Included on the axis are predicted data for $\Updelta H_{\text{subl}} \left( {{\text{RfCl}}_{ 4} } \right)$ from [44] (crossed triangle) and r _(cryst.) $\text{Rf}^{4{+}}$ (crossed diamond) from [59]

2.2.3 Empirical Correlations Between Adsorption Properties of Single Atoms or Molecules and the Volatility of Pure Substances

Empirical extrapolative predictions regarding the stability and the volatility of transactinides and of their compounds are, in the next step, followed by the determination of empirical correlations between adsorption properties of extremely small amounts of these elements or compounds and the volatility of the pure macroscopic solid phase of this substance, respectively.

It is assumed that the molar binding energy of an adsorbed single molecule to the surface approximately equals its partial molar adsorption enthalpy at zero surface coverage. In the adsorbed state at zero surface coverage, however, the individual variations of the entropy are partly but not completely suppressed.

The adsorption enthalpy can be expressed as a sum of two independent terms (see also Sect 1.3.2):

• The net adsorption enthalpy, as the enthalpy difference between a pure solid compound and its adsorbed state on a surface at zero surface coverage;

• The desublimation enthalpy.

Hence, for small or equal net adsorption enthalpies it can be expected that the adsorption enthalpy is proportional to the standard sublimation enthalpy, which characterizes the volatility properties of pure solid phases as an integral value:

$$ - \Updelta H_{\text{ads}}^{o} \propto - \Updelta H_{\text{subl}}^{o} $$

(49)

Small net adsorption enthalpies represent a prerequisite to reversible non-localized mobile adsorption processes without reactions of the species adsorbed on the stationary surfaces. The experimental proof of such correlations for defined classes of pure substances is essential for the prediction of adsorption properties of transactinides and their compounds. Therefore, a variety of gas adsorption chromatographic experiments were carried out with carrier free amounts of different radioisotopes using selected modified surfaces as stationary phases. The use of carrier free amounts is necessary to experimentally obtain adsorption conditions at nearly zero surface coverage.

The selection of proper elements and compounds needs to take into account a very complex situation. The chemical state of the radioisotopes in the reactive carrier gas and the standard sublimation enthalpy of this chemical state must be known. In principle all compounds that are unstable under the selected conditions, have to be rejected as well as compounds that undergo diffusion processes or irreversible reactions with the stationary phase. The following correlations were obtained experimentally for elements and selected compound classes; see Table 1 and Fig. 7 a–d:

Fig. 7

Correlation of the molecular property adsorption enthalpy $\Updelta H^{o}_{\text{ads}}$ with the property of the macroscopic solid phase sublimation enthalpy $\Updelta H^{o}_{\text{subl}}$ for different gas phase chemical systems: Panel (a) for elements in H₂ on quartz adapted from [63]; b for elements in H₂ or He on gold adapted from [65]; c for chlorides and oxychlorides in Cl₂, HCl, CCl₄ (O₂) on quartz adapted from [35], and d for oxides and oxyhydroxides in O₂ (H₂O) on quartz adapted from [63]

• 1. A: For elements (with H₂, (H₂O)) on quartz [62, 63]

  $$ - \Updelta H_{\text{ads}}^{o} = \left( { 2. 9\pm 1 6} \right) + \left( { 0. 7 3\pm 0. 1} \right) \cdot \Updelta H_{\text{subl}}^{o} ,\;{\text{kJ}}/{\text{mol}} $$

  (50)

• 2. B: For elements (with He or H₂) on gold [64, 65]

  $$ - \Updelta H_{\text{ads}}^{o} = \left( { 1 0. 3\pm 6. 4} \right) + \left( { 1. 0 8\pm 0. 0 5} \right) \cdot \Updelta H_{\text{subl}}^{o} ,\;{\text{kJ}}/{\text{mol}} $$

  (51)

• 3. C: For chlorides and oxychlorides (with Cl₂, SOCl₂, HCl, (O₂)) on quartz [35]

  $$ - \Updelta H_{\text{ads}}^{o} = \left( { 2 1. 5\pm 5. 2} \right) + \left( {0.600 \pm 0.025} \right) \cdot \Updelta H_{\text{subl}}^{o} ,{\text{ kJ}}/{\text{mol}} $$

  (52)

• 4. D: For oxides and oxyhydroxides (with O₂, (H₂O)) on quartz [63, 66]

  $$ - \Updelta H_{ads}^{o} = \left( { 7. 0\pm 8. 0} \right) + \left( {0.697 \pm 0.03} \right) \cdot \Updelta H_{\text{subl}}^{o} ,\;{\text{kJ}}/{\text{mol}} $$

  (53)

Experimental observations of these empirical correlations clearly prove the postulated proportionality. These correlations suggest a similarity between the bond (with lower coordination) of the adsorbed particles to the modified surface and the bond to the surface of the pure macroscopic phase of the compound, which is relevant for the desublimation process. The adsorption behavior of atoms and compounds for most of the experiments used in the described correlations were evaluated using differently defined standard adsorption entropies [65–70]. Adsorption data from more recent experimental results were evaluated applying the model of mobile adsorption [4]. Hence, data from previous experiments were re-evaluated using the latter model. These correlations based on estimated standard sublimation enthalpies allow predictions of adsorption enthalpies for selected compounds for the case of zero surface coverage. These results are only valid for experimental conditions using the same reactive gases, and thus, similarly modified stationary surfaces.

For gas chemical studies we assume, in the simplest case, that under a given experimental condition the most stable chemical state is formed and that this state remains unchanged during the entire experiment. Pure and very reactive gas mixtures are used at high concentrations to obtain and stabilize this chemical state. During gas adsorption processes the reactive carrier gas determines the chemical state of the investigated elements and modifies the surface of the stationary chromatographic phase in a characteristic way. Thus, in the case of dissociation reactions, which may occur with very high reaction rates especially at high temperatures, the restoration of the most stable chemical state is possible and is very fast. All deposition and volatilization processes of single atoms or single molecules (the nearest case to zero surface coverage) are basically adsorption and desorption processes, respectively.

Two fundamentally different types of reversible processes can occur in the gas-adsorption chromatography:

• Adsorption and desorption of molecules without any change in the oxidation state and no change in the ligands. For example, the simple adsorption of a group-5 oxychloride can be described by the following equation:

  $$ {\text{MOCl}}_{ 3} \left( {\text{g}} \right) \Leftrightarrow {\text{MOCl}}_{ 3} \left( {\text{ads}} \right) $$

  (54)

  If the interaction of the molecule with the surface is weak (Van der Waals forces), the molecule remains nearly unchanged regarding its binding structure, oxidation state, and number of ligands. This is typical for physisorption. Adsorption processes of molecules with non-saturated coordination show much stronger interactions. These adsorption processes are accompanied by changes of the coordination number of the central atom and, in some cases, by changes of the type of binding (e.g., break-up of a double bond). The oxidation state remains unchanged. This depicts a typical case of chemisorption.

• In a different kind of chemisorption process, a reversible change of the oxidation state or a reversible change of the number of ligands or even the nature of ligands may occur during the adsorption and the desorption of a molecule. For example, for the oxyhydroxide compound of a metal ion in the oxidation state $+6$ the dissociative adsorption can be described by [7]:

  $$ {\text{H}}_{ 2} {\text{MO}}_{ 4} \left( {\text{g}} \right) \Leftrightarrow {\text{MO}}_{ 3} \left( {\text{ads}} \right) \, + {\text{ H}}_{ 2} {\text{O}}\left( {\text{g}} \right) $$

  (55)

  As another example, the substitutive adsorption of a group 4 tetrachloride is described in [8] by:

  $$ {\text{MCl}}_{ 4} \left( {\text{g}} \right) \, + { 1}/ 2 {\text{O}}_{ 2} \left( {\text{g}} \right) \Leftrightarrow {\text{MOCl}}_{ 2} \left( {\text{ads}} \right) \, + {\text{ Cl}}_{ 2} \left( {\text{g}} \right) $$

  (56)

The knowledge of the type of the adsorption process is crucial for the determination of the adsorption enthalpy from experimental results (see Part I of this chapter, Eqs. 7 and 8). One experimental approach to assess the reaction type is the variation of the partial pressure of the reactive carrier gas, which is supposed to be involved in the mechanism of the adsorption reaction [8].

2.3 Empirical Calculation Methods for Adsorption Enthalpies

2.3.1 Physisorption of Noble Gases and Noble-Gas Like Elements

2.3.1.1 Van der Waals Model

Depending on the type of interaction between an adsorbed particle and a non-metallic solid state surface, there are cases, where adsorption enthalpies can be calculated using empirical and semi-empirical relations. In the cases of atoms with a noble-gas like ground-state configuration and of symmetrical molecules the binding energy (E _B ) to a solid surface can be calculated as a function of the polarizability (α), the ionization potential (IP), the distance (r) between the adsorbed atom or molecule and surface, and the relative dielectric constants (ε) [37, 73, 74]:

$$ E_{B} = \frac{ 3}{ 1 6} \cdot \alpha \cdot \frac{{\left( {\varepsilon - 1} \right)}}{{\left( {\varepsilon + 2} \right)}} \cdot \frac{{ 1. 5 7\cdot \left( {IP_{ 1} \cdot IP_{ 2} } \right)}}{{\left( {IP_{ 1} + IP_{ 2} } \right)}} \cdot \frac{ 1}{{r^{ 3} }} $$

(57)

The binding energy E _B approximately equals the adsorption enthalpy $\Updelta H^\circ_{\text{ads}}$ as:

$$ \Updelta H^{o}_{\text{ads}} = - E_{B} - 0. 5\cdot R \cdot T $$

(58)

If the elements Cn and Fl (element 114) have a noble-gas like character [54], then, in a fictitious solid state, they would form non-conducting colorless crystals. A physisorptive type of adsorption may occur and their adsorption properties, for example on quartz, can be calculated with this method, see Table 3. For physisorbed noble gas atoms a roughly uniform distance to different surfaces of about 2.47 ± 0.2 Å was deduced from experimental results [47]. A predicted value of the adsorption properties of HsO₄ was based on this model in [37]. In conjunction with molecular and elemental data, which were calculated using density functional theory, this model yields valuable predictive results; see chapter “Theoretical Chemistry of the Heaviest Elements”.

Table 3 Adsorption enthalpies on quartz estimated using the physisorption model for Cn, Fl, 118, and HsO₄ [37]

2.3.1.2 The Law of the Corresponding States

Another approach to assess volatility data for noble gas like elements is based on the fundamental law of corresponding states, which was established by van der Waals in 1880. It suggests the existence of a universal equation of states (59) valid for various liquids and gases with high similarity of bonding, interaction, and structure. In the case of the heavier noble gases these requirements are indeed fulfilled. The variables in this equation are the parameters of state reduced by the critical values.

$$ {\text{f}}\left( {\frac{p}{{p_{c} }},\frac{V}{{V_{c} }},\frac{T}{{T_{c} }}} \right) = 0 $$

(59)

Based on ab initio atomic calculations, Pitzer [54] first suggested for the elements Cn, Fl, and 118 the possibility of a noble-gas like behavior. Based on these predictions the potential diagrams of these elements should be similar to the noble gas potentials. Hence, in the condensed states van der Waals interactions predominate. Grosse [23] estimated the critical constants (p _c , V _c , T _c ) for the elements Rn and 118 using extrapolations along the group 18 based on the period numbers. Improved extrapolations, using the atomic weight and the radii of the outermost electron orbitals [55] have been suggested in [49]; see, e.g., Fig. 8. Note the clear divergence for the light noble gases in [49] for the correlations with p _c and V _c , which is in agreement with the observation of remarkable deviations of He and Ne from the law of corresponding states, having non-zero energy states at 0 K [76]. However, using the data for the heavy noble gases (Ar–Rn) a complete set of critical data is deduced, which is used as follows to describe the volatilization process of the noble gas element 118 and of the potentially noble-gas like elements Cn and Fl. The reduced vapor pressure equation is given as:

Fig. 8

Correlation of T _c with r_max·ln(A) for noble gases (black squares) (adopted from [49]). Linear regression (black line) and extrapolated data (open circles) are indicated

$$ { \log }\left( {{\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p {p_{c} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${p_{c} }$}}} \right) = - \frac{\text{a}}{{{\raise0.7ex\hbox{$T$} \!\mathord{\left/ {\vphantom {T {T_{c} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{c} }$}}}} + {\text{b}} $$

(60)

Using the given and extrapolated critical temperatures T _c and critical pressures p _c the coefficients a and b can be deduced. According to the law of corresponding states, these coefficients shall be nearly identical for all ideal solids and liquids. This is true for the heavy noble gases Ar, Kr, Xe, and Rn. Thus, those coefficients a and b from the heavy noble gases are used for predicting vapor pressure coefficients for the elements Cn, Fl, and 118 as:

$$ { \log }\left( \frac{p}{kPa} \right) = - \frac{{a \cdot T_{c} }}{T} + b + { \log }\left( {\frac{{p_{c} }}{kPa}} \right) = - \frac{\text{A}}{\text{T}} + {\text{B}} $$

(61)

From the vapor pressure coefficients of the solid phase the sublimation enthalpy can be estimated; see Table 4, $\Updelta H^{0}_{\text{subl}}$(1).

Table 4 Thermochemical data of noble gases and empirically deduced data for elements Rn, Cn, Fl, and 118

A third approach, assumes the potential curves are very similar for elements revealing only van der Waals interaction in the solid state according to Lennard-Jones [77]. A constant proportionality between the depths of the potentials k ^Θ and the critical temperatures is expected according to:

$$ k^{\Uptheta } = \frac{{\left( {\Updelta H^{0}_{\text{subl}} + U_{0} } \right)}}{{R \cdot T_{c} }} = 6. 5 $$

(62)

with [76, 78]:

$$ U_{0} = \frac{ 9}{ 8} \cdot R \cdot \Uptheta_{D} $$

(63)

The Debye temperature (Θ _D ) can be calculated [76, 78] as:

$$ \Uptheta_{D} = \frac{ 1 6 3}{{V_{0K}^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}} }} \cdot \sqrt {\frac{b.p.}{A}} $$

(64)

The boiling points (b.p.) are estimated according to [23] from the critical temperatures as:

$$ b.p. = 0. 5 8\cdot T_{c} $$

(65)

The atomic volume at 0 K (V _0K) was shown to be proportional to the critical volume [79]:

$$ V_{{ 0 {\text{K}}}} = 0. 2 3 9\cdot V_{c} $$

(66)

For the heavy noble gases very similar k ^Θ ~ 6.8 are deduced. Hence, the sublimation enthalpies at 0 K (Table 4, $\Updelta H^{0}_{\text{subl}}$(2)) for the elements Rn, Cn, Fl, and 118 can be calculated using their critical data. The obtained sublimation enthalpies can be used to predict the potential adsorption behavior of the elements Cn, Fl, and 118 using empirical correlations (Eqs. 50 and 51).

2.3.1.3 The Adhesion Model

Under the assumption that elements Cn, Fl, and 118 behave as typical heavy noble gases [54] and undergo pure van der Waals interaction with metal surfaces, the adhesion model based on an approach given in [80] can be applied. The adsorption enthalpy of noble-gas like elements was related to the adhesion of atoms to a surface with a certain surface energy. This approach revealed that the adsorption enthalpy of a noble gas (E) on a metal surface is linearly correlated to the adsorption enthalpy of Xe on the same metal.

$$ \Updelta H^{o}_{\text{ads}} (E )= C ( {\text{E)}} \cdot \Updelta H^{o}_{\text{ads}} ( {\text{Xe)}} $$

(67)

This model was successfully applied to reproduce experimental adsorption data for noble gases on metal surfaces. Furthermore, this model was empirically extended in [16] to predict adsorption enthalpies of the elements Rn, Cn, and Fl on metal surfaces, assuming a physisorptive adsorption; see Fig. 9. Please note that two different dipole polarizabilities of Cn as given in the literature lead to two slightly different predictions for the Cn (I, II) adsorption enthalpy on gold, based on correlation depicted in Fig. 9 (upper panel); see Table 5. Experimental adsorption data measured for Rn on various metal surfaces [16] revealed a fairly good agreement with these predictions.

Fig. 9

Empirical correlations of C(E) with the atomic dipole polarizabilities α (upper panel) and with the ionization potentials IP (lower panel) (adopted from [16])

Table 5 Proportionality factors C(E) connecting the adsorption enthalpy of noble-gas like elements on metals with the adsorption enthalpy of Xe on these metals

2.3.2 Adsorption of Metals on Metal Surfaces

2.3.2.1 The Metallic Character

One general rule of trends in the Periodic Table is that the metallic character of all elements increases with increasing atomic number Z along the groups of s-elements and of p-elements. The relative difference between the dissociation enthalpy of two-atomic molecules, $\Updelta H^\circ_{\text{diss}}$, and the sublimation enthalpy, $\Updelta H^\circ_{\text{subl}}$, of these elements (both values are given in [81] and from section “Thermochemical Data”) can be used to estimate the metallic character (m) of elements (see Fig. 10) as suggested in [30]:

Fig. 10

The metallic character m of s- and p-elements as a function of the atomic number Z and its extrapolation (in italics) along the groups of the periodic table to the region of transactinides (adopted from [30])

$$ m = \left( {\Updelta H_{\text{subl}}^{o} - 0. 5\cdot \Updelta H_{\text{diss}}^{o} } \right)/\Updelta H_{\text{subl}}^{o} $$

(68)

For real metals this value is close to 1. This value qualitatively shows whether the association to homo-nuclear two-atomic molecules $\left( {\Updelta H^{o}_{\text{diss}} } \right)$ (non-metallic character) is energetically preferred over the formation of a coordination lattice (metallic character) and vice versa. According to this relation, a metallic character can also be expected for the elements Cn and Fl [30]. Element 117, for example, can be assumed to have a semi-metallic character. However, special precaution is required since the data available for some elements show a large spread. As an example, the range of the literature data on the sublimation of Po is shown here to be limited by two sublimation enthalpies Po(A) [82] and Po(B) [83], which yield quite large variations in the prediction for the Lv character; see Fig. 10.

2.3.2.2 The Semi-Empirical Macroscopic Adsorption Model

This model assumes that the adsorption enthalpy can be divided into two independent energetic parts—the desublimation enthalpy and the net adsorption enthalpy. Hence, the net adsorption enthalpy characterizes the interaction, which depends on the nature of both metals. On the other hand, the desublimation enthalpy is an exclusive property of the adsorbate. For the calculation of the net adsorption enthalpies of transactinides on metal surfaces the partial molar enthalpies of solution of the adsorbate in the solid stationary material at infinite dilution and the enthalpy of displacement are required. These values can be obtained using (i) the semi-empirical Miedema model [84–87]; (ii) the Volume-Vacancy model, if the adsorption occurs in the surface; or (iii) the Surface-Vacancy model, if the adsorption occurs on the surface [88–90]. The magnitude of the partial molar enthalpy of solid solution is decisive to the selection of the appropriate Vacancy model. This value is illustrated in Fig. 11. It can be clearly seen from the given figure where the strongest interactions occur between the elements of the Periodic Table.

Fig. 11

The partial molar enthalpy of solution $\left( {\Updelta H_{\text{sol}} } \right)$ of element Z _A in element Z _B calculated based on a semi-empirical model suggested by Miedema [82–85]. The halogens and noble gases are excluded

The data required for the calculations of the net adsorption enthalpies of transactinides are estimated from trends in the Periodic Table [45, 46]. The net adsorption enthalpies and the predicted sublimation enthalpies, e.g., [50], were combined to predict the adsorption enthalpies of transactinides on selected metal surfaces [45, 46, 90]. The metals, which are presented in Table 6, can be used as the stationary phase in gas-adsorption chromatographic experiments for selective gas chemical separations or, in the case of high adsorption interaction, as strong fixation materials for the sample preparation in the measurement of transactinides.

Table 6 Predicted adsorption data of metallic s- and p-elements on different stationary phases