PHYSICAL CHEMISTRY

V. A. LAVRENKO, L. P. KHOROSHUN, Academician of the Academy of Sciences of the Ukrainian SSR I. N. FRANTSEVICH

THERMODYNAMICS OF HETEROGENEOUS CATALYSIS PROCESSES

RECOMBINATION OF GAS ATOMS ON THE SURFACE OF SOLIDS

Various attempts to create a unified modern theory of catalysis, with the aim of developing scientific foundations for the selection of catalysts with a preassigned combination of catalytic properties, have so far not been crowned with success. Therefore the cautious attitude toward all-embracing concepts that has developed in recent years seems justified, as does the reasonable grouping of processes and catalysts into groups and classes according to the similarity of the elementary mechanism \((^{1,2})\).

For solving problems of catalysis occurring on the surface of a solid phase, the phenomena that are connected with considerable energy effects are, above all, very important. Therefore the most general approach here is the quantum-mechanical consideration \((^3)\) of the chemical properties of atoms and molecules, as well as of the mechanism of chemical reactions on the surface of the crystal lattice of solids.

Fig. 1

Another possible general approach is the application, to catalysis processes possessing extraordinary simplicity and generality, of the methods of the modern thermodynamics of irreversible processes \((^{4,5})\). Most heterogeneous chemical reactions are associated with nonequilibrium states and irreversible processes, and therefore only as a first approximation could they be described by classical thermodynamics, developed predominantly for equilibrium systems.

In the present work, an irreversible catalysis process is considered and the corresponding values of the parameters are determined in the study of the rather widespread \((^6)\) elementary reaction of recombination of gas atoms on the surface of solids.

Let us consider an open thermodynamic system (Fig. 1) consisting of two phases (subsystems). For a concrete case, phase 1 may be an atomic-molecular mixture of a diatomic gas arriving from a discharge tube into the reaction system; phase 2 is a solid body catalyzing the process of atom recombination.

The internal energy \(E\) of a thermodynamic system characterizes the possibility that the system can perform specific kinds of work and deliver to the external medium a certain amount of heat. This possibility is due to the fact that the internal thermodynamic forces and the absolute temperature of the system are different from absolute zero. In particular, for gaseous phase 1, consisting in the general case of \(r\) components, the internal thermodynamic forces are the chemical potentials \(\mu_\gamma(\gamma = 1,\ldots,r)\) and the gas pressure \(p\), while the numbers of moles \(n_\gamma(\gamma = 1,\ldots,r)\) and the volume \(V\) will be the corresponding conjugate parameters. For solid phase 2, the thermodynamic forces may be elastic stresses, forces characterizing work hardening (strengthening) and leading to various kinds of distortions of the crystal lattice of the material, chemical potentials of the components in the case of a solid solution, and so on. For convenience we shall denote the latter by the general symbol \(B_k\) \((k = 1,\ldots,s)\), and the parameters conjugate to them by \(b_k\) \((k = 1,\ldots,s)\).

Thus, the Gibbs equations for phases 1 and 2, respectively, will be

\[ dE_1=\sum_{\gamma=1}^{r}\mu_\gamma\,dn_{\gamma i}-p\,dV+T_1\,d\eta_1,\qquad dE_2=\sum_{k=1}^{s}B_k\,db_k+T_2\,d\eta_2, \tag{1} \]

where \(\eta_1,\eta_2\) are the corresponding entropies of the phases; the index \(i\) indicates internal changes in \(n_\gamma\) due to the chemical reaction.

The interaction of subsystems 1 and 2 with the surroundings is characterized by the energy-conservation law

\[ dE_1=\delta Q_1-\sum_{\gamma=1}^{r}\mu_\gamma\,dn_{\gamma e}-p\,dV,\qquad dE_2=\delta Q_2. \tag{2} \]

Here \(\delta Q_1\) and \(\delta Q_2\) are the heats received by subsystems 1 and 2 from outside, and \(dn_{\gamma e}\) are increments in the number of moles of the substances due to inflow from outside. From (1) and (2) we obtain

\[ d\eta_1=\frac{1}{T_1}\,\delta Q_1-\frac{1}{T_1}\sum_{\gamma=1}^{r}\mu_\gamma\,(dn_{\gamma i}+dn_{\gamma e}), \]

\[ d\eta_2=\frac{1}{T_2}\,\delta Q_2-\frac{1}{T_2}\sum_{k=1}^{s}B_k\,db_k. \tag{3} \]

Then the equation for the overall entropy balance has the form

\[ d\eta=d\eta_1+d\eta_2= \frac{1}{T_1}\,\delta Q_1+\frac{1}{T_2}\,\delta Q_2 -\frac{1}{T_1}\sum_{\gamma=1}^{r}\mu_\gamma\,(dn_{\gamma i}+dn_{\gamma e})- \]

\[ -\frac{1}{T_2}\sum_{k=1}^{s}B_k\,db_k. \tag{4} \]

Since heterogeneous recombination of atoms is mainly governed by the transfer of thermal energy from phase 1 to phase 2 \({}^{(6)}\) while the other parameters \((B_k,b_k)\) of the solid remain constant, the last term of equation (4) may be neglected. Taking into account external heating of the catalyst (heat influx \(\delta Q_3\)),

\[ \delta Q_2=-\delta Q_1+\delta Q_3. \tag{5} \]

Expressing the chemical potentials through the chemical affinity of the reaction \(A\),

\[ \mu_\gamma=\varkappa_\gamma-A\left/\sum_{\gamma=1}^{r}\nu_\gamma\right., \tag{6} \]

where

\[ \varkappa_\gamma =RT_1\ln v_\gamma \left[ \frac{K(p,T_1)}{v_1^{\nu_1}\cdots v_r^{\nu_r}} \right]^{ 1\left/\sum_{\gamma=1}^{r}\nu_\gamma\right. } +\xi_\gamma(p,T_1), \]

\(\nu_\gamma\) is the stoichiometric coefficient of component \(\gamma\); \(K\) is the equilibrium constant; \(\xi_\gamma\) is the part of the chemical potential independent of composition; and also using the relations \(dn_{\gamma i}=\nu_\gamma\,d\xi\) (\(\xi\) is the degree of reaction completeness), \(-\sum_{\gamma=1}^{r}\mu_\gamma\nu_\gamma=A\), and replacing in (4) differentials by derivatives with respect to time, we obtain

\[ \dot{\eta}= \frac{1}{T_2}\dot{Q}_3+ \left(\frac{1}{T_1}-\frac{1}{T_2}\right)\dot{Q}_1+ \frac{1}{T_1}A \left( \dot{\xi}+\sum_{\gamma=1}^{r}\dot{n}_{\gamma e}\left/\sum \nu_\gamma\right. \right) -\frac{1}{T_1}\sum_{\gamma}\varkappa_\gamma\dot{n}_{\gamma e}. \tag{7} \]

Since the heat flux \(q\) from phase 1 to phase 2 through a unit surface \(S\) of the catalyst is equal to

\[ q=-\frac{1}{S}\dot Q_1, \]

the production of entropy \(\sigma\), according to (7), is written in the form

\[ \sigma=\left(\frac{1}{T_2}-\frac{1}{T_1}\right)qS+\frac{1}{T_1}A\left(\dot{\xi}+\sum_{\gamma=1}^{r}\dot n_{\gamma e}\,\middle/\,\sum_{\gamma=1}^{r}\nu_\gamma\right) -\frac{1}{T_1}\sum_{\gamma=1}^{r} x_\gamma \dot n_{\gamma e}. \tag{8} \]

From the condition that \(\sigma\) be nonnegative it follows that between the fluxes and the thermodynamic forces conjugate to them there exist functional dependences which, near the equilibrium state of the system, are linear:

\[ qS=L_{11}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)+L_{12}\frac{A}{T_1}; \]

\[ \dot{\xi}+\sum_{\gamma=1}^{r}\dot n_{\gamma e}\,\middle/\,\sum_{\gamma=1}^{r}\nu_\gamma =L_{21}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)+L_{22}\frac{A}{T_1}; \tag{9} \]

\[ -\dot n_{\gamma e}=\sum_{\beta=1}^{r}P_{\gamma\beta}\frac{x_\beta}{T_1} \quad(\gamma=1,\ldots,r). \]

The last term in expression (8) characterizes the deviation of the components entering subsystem 1 from the stoichiometric ratio; therefore the fluxes \(\dot n_{\gamma e}\) in (9) do not couple with the others. The coefficients entering the dependences (9) satisfy Onsager’s reciprocity principle

\[ L_{12}=L_{21},\qquad P_{\gamma\beta}=P_{\beta\gamma}\quad(\gamma,\beta=1,\ldots,r). \tag{10} \]

Expressing in (3) the entropies \(\eta_1,\eta_2\) through the corresponding temperatures according to

\[ \dot\eta_1=C_{1v}\frac{\dot T_1}{T_1}+p\frac{\dot V}{T_1};\qquad \dot\eta_2=\frac{C_2\dot T_2}{T_2}, \tag{11} \]

where \(C_{1v}, C_2\) are the heat capacities of phases 1 and 2, we obtain, taking (9) into account, the heat-balance equations of phases 1 and 2:

\[ C_{1v}\dot T_1+p\dot V =-\left[L_{11}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)+L_{12}\frac{A}{T_1}\right] +A\left[L_{21}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)+L_{22}\frac{A}{T_1}\right] +\sum_{\gamma,\beta=1}^{r}P_{\gamma\beta}\frac{x_\gamma x_\beta}{T_1}; \tag{12} \]

\[ C_2\dot T_2=\dot Q_3+\left[L_{11}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)+L_{12}\frac{A}{T_1}\right], \]

which, together with the last two equations of (9), form a system of kinetic equations for the process under consideration.

Let us consider the experimentally observed case of a stationary process. Since in this case

\[ \dot n_{\gamma e}=-\nu_\gamma N, \tag{13} \]

where \(N\) is the number of moles reacting per unit time, from (9) and (12), using (10), we have

\[ -\frac{\dot Q_3}{S}=L_{11}^{*}(T_1-T_2)+L_{12}^{*}A,\qquad -N=L_{12}^{*}\frac{S}{T_2}(T_1-T_2)+L_{22}^{*}A,\qquad A=\frac{\dot Q_3}{N}. \tag{14} \]

Here \(L_{11}^{*}=L_{11}/ST_1T_2\) is the heat-transfer coefficient between phases 1 and 2 \((\mathrm{cal}/\mathrm{cm}^2\cdot\mathrm{sec}\cdot\mathrm{deg})\); \(L_{12}^{*}=L_{12}/ST_1\) is the rate constant of the catalytic

process, referred to unit surface area of the catalyst (mol/cm\(^2\)·sec); \(L_{22}^{*}=L_{22}/T_1\) is the coefficient characterizing the homogeneous reaction in the gaseous phase.

Equations (14) were used by us to determine the coefficients \(L_{11}^{*}\) and \(L_{12}^{*}\) in an experimental study of the catalytic reaction of recombination of hydrogen atoms (\(P_{\mathrm H}=0.01\)--0.05 mm) on the surface of annealed specimens of refractory metals—molybdenum and tungsten—in the temperature range from 50 to 850°. The temperature dependences, obtained by the method of S. Z. Roginsky and A. B. Schechter\(^7\), of the number \(\dot N_r\) of H atoms reacting per unit time on unit surface area of the catalyst are presented in Fig. 2. Since, in the indicated range of partial pressures of atomic gases, homogeneous recombination of atoms is usually neglected (\(L_{22}^{*}A=0\)), for our case from (14) we have

\[ L_{12}^{*}=-N_r\frac{T_2}{T_1-T_2};\quad L_{11}^{*}=-\frac{\dot Q_3(T_1-2T_2)}{S(T_1-T_2)^2}. \tag{15} \]

Fig. 2. Temperature dependence of the number of recombining hydrogen atoms:
1—on tungsten at \(P_{\mathrm H}=0.0114\) mm;
2—on molybdenum at \(P_{\mathrm H}=0.0469\) mm

Here \(\dot Q_3\) is the thermal power supplied to the metal wire when an electric current is passed directly through it (cal/sec).

The values of the rate constants \(L_{12}^{*}\) of heterogeneous recombination of hydrogen atoms on the surface of molybdenum and tungsten in the indicated temperature range, calculated according to (15), are constant. Their mean values are, respectively, \(8.45\cdot10^{-6}\) and \(1.02\cdot10^{-6}\) mol/cm\(^2\)·sec. The mean errors in determining these coefficients are, respectively, 8.7 and 5.7%.

The values, determined according to (15), of the heat-transfer coefficients \(L_{11}^{*}\) under the experimental conditions for the recombination of H on recrystallized molybdenum are given in Table 1, which shows their dependence on temperature.

Table 1

Institute of Problems of Materials Science
Academy of Sciences of the Ukrainian SSR

Institute of Mechanics
Academy of Sciences of the Ukrainian SSR

Received
23 VII 1964

REFERENCES

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