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Reports of the Academy of Sciences of the USSR
- Vol. 139, No. 5
PHYSICAL CHEMISTRY
A. V. KISELEV and D. P. POSHKUS
STATISTICAL-THERMODYNAMIC CALCULATION OF THE ADSORPTION EQUILIBRIUM OF BENZENE ON GRAPHITE
(Presented by Academician A. N. Frumkin on 18 III 1961)
The calculation we carried out \((^{1})\) of the change in the chemical potential \(\Delta \mu\) of argon on transfer from the gas to the surface of graphite showed that, by a statistical method, based on the theoretical calculation of the potential energy of adsorption, it is possible, to a satisfactory approximation, to calculate \(\Delta \mu\) for monatomic molecules at small surface coverages \(\theta\). In the present work a calculation has been made of \(\Delta \mu\) for the polyatomic benzene molecule upon adsorption on graphite.
At sufficiently small \(\theta\), for both localized and nonlocalized adsorption \((^{2,1})\),
\[ \Delta \mu = -RT \ln \frac{f_a/N_a}{f/N}, \tag{1} \]
where \(f_a, f\) and \(N_a, N\) are the state functions and the numbers of molecules in the adsorbed layer and in the gas. According to the approximation \((^{3})\)
\[ f_a = f_{a\,\text{class}} \nu^{**}, \tag{2} \]
where
\[ f_{a\,\text{class}}=\frac{1}{h^n}\int \cdots \int e^{-H/kT}\,dp_1 \cdots dp_n dq_1 \cdots dq_n \tag{3} \]
(\(n\) is the number of degrees of freedom of the adsorbate molecule, \(H\) is the Hamiltonian function; \(p_n\) and \(q_n\) are components of momentum and coordinates), and
\[ \nu^{**}=f_{\text{harm. osc. quant.}}/f_{\text{harm. osc. class}}, \tag{4} \]
where \(f_{\text{harm. osc. quant.}}\) and \(f_{\text{harm. osc. class}}\) are the state functions of harmonic oscillators, respectively in the quantum-mechanical and classical forms, calculated from the shape of the potential-energy surface of the adsorbed molecule near the minimum.
For an adsorbed benzene molecule \((^{4,5})\)
\[ H=\frac{p_x^2+p_y^2+p_z^2}{2m}+\frac{p_\vartheta^2}{2A} +\frac{(p_\varphi-p_\psi \cos \vartheta)^2}{2A\sin^2\vartheta} +\frac{p_\psi^2}{2C}+\Phi, \tag{5} \]
where \(m\) is the mass of the molecule; \(A\) and \(C\) are its moments of inertia, respectively with respect to a diameter and with respect to the axis perpendicular to the plane of the benzene ring and passing through its center; \(p_x, p_y, p_z, p_\varphi, p_\vartheta, p_\psi\) are components of momentum; \(x, y, z\) are the Cartesian coordinates of the center of the benzene molecule (the plane \(x, y, z=0\) passes through the centers of the carbon atoms of the basal face of graphite); \(\varphi, \vartheta, \psi\) are the usual Euler angles, and \(\Phi=\Phi(x,y,z,\varphi,\vartheta,\psi)\) is the potential energy of the adsorbed benzene molecule. We obtained the function \(\Phi\) by summing the potential energies of interaction of the force centers of the benzene molecule with the force centers of the adsorbent. We neglected the potential energy of interaction between adsorbed molecules.*
* In \((^{4})\), \(\Phi\) was obtained by integrating the interaction energy of the volume elements of the benzene molecule with the volume elements of the graphite lattice.
** Measurements of the differential heats of adsorption of benzene on graphitized thermal blacks with a homogeneous surface showed that the energy of interaction adsorbate–adsorbate in the case of benzene is indeed small \((^{6})\).
Calculations of \(\Phi\) for different positions of a benzene molecule above the basal face of graphite and different \(z\) at \(\vartheta = 0\) \({}^{(7,8)}\) showed that, for a given \(z\) and \(\vartheta = 0\), \(\Phi\) depends only weakly on the position of the benzene molecule. In the present work we neglected the dependence of \(\Phi\) and of the equilibrium distance \(z_0\) on \(x\), \(y\), and \(\varphi\). In deriving the dependence of \(\Phi\) on \(z\), \(\vartheta\), and \(\psi\), we considered the benzene molecule as a regular plane hexagon, at whose vertices are located the force centers \(i\)—the CH groups. For the dependence of the potential energy \(\Phi_i\) of interaction of center \(i\) (the CH group) of the adsorbate with the entire lattice of the adsorbent on \(z_i\), the expression \({}^{(1)}\) was adopted
\[ \Phi_i = u_{01}\alpha_i^{-q_1} + u_{02}\alpha_i^{-q_2} + u_{0\rho}e^{-(\alpha_i-1)z_{i0}/l}, \tag{6} \]
where
\[ u_{01} = - C_{1i}p_1z_{i0}^{-q_1}, \qquad u_{02} = - C_{2i}p_2z_{i0}^{-q_2}, \tag{7} \]
\[ u_{0\rho} = B_i k e^{-z_{i0}/l} = - \frac{l}{z_{i0}}(u_{01}q_1 + u_{02}q_2), \qquad \alpha_i = z_i/z_{i0}. \]
The constants of dispersion attraction for the benzene molecule \(C_1 = 6C_{1i}\) and \(C_2 = 6C_{2i}\) were calculated in \({}^{(7)}\) (respectively by the Kirkwood–Müller formula \({}^{(9)}\)) and by an analogous formula \({}^{(10,7)}\), and the constants \(p_1, q_1, p_2, q_2\), and \(l\)—in \({}^{(1)}\). For CH, \(z_{i0}\) was taken equal to the sum of the effective van der Waals radius of CH (1.85 Å, i.e., one-half of the van der Waals thickness of the benzene molecule) and one-half of the interplanar spacing of the lattice (1.70 Å), i.e., equal to 3.55 Å.
The energy \(\Phi\) is found by summing the \(\Phi_i\):
\[ \Phi(\alpha,\beta,\psi) = u_{01}\sum_{i=1}^{6}(\alpha+\beta\sin\psi_i)^{-q_1} + u_{02}\sum_{i=1}^{6}(\alpha+\beta\sin\psi_i)^{-q_2} + u_{0\rho}\sum_{i=1}^{6} e^{-(\alpha+\beta\sin\psi_i-1)z_{i0}/l}, \tag{8} \]
where
\[ \alpha = z/z_{i0}; \qquad \beta = \frac{d}{z_{i0}}\sin\vartheta^*; \qquad \psi_i = (i-1)\frac{\pi}{3}+\psi. \tag{9} \]
Substitution of (5) into (3), integration, and allowance for the symmetry number of the molecule \(\sigma\) and for the state function of its internal vibrations \(j_v\) lead to expression (4):
\[ f_{a\,\mathrm{class}} = \frac{1}{\sigma}j_v\,2\pi \left(\frac{2\pi m kT}{h^2}\right)^{3/2} s\,\frac{2\pi A kT}{h^2} \left(\frac{2\pi C kT}{h^2}\right)^{1/2} \frac{2z_0^2}{d}\,Q, \tag{10} \]
where
\[ Q = \int_{0}^{2\pi} d\psi \int_{0}^{d/z_{i0}} \frac{\beta\,d\beta}{\left[(d/z_{i0})^2-\beta^2\right]^{1/2}} \int_{\Phi \le 0} \exp\left(-\frac{\Phi}{kT}\right) \operatorname{erf}\left(-\frac{\Phi}{kT}\right)^{1/2} \,d\alpha, \tag{11} \]
and \(s\) is the surface area of the adsorbent. The value of integral (11) was calculated graphically; the results are given in Table 1. As can be seen from the table, the subinte—
* The distance of the center of the CH group from the center of the hexagon of carbon atoms, \(d\), was estimated from the relation \({}^{(4)}\)
\[ d = r_{\mathrm{C-C}} + r_{\mathrm{C-H}} \frac{\Phi_{0\mathrm{H}}}{\Phi_{0\mathrm{H}}+\Phi_{0\mathrm{C}}} \simeq 1.87\ \text{Å}, \]
where \(r_{\mathrm{C-C}} = 1.39\) Å is the C—C bond length; \(r_{\mathrm{C-H}} = 1.08\) Å is the C—H bond length; \(\Phi_{0\mathrm{H}}\) and \(\Phi_{0\mathrm{C}}\) are the potential energies of interaction of the H and C atoms of the molecule with the basal face of graphite at \(z_i=z_{i0}\) (\(z_i\) is the distance of force center \(i\) from the \(x,y\) plane, \(z=0\); \(z_{i0}\) is the corresponding equilibrium distance). \(\Phi_{0\mathrm{H}}\) and \(\Phi_{0\mathrm{C}}\) were estimated from formulas (6) and (7). It was assumed that the polarizability \(\alpha_{\mathrm{CH}}\) and the diamagnetic susceptibility \(\chi_{\mathrm{CH}}\) of the aromatic CH group are additively composed of the \(\alpha\) and \(\chi\) of the C and H atoms, and that the ratios \(\alpha_{\mathrm{C}}/\alpha_{\mathrm{H}}\) and \(\chi_{\mathrm{C}}/\chi_{\mathrm{H}}\) in the aromatic CH group are equal to the corresponding ratios in the aliphatic CH group. For the H and C atoms of the benzene molecule, \(z_{i0}\) were taken equal to 2.90 and 3.55 Å.
Table 1
Results of calculating integral (11) for a benzene molecule on the surface of graphite at
\[ T=293^\circ\mathrm{K};\quad f(\beta,\psi)=\int \exp\left(-\frac{\Phi}{kT}\right)\operatorname{erf}\left(-\frac{\Phi}{kT}\right)^{1/2}\,d\alpha \]
and
\[ f(\psi)=\int_0^{d/z_{i0}} \frac{f(\beta,\psi)\,\beta\,d\beta} {\left[\left(\frac{d}{z_{i0}}\right)^2-\beta^2\right]^{1/2}} . \]
| \(\beta\) | \(\dfrac{\beta}{\left[(d/z_{i0})^2-\beta^2\right]^{1/2}}\) | \(\psi=0^\circ\): \(f(\beta,\psi)\) | \(\psi=0^\circ\): \(\dfrac{f(\beta,\psi)\beta}{\left[(d/z_{i0})^2-\beta^2\right]^{1/2}}\) | \(\psi=30^\circ\): \(f(\beta,\psi)\) | \(\psi=30^\circ\): \(\dfrac{f(\beta,\psi)\beta}{\left[(d/z_{i0})^2-\beta^2\right]^{1/2}}\) |
|---|---|---|---|---|---|
| 0 | 0.00 | \(2.21\cdot10^6\) | 0.00 | — | — |
| 0.025 | 0.0475 | \(1.92\cdot10^6\) | \(0.0912\cdot10^6\) | — | — |
| 0.050 | 0.0953 | \(1.53\cdot10^6\) | \(0.146\cdot10^6\) | — | — |
| 0.100 | 0.1934 | \(0.556\cdot10^6\) | \(0.108\cdot10^6\) | \(0.556\cdot10^6\) | \(0.108\cdot10^6\) |
| 0.20 | 0.4103 | \(0.049\cdot10^6\) | \(0.020\cdot10^6\) | — | — |
| 0.30 | 0.6926 | \(0.0072\cdot10^6\) | \(0.0050\cdot10^6\) | \(0.0063\cdot10^6\) | \(0.0044\cdot10^6\) |
| 0.50 | 3.008 | \(0.00079\cdot10^6\) | \(0.0024\cdot10^6\) | \(0.00045\cdot10^6\) | \(0.0014\cdot10^6\) |
| \(f(\psi)=0.185\cdot10^5\) | \(f(\psi)=0.185\cdot10^5\) |
The integral function \(f(\psi)\) is practically independent of \(\psi\). Hence \(Q=2\pi 0.185\cdot10^5\) and \(f_{a\,\mathrm{class}}=0.74\cdot10^{29}s j_\sigma/\sigma\).
Further, in our case (4)
\[ \nu^{**}=(h\nu_z/kT)(h\nu_\vartheta/kT)^2 \left[1-\exp(-h\nu_z/kT)\right]^{-1} \left[1-\exp(-h\nu_\vartheta/kT)\right]^{-2}, \tag{12} \]
where \(\nu_z\) is the frequency of vibration of the center of the benzene molecule perpendicular to the surface, and \(\nu_\vartheta\) is the frequency of its torsional vibrations. Near the potential minimum
\[ \Phi=\Phi_{\substack{z=z_{i0}\\ \vartheta=0}} +\frac{3D}{z_{i0}^2}(z-z_{i0})^2 +\frac{3}{2}\frac{Dd^2}{z_{i0}^2}\vartheta^2, \tag{13} \]
where
\[ D=u_{01}q_1(q_1+1)+u_{02}q_2(q_2+1)+u_{0p}\frac{z_{i0}^2}{l^2}. \tag{14} \]
Hence
\[ \nu_z=\frac{1}{2\pi z_{i0}}\sqrt{\frac{6D}{m}} =2.08\cdot10^{12}\ \mathrm{sec}^{-1}, \tag{15} \]
\[ \nu_\vartheta=\frac{d}{2\pi z_{i0}}\sqrt{\frac{3D}{A}} =2.55\cdot10^{12}\ \mathrm{sec}^{-1}. \tag{16} \]
Fig. 1. Dependence of the change in the chemical potential of benzene on the filling of the graphite surface at \(20^\circ\); 1 — calculated from experiments with graphitized carbon black, 2 — calculated theoretically.
Substituting these values into (12) and (2), we obtain \(\nu^{**}=1.8\) and \(f_a=1.33\cdot10^{29}s j_\sigma/\sigma\).
The partition function for a benzene molecule in the gas phase \((\Phi=0)\)
\[ f=\frac{j_v}{\sigma}\,8\pi^2 \left(\frac{2\pi m kT}{h^2}\right)^{3/2} v\left(\frac{2\pi A kT}{h^2}\right) \left(\frac{2\pi C kT}{h^2}\right)^{1/2}, \tag{17} \]
where \(v\) is the volume of the gas. Substitution of (17) and (10), taking (2) into account, into (1), under the assumption that \(\sigma\) and \(j_v\) do not change upon adsorption, gives
\[ \Delta\mu = -RT\ln \frac{z_{i0}^{2}Q\nu^{**}p^0\omega_m}{dkT} +RT\ln\theta, \tag{18} \]
where \(p^0=760\) mm Hg is the standard gas pressure; \(\omega_m=40\ \text{\AA}^2\) \((^{11,6})\) is the area occupied by a benzene molecule in a dense monolayer, and \(\theta=\omega_m/s/N_a\).
Substituting the corresponding values in (18), we obtain, for the transfer of benzene from the gas at 760 mm to the surface of the basal face of graphite at \(293^\circ\mathrm{K}\),
\[ \Delta\mu=-3.15+1.34\lg\theta\ \text{kcal/mole}. \tag{19} \]
In Fig. 1 the initial portions of the experimental and calculated dependences \(-\Delta\mu\) on \(\theta\) are compared. Experimental curve 1 was calculated from the adsorption isotherm obtained in \({}^{(6)}\), using the thermodynamic formula \(-\Delta\mu=RT\ln 760/p(\theta)\). Calculated curve 2 lies close to the experimental one. Thus, also in the case of a polyatomic molecule, the complete theoretical calculation gave values of \(\Delta\mu\) close to the experimental ones.
Table 2
Results of approximate calculations
| No. | Character of the approximations | \(f_{a\,\mathrm{class}}/10^{29} S j_\sigma\) | Error in \(f_{a\,\mathrm{class}}\) | Error in \(\Delta\mu\), kcal/mole |
|---|---|---|---|---|
| 1 | The most rigorous variant: the form \(\Phi=\Phi(z,\vartheta,\psi)\) is given by expression (8) | 0.74 | 0 | 0 |
| 2 | Motion perpendicular to the surface is a harmonic vibration and \(\Phi\) does not depend on \(\psi\): \(\displaystyle \Phi(z,\vartheta)=\Phi(z,\vartheta)_{z=z_0}+\frac{k_z(\vartheta)}{2}(z-z_0)^2\) |
0.72 | 3% | 0.016 |
| 3 | As in variant 2, but it is assumed that \(k_z\ne f(\vartheta)\), and is equal to its value at \(\vartheta=0\) | 0.68 | 9% | 0.05 |
| 4 | Harmonic vibration perpendicular to the surface, and harmonic torsional vibrations: \(\displaystyle \Phi(z,\vartheta)=\Phi_{\substack{z=z_0\\ \vartheta=0}}+\frac{k_z(z-z_0)^2}{2}+\frac{k_\theta}{2}\vartheta^2\) |
0.23 | by a factor of 3.2 | 0.68 |
| 5 | Harmonic vibration perpendicular to the surface and free rotation \(\displaystyle \Phi(z)=\Phi_{\substack{z=z_0\\ \vartheta=0}}+\frac{k_z(z-z_0)^2}{2}\) |
44.5 | by a factor of 60 | 2.4 |
Let us now consider some other variants of the approximate calculation. In Table 2 the result of calculating \(f_{a\,\mathrm{class}}\) with expression (8) for \(\Phi\) is compared with the results of calculations of \(f_{a\,\mathrm{class}}\) for various assumptions about the state of the benzene molecule at the graphite surface. Also given are the errors in \(f_{a\,\mathrm{class}}\) and in \(\Delta\mu\) arising under these assumptions, relative to the corresponding values of the first calculation variant. The table shows the admissibility of approximations Nos. 2 and 3, which greatly facilitates the calculations.
Institute of Chemistry and Chemical Technology
Academy of Sciences of the Lithuanian SSR
Institute of Physical Chemistry, Academy of Sciences of the USSR
Received
16 III 1961
REFERENCES
- A. V. Kiselev, D. P. Poshkus, DAN, 132, 876 (1960).
- T. L. Hill, J. Chem. Phys., 14, 441 (1946).
- K. S. Pitzer, W. D. Gwinn, J. Chem. Phys., 10, 428 (1942).
- J. W. Drenan, T. L. Hill, J. Chem. Phys., 17, 775 (1949).
- J. E. Mayer, M. G. Mayer, Statistical Mechanics, N. Y., 1940.
- A. A. Isirikyan, A. V. Kiselev, J. Phys. Chem., 65, 601 (1961); A. V. Kiselev, ZhFKh, 35, 233 (1961).
- N. N. Avgul, A. A. Isirikyan et al., Izv. AN SSSR, OKhN, 1957, 1314.
- N. N. Avgul, A. V. Kiselev et al., Izv. AN SSSR, OKhN, 1959, 1196.
- J. G. Kirkwood, Phys. Zs., 33, 57 (1932); A. Müller, Proc. Roy. Soc., A154, 624 (1936).
- A. V. Kiselev, D. P. Poshkus, ZhFKh, 32, 2824 (1958).
- N. N. Avgul, G. I. Berezin et al., Izv. AN SSSR, OKhN, 1956, 1304.