where the length a is an atomic dimension (a25A/N0). The comparison of Langmuir and 2D gas isotherms is illustrated in ®gure 4.1(b) for Xe/graphite parameters at T580K, using the reasonably realistic value of the well depth (E0 2 Ea)536 K (Kariotis et al. 1988). Note that in both models the coverage varies linearly with pressure at low coverage. However, as shown here, the 2D gas model is most appropriate, but if the well depth were much larger, the localized model with vibrations would be a better description. The model without vibrations is numerically quite poor in all such situations.

The second equality (4.11b) is only true for the Einstein model at high temperature. In this limit, where equipartition of energy holds (no term in h), the following argument can be made. Localized atoms vibrate with amplitude x, and 4p2mx2nd2 is the energy associated with this 2D oscillation, which is equal to 2kT at high temperature, if a harmonic approximation is good enough (a big if ). Thus, the pre-exponential is just a ratio of free areas (a2/px2), the numerator associated with the 2D gas, and the denominator with the potential well in which the adatom vibrates. Clearly the vibrational model starts to fail as x increases towards px25a2.

4.2.4Interactions and vibrations in higher density adsorbates

To consider the statistical mechanics of higher density adsorbates, we need both the interaction potentials and suitable models of the atomic vibrations. In analogy with the 3D case, moderate densities in a 2D ¯uid phase can be described by virial expansions (Hill 1960, chapter 15, Bruch et al. 1997, section 4.2.2). The spreading pressure is given by

in which the ®rst term in an expansion in powers of the 2D density (Na/A) is the second virial coeYcient, B2(T), given by

where the interaction potential U(r) is between two atoms; the 2D integral is performed over the substrate area A, where for cylindrical symmetry dr; 2prdr. In a relatively low-density gas at high T, this integral is small due to the fact that the atoms spend most of their time outside the range of in¯uence of U(r).

There is a continuous line of reasoning between the argument leading to (4.11), (4.12) and the cell model of lattice vibrations. This model was originally introduced by Lennard-Jones and Devonshire (1937, 1938) as an approximation of the 3D liquid state, and described e.g. by Hill (1960, chapter 16) and Bruch et al. (1997, chapter 5).

The free area, Af5px2 in the discussion following (4.11), is de®ned by integrating the Boltzmann factor over the `cell' in which the atom vibrates, namely