## 4. Thermodynamic properties of the MgO clusters## 4.1. Calculation of the free enthalpyThe thermochemical properties of the MgO molecule and of its
clusters can be determined once the enthalpy of
formation of the clusters and their entropy is
known. The enthalpy of formation of a cluster of where is given in Table (4) and is the heat content of the internal degrees of freedom of the cluster. The contribution of the vibrational degrees of freedom to the
internal energy with being the Avogadro number (note that
The translational, rotational, vibrational and electronic contributions to the partition function of a cluster are assumed to be independent of each other. The entropy in this case is additive in these contributions and we have for the contribution of the translational degrees of freedom, for the contribution of the rotational degree of freedom of a
for the contribution of the rotational degrees of freedom for
for the vibrational degrees of freedom (see e.g. Benson 1976).
The contribution of the vibrational degrees of freedom to the
entropy From the data given in Table 4 and the equations (34), ...,
(38) we can calculate the equilibrium abundance of a cluster of size
is the difference between the entropy of
one cluster of size ## 4.2. The cluster size spectrumUsing the results for the bond energy of the clusters, their
moments of inertia the symmetry numbers and the vibrational
frequencies we can calculate the abundance of the
clusters in a state of thermodynamic
equilibrium. The result for a representative H
Application of the law of mass action to the heterogeneous equilibrium between the monomer and the solid yields where is the enthalpy of formation of the solid from the monomers and the corresponding entropy change. is the partial pressure of the monomers in equilibrium with the condensed phase. We define the supersaturation ratio as is the difference in the bond energies of
monomers between the cluster and the solid. This difference is
positive since the bond energy of monomers in the solid is higher than
in a cluster built from a finite number of monomers. This can clearly
be seen from Table 4 or Fig. 1. This bond energy defect
results from the reduced attractive interaction energy of monomers
with their environment for particles close to the surface as compared
to interior particles. In a small cluster, all particles are close to
the surface while for the solid such particles form a completely
negligible fraction of all particles. In principle, the bond energy
defect is roughly proportional to the surface area of the cluster,
which in turn for compact structures varies roughly as
with the number For small cluster sizes is a big number
since the energy defect for small clusters is of the order of 2 to 3
eV per particle (cf. Fig. 1) while The entropy difference in (43) is dominated by the large and
positive contribution of the translational degrees of freedom (cf.
Eqs. (36), (37) and (38)), which shows only a weak logarithmic
dependence on The term in (43) varies linearly with the
cluster size - . All three terms in the exponential in
Eq. (43) are negative. The equilibrium pressure of clusters relative
to monomers decreases with increasing size - . The term proportional to
increases linearly with increasing then the exponential in (43) first is dominated for small - and inequality (45) not satisfied. The
exponential is always dominated by the term
and only increases with increasing In any case, for a supersaturated vapour we have the following
asymptotic behaviour of the equilibrium density of clusters for large
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |