If we know the free energy of an adsorbed gas as a function of areal density, then we can also calculate the pressure. This arises because two phases in equilibrium have the same chemical potential, thus in this case the free energy of the adsorbed phase 'mu-a' is equal to 'mu-g'. We already have an expression for mu-a, and mu-g is related to the pressure via the statistical mechanics of the perfect gas, as set out at the beginning of section C2 of the Surface Physics course. We can see qualitatively that the lattice parameter of the adsorbed layer can effect the pressure, via the atomic and entropic forces between the atoms. The smaller the lattice parameter the closer together are the atoms adsorbed on the graphite, which generates a greater repulsive force. In order to balance the system so that a specific lattice parameter can be maintained there must be a pressure to counteract this repulsion; the 2-dimensional pressure on the surface is called the spreading pressure, and this is generated by an increase in the 3-dimensional pressure within the gas phase.
Quantitatively, the chemical potential is the derivative of free energy with respect to number density; in this 2-dimensional problem this is dF/dn, where n is the areal density in the film. For a close-packed solid n ~ a to the power (-2), so ndF/dn can be expressed as -(a/2)dF/da. Thus this change in energy can be used in conjunction with the free energy values for a given lattice parameter to determine the pressure of the system. This is shown by using equations (1) and (2) to obtain an expression for the pressure, p. This expression (with a conversion of units from Pascal to Torr) can be seen in equation (3). The expression of Delta-F in this final equation (3) represents the energy change resulting from the changing atomic and entropic forces as the lattice spacing is changed.
To determine the change in the system pressure due to variations in the substrate coverage parabolas were fitted to the free energy as a function of lattice parameter at the given temperature. From this the first derivative of the curves provided the change in the chemical potential with respect to the substrate coverage, Delta-F. This analytic derivative for the three different lattice parameter results in equations (4a), (4b), and (4c) which represent the chemical potential change for lattice parameters 3.09A, 3.19A, and 3.29A respectively.
Equations (4a), (4b), and (4c) can then be substituted into equation (3) in order to determine the pressure of the system at each lattice parameter. This resulted in the data shown in the table and a full range of this data over the examined temperature range produced the plot.
(100/T) (K-1) |
Pressure 3.09 (Torr) |
Pressure 3.19 (Torr) |
Pressure 3.29 (Torr) |
|---|---|---|---|
8 |
5.85E-12 |
3.31E-13 |
2.22E-14 |
7.84 |
1.60E-11 |
9.46E-13 |
6.63E-14 |
7.69 |
4.20E-11 |
2.60E-12 |
1.90E-13 |
7.54 |
1.06E-10 |
6.88E-12 |
5.24E-13 |
7.40 |
2.60E-10 |
1.76E-11 |
1.39E-12 |
7.27 |
6.16E-10 |
4.33E-11 |
3.55E-12 |
7.14 |
1.41E-9 |
1.03E-10 |
8.78E-12 |
7.01 |
3.15E-9 |
2.39E-10 |
2.10E-11 |
6.89 |
6.83E-9 |
5.36E-10 |
4.89E-11 |
This data table only indicates the low T end of the data shown on the plot. The next stage is to check these values in detail, and to compare them with the experimental results obtained by Suzanne et al. (1982).
Continue to the following section 3, or alternatively return to section 1.
