the first layer are occupied. To derive Langmuir’s formula for adsorption, one starts
with the following reaction:
M G þ V ! N À V
ð12:3Þ
where M G is the quantity of gas to be adsorbed, N is the number of possible sites
for adsorption, and V is the number of vacant site for adsorption at the surface.
Equation (12.3) assumes that there is no interaction between the adsorbed molecules
and that the reaction is not influenced by the coverage, or the enthalpy of adsorption is
independent of the coverage. The equilibrium constant K of Eq. (12.3) is:
K ¼
N À V
VM G
ð12:4Þ
The relative amount of adsorbate is H ¼ ðN À VÞ=N and the number of adsorbed
gas molecules M G is proportional to the gas pressure p; therefore, one may assume
M G ¼ ap. The number of vacancies is V ¼ Nð1 À HÞ; hence, one obtains for the
equilibrium constant:
K ¼
H
ð1 À HÞap
or modified b ¼
H
ð1 À HÞp
ð12:5Þ
The amount of gas adsorbed at the surface is experimentally accessible; therefore,
the following expression derived from Eq. (12.5) is used:
H ¼
bp
1 þ bp
ð12:6Þ
Equation (12.6) is the famous Langmuir adsorption isotherm. For large values of
the gas pressure p, H approaches asymptotically 1. This is independent of the
temperature. The factor b is determined by measuring the adsorption isotherm at
different temperatures; it is a function of the enthalpy of adsorption DH ads :
b ¼ exp
DH ads
RT
ð12:7Þ
where R is the molar gas constant and T is the temperature. From Eq. (12.7) it is
obvious that b increases with decreasing temperature. To obtain the asymptotic value
of H it is advised that these measurements be made, if possible, at low temperatures.
In most cases, nitrogen is selected as the adsorbate and therefore it is not too difficult
to make the measurements at the temperature of boiling nitrogen. Figure 12.2
displays Langmuir’s adsorption isotherms for different values of the temperaturedependent equilibrium constant b. Figure 12.2 also demonstrates, drastically, the
advantage of applying low temperatures for adsorption measurements.
At first glance, Eq. (12.6) seems to be ready for evaluation and for determining
specific areas. However, one of the crucial assumptions – the adsorption of one
complete monolayer – is a too far-reaching simplification. Brunauer et al. [1]
expanded Langmuir’s theory of adsorption isotherms by taking the possibility of
multiple layers of adsorbed gas atoms into account. As did Langmuir, these authors
assumed that there is no interaction between the layers of adsorbed molecules and,
12.2 Global Methods for Characterization j337
with the following reaction:
M G þ V ! N À V
ð12:3Þ
where M G is the quantity of gas to be adsorbed, N is the number of possible sites
for adsorption, and V is the number of vacant site for adsorption at the surface.
Equation (12.3) assumes that there is no interaction between the adsorbed molecules
and that the reaction is not influenced by the coverage, or the enthalpy of adsorption is
independent of the coverage. The equilibrium constant K of Eq. (12.3) is:
K ¼
N À V
VM G
ð12:4Þ
The relative amount of adsorbate is H ¼ ðN À VÞ=N and the number of adsorbed
gas molecules M G is proportional to the gas pressure p; therefore, one may assume
M G ¼ ap. The number of vacancies is V ¼ Nð1 À HÞ; hence, one obtains for the
equilibrium constant:
K ¼
H
ð1 À HÞap
or modified b ¼
H
ð1 À HÞp
ð12:5Þ
The amount of gas adsorbed at the surface is experimentally accessible; therefore,
the following expression derived from Eq. (12.5) is used:
H ¼
bp
1 þ bp
ð12:6Þ
Equation (12.6) is the famous Langmuir adsorption isotherm. For large values of
the gas pressure p, H approaches asymptotically 1. This is independent of the
temperature. The factor b is determined by measuring the adsorption isotherm at
different temperatures; it is a function of the enthalpy of adsorption DH ads :
b ¼ exp
DH ads
RT
ð12:7Þ
where R is the molar gas constant and T is the temperature. From Eq. (12.7) it is
obvious that b increases with decreasing temperature. To obtain the asymptotic value
of H it is advised that these measurements be made, if possible, at low temperatures.
In most cases, nitrogen is selected as the adsorbate and therefore it is not too difficult
to make the measurements at the temperature of boiling nitrogen. Figure 12.2
displays Langmuir’s adsorption isotherms for different values of the temperaturedependent equilibrium constant b. Figure 12.2 also demonstrates, drastically, the
advantage of applying low temperatures for adsorption measurements.
At first glance, Eq. (12.6) seems to be ready for evaluation and for determining
specific areas. However, one of the crucial assumptions – the adsorption of one
complete monolayer – is a too far-reaching simplification. Brunauer et al. [1]
expanded Langmuir’s theory of adsorption isotherms by taking the possibility of
multiple layers of adsorbed gas atoms into account. As did Langmuir, these authors
assumed that there is no interaction between the layers of adsorbed molecules and,
12.2 Global Methods for Characterization j337