9.9 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Adsorption | 2/6 | https://en.wikipedia.org/wiki/Adsorption | reference | science, encyclopedia | 2026-05-05T10:45:47.677105+00:00 | kb-cron |
where
P
{\displaystyle P}
is the partial pressure of the gas or the molar concentration of the solution. For very low pressures
θ
≈
K
P
{\displaystyle \theta \approx KP}
, and for high pressures
θ
≈
1
{\displaystyle \theta \approx 1}
. The value of
θ
{\displaystyle \theta }
is difficult to measure experimentally; usually, the adsorbate is a gas and the quantity adsorbed is given in moles, grams, or gas volumes at standard temperature and pressure (STP) per gram of adsorbent. If we call vmon the STP volume of adsorbate required to form a monolayer on the adsorbent (per gram of adsorbent), then
θ
=
v
v
mon
{\displaystyle \theta ={\frac {v}{v_{\text{mon}}}}}
, and we obtain an expression for a straight line:
1
v
=
1
K
v
mon
1
P
+
1
v
mon
.
{\displaystyle {\frac {1}{v}}={\frac {1}{Kv_{\text{mon}}}}{\frac {1}{P}}+{\frac {1}{v_{\text{mon}}}}.}
Through its slope and y intercept we can obtain vmon and K, which are constants for each adsorbent–adsorbate pair at a given temperature. vmon is related to the number of adsorption sites through the ideal gas law. If we assume that the number of sites is just the whole area of the solid divided into the cross section of the adsorbate molecules, we can easily calculate the surface area of the adsorbent. The surface area of an adsorbent depends on its structure: the more pores it has, the greater the area, which has a big influence on reactions on surfaces. If more than one gas adsorbs on the surface, we define
θ
E
{\displaystyle \theta _{E}}
as the fraction of empty sites, and we have:
θ
E
=
1
1
+
∑
i
=
1
n
K
i
P
i
.
{\displaystyle \theta _{E}={\dfrac {1}{1+\sum _{i=1}^{n}K_{i}P_{i}}}.}
Also, we can define
θ
j
{\displaystyle \theta _{j}}
as the fraction of the sites occupied by the j-th gas:
θ
j
=
K
j
P
j
1
+
∑
i
=
1
n
K
i
P
i
,
{\displaystyle \theta _{j}={\dfrac {K_{j}P_{j}}{1+\sum _{i=1}^{n}K_{i}P_{i}}},}
where i is each one of the gases that adsorb. Note:
- To choose between the Langmuir and Freundlich equations, the enthalpies of adsorption must be investigated. While the Langmuir model assumes that the energy of adsorption remains constant with surface occupancy, the Freundlich equation is derived with the assumption that the heat of adsorption continually decrease as the binding sites are occupied. The choice of the model based on best fitting of the data is a common misconception.
- The use of the linearized form of the Langmuir model is no longer common practice. Advances in computational power allowed for nonlinear regression to be performed quickly and with higher confidence since no data transformation is required.
=== BET ===
Often molecules do form multilayers, that is, some are adsorbed on already adsorbed molecules, and the Langmuir isotherm is not valid. In 1938 Stephen Brunauer, Paul Emmett, and Edward Teller developed a model isotherm that takes that possibility into account. Their theory is called BET theory, after the initials in their last names. They modified Langmuir's mechanism as follows:
A(g) + S ⇌ AS, A(g) + AS ⇌ A2S, A(g) + A2S ⇌ A3S, and so on.
The derivation of the formula is more complicated than Langmuir's (see links for complete derivation). We obtain:
x
v
(
1
−
x
)
=
1
v
mon
c
+
x
(
c
−
1
)
v
mon
c
,
{\displaystyle {\frac {x}{v(1-x)}}={\frac {1}{v_{\text{mon}}c}}+{\frac {x(c-1)}{v_{\text{mon}}c}},}
where x is the pressure divided by the vapor pressure for the adsorbate at that temperature (usually denoted
P
/
P
0
{\displaystyle P/P_{0}}
), v is the STP volume of adsorbed adsorbate, vmon is the STP volume of the amount of adsorbate required to form a monolayer, and c is the equilibrium constant K we used in Langmuir isotherm multiplied by the vapor pressure of the adsorbate. The key assumption used in deriving the BET equation that the successive heats of adsorption for all layers except the first are equal to the heat of condensation of the adsorbate. The Langmuir isotherm is usually better for chemisorption, and the BET isotherm works better for physisorption for non-microporous surfaces.
=== Kisliuk ===
In other instances, molecular interactions between gas molecules previously adsorbed on a solid surface form significant interactions with gas molecules in the gaseous phases. Hence, adsorption of gas molecules to the surface is more likely to occur around gas molecules that are already present on the solid surface, rendering the Langmuir adsorption isotherm ineffective for the purposes of modelling. This effect was studied in a system where nitrogen was the adsorbate and tungsten was the adsorbent by Paul Kisliuk (1922–2008) in 1957. To compensate for the increased probability of adsorption occurring around molecules present on the substrate surface, Kisliuk developed the precursor state theory, whereby molecules would enter a precursor state at the interface between the solid adsorbent and adsorbate in the gaseous phase. From here, adsorbate molecules would either adsorb to the adsorbent or desorb into the gaseous phase. The probability of adsorption occurring from the precursor state is dependent on the adsorbate's proximity to other adsorbate molecules that have already been adsorbed. If the adsorbate molecule in the precursor state is in close proximity to an adsorbate molecule that has already formed on the surface, it has a sticking probability reflected by the size of the SE constant and will either be adsorbed from the precursor state at a rate of kEC or will desorb into the gaseous phase at a rate of kES. If an adsorbate molecule enters the precursor state at a location that is remote from any other previously adsorbed adsorbate molecules, the sticking probability is reflected by the size of the SD constant. These factors were included as part of a single constant termed a "sticking coefficient", kE, described below:
k
E
=
S
E
k
ES
S
D
.
{\displaystyle k_{\text{E}}={\frac {S_{\text{E}}}{k_{\text{ES}}S_{\text{D}}}}.}