Many chemical reactions are facilitated by the presence of so called catalysts, sub- stances that take part in the reaction but are not consumed themselves. The mode of action of a catalyst varies, but the end result is that the energy of activation of the reaction is lowered, which leads to an increased reaction rate. The catalysis of chemical reaction is essential for living beings. Most people associate catalysts with cars, where they aid in cleaning the exhausts, but as a matter of fact nearly all of the
© Springer International Publishing Switzerland 2016
P. Gerlee and T. Lundh,Scientific Models, DOI 10.1007/978-3-319-27081-4_5
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68 Worked Examples chemical reactions that take place in our bodies are controlled by a certain type of proteins, called enzymes, that serve as catalysts of biochemical reactions.
Catalysis is also an important tool in the production of chemical products. For example, iron is used as a catalyst when the elements nitrogen and hydrogen react to form ammonium, and nickel is used in the production of margarine, in a process where hydrogen gas reacts with unsaturated fatty acids. These two catalytic reactions share a common feature in that they occur on the surface of the catalysts onto which the reactants bind and react with one another. The products of the reaction then disassociate from the surface and can be harvested.
In order to control and optimise reactions such as these it is important to understand how a chemical compound is adsorbed (binds) and desorbed (is released) from a surface, and how these processes are influenced by external factors such as pressure and temperature.1A complete description of these relationships would be immensely complex and formulated in terms of quantum mechanical wave functions, but luckily there are models that provide simplified descriptions that for most purposes are sufficient. A common model of this phenomenon is so called Langmuir adsorption, first put forward by Irving Langmuir at the beginning of the 20th century.2In its most basic form, which we will discuss here, it describes, with the aid of a mathematical formula, how much a surface can adsorb of a certain compound as a function of the concentration or partial pressure of the compound.3It is a symbolic model that relates a number of physical quantities to each other.
Let us consider a gas (A) that is in contact with a surface (M) and assume that the gas is in a state of dynamic equilibrium (see Fig.1). By this we mean that despite the fact that gas molecules are constantly being adsorbed and desorbed at the surface, the mean number of molecules bound to the surface is constant. This type of relationship is usually written:
A(gas)+M(surface)ka
kd
AM(surface) (1)
where the reaction to the right occurs with rateka and the one to the left with rate kd. These constants are known as the adsorption and desorption rates and describe how fast each reaction occurs.
In addition to the assumption about dynamic equilibrium we will also make the following simplifying assumptions:
1. The gas molecules are assumed to only cover the surface in a single layer, i.e.
they form a so called monolayer.
1Please note the difference between adsorption and absorption: the former case refers to the binding on a surface, while the latter corresponds to the uptake of a substance in another gas, liquid or solid substance.
2Langmuir, I. (1916). The constitution and fundamental properties of solids and liquids. Part I.
Solids. Journal of the American Chemical Society 38(11): 2221–2295.
3The partial pressure is the pressure a gas would exert on a surface if all other gases (such as air) were absent.
Fig. 1 A schematic image of a gas being adsorbed and desorbed from a surface. The constantskaandkddescribe the rate with which the two processes occur
surface gas
k
dk
a2. The surface is assumed to be homogenous and uniform. Hence we assume that it is irrelevant where on the surface a gas molecule is bound.
3. We assume that there are no interactions between the bound molecules. The like- lihood of a molecule being bound in a specific location is therefore independent of the state of the neighbouring locations.
Let us now consider a part of the surface that containsN(independent) locations where adsorption can occur, and letθdenote the fraction of locations that are occu- pied. The gas Ais adsorbed onto the surface with a rate that is proportional to the partial pressure of the gasp(this is because the number of collisions occurring per unit time is proportional to the pressure). But the rate of adsorption is also proportional to the number of unoccupied locations given by(1−θ)N, and lastly to the adsorption constant ka. Therefore the total rate of adsorption is given byka(1−θ)N p. The rate of desorption on the other hand is given by the number of bound moleculesθN multiplied by the desorption constantkd. At dynamic equilibrium these quantities must be equal, which leads to the relation
ka(1−θ)N p=kdNθ (2)
which, with a bit of manipulation can be rewritten as
K p(1−θ)=θ (3)
where K =ka/kd. This expression can finally be written as θ= K p
1+K p. (4)
70 Worked Examples
Fig. 2 The Langmuir isotherm describes how large a fraction of a surface is covered by adsorbed molecules as a function of the pressurepin the gas.
K =ka/kdis the ratio of the adsorption and desorption constants
0 1 2 3 4 5 6 7 8 9 10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
pressure p
fraction of occupied locations
K = 4
This relationship is usually called the Langmuir isotherm, and describes the fraction of the surface that is coveredθ(when the temperature is constant) as a function of K and the pressure p. Figure2shows it as a function of p when K =4, and we can see that for small values of pthe number of occupied locations increases almost linearly (θ ≈K p), while when the pressure is large (K p1)θis roughly constant and equal to unity.
This very simple model with its strict assumptions works surprisingly well and is often used as a first approximation. It can also be extended in order to describe catalytic systems where two or more gases are adsorbed to and react on a surface, and is then referred to as the Langmuir–Hinshelwood model. There are also more advanced models, such as the BET-model that allows for multiple layers of adsorbed molecules and therefore provides more accurate predictions at high pressures.4