Download Diffusion: Thermal Activation Controlling Reaction Rates in Chemistry & Technology and more Study notes Chemistry in PDF only on Docsity!
CHGN 124
Supplementary Material DIFFUSION To this point we have assumed the rate-limiting step in a chemical reaction results from collisions between reacting molecules. While this is often an excellent assumption, it is not always true. Frequently, atoms or molecules moving from a region of high to low concentration controls the rate-limiting step in a reaction. In other words, moving against a concentration gradient. This motion is referred to as diffusion. We have all experienced diffusion. For example, when a bottle of perfume is opened, it takes only a few seconds before someone across the room is aware of the smell. The molecules of the perfume vapor have diffused across the room and reacted with molecules in the olfactory lobes of the observer’s nose. This reaction is fast compared to the time needed for the perfume molecules to cross the room and so in this process the rate-limiting step is the diffusion step. A more technologically interesting example is offered by the processing of silicon chips. Before the chips in your calculator and computers will function properly they must be processed to create what is called a p-n junction. Essentially this involves allowing a “dopant” like boron to diffuse into a silicon wafer to a specified depth.
Figure 1 To make a p-n junction one side of a silicon wafer is exposed to hot boron gas. The boron atoms diffuse into the silicon. The process is stopped when the boron atoms have diffused to a known depth by removing the born atmosphere and cooling the wafer. In this and many more processes the depth to which the diffusing species penetrates must be controlled exactly. Such control requires a detailed understanding the rate laws governing diffusion. And like all reaction rates, the key to understanding is the mechanism and it’s associated activation energy.
Interstitial diffusion in a solid The atoms of solids are much less mobile than those of liquids and consequently are less able to accommodate the motion of diffusing species. However, some small atoms, which are located in the interstices between solid atoms and are consequently called interstitials, have only moderate activation energies. I like to think of interstitial diffusion like a motorcycle moving through congested traffic. And like a motorcycle small atoms such as boron, carbon, hydrogen and helium diffuse comparatively rapidly through many solids. Activation energies for interstitial diffusion are on the order of 50 to 100 kJoules/mole. Diffusion in a crystalline solid On the other hand, large atoms and molecules are constrained and unable to move through crystalline solids, as the activation energy required for such jumps is huge on the order of thousands of kJoules/mole. However, measurement of jump rates in solids suggests that large atoms move more readily than is expected.
Diffusion via vacancies vacancy vacancy To explain this unexpected observation it has been proposed that large atoms diffuse through crystalline solids via naturally occurring defects called vacancies. The activation energies associated with vacancy-assisted diffusion are just a little larger than those associated with interstitial diffusion (200 to 400 kJoules/mole) making vacancy diffusion a significant process at elevated temperatures. Diffusion in general displays first order kinetics, i.e. rate = k [Sd], where k is the rate constant and [Sd] is the concentration of the diffusing species. In the case of vacancy assisted diffusion, the rate for the overall process obeys second order kinetics, i.e. rate = k [Sd][V], where here [V] is the concentration of vacancies. However, because vacancy concentration is a constant at a given temperature and a given solid composition, the apparent rate law is first order. In this mechanism vacancies are acting as catalysts and anything that increases the concentration of vacancies will increase the diffusion rate. This is a subject to which we will return at the end of the semester. Like chemical reactions, diffusion is a thermally activated process and obeys an “Arrhenius-type” equation:
D= Do e
-Ea/RT
In the case of diffusion we need to know not only the concentration as a function of time, but also the concentration as a function of distance from the source. The instantaneous form of our rate law can be expressed as a differential equation called Fick’s Second Law: € ∂[ Sd ] ∂ t
= D
∂^2 [ Sd ] ∂ x^2 Here, as before, [ Sd ] is the concentration of the diffusing species, t is time and x is the depth of penetration. The solution of this differential equation, which will give us the counterpart of the integrated rate law, depends on the initial distribution of diffusing species—a point source will diffuse differently than a planar source, as in the example of boron atoms diffusing into a silicon wafer. While a general solution to Fick’s Second Law will take us too far a-field we will give the solution to the problem of planar diffusion from a surface with a constant concentration of the diffusing species at the surface, [S d ]0,0 where the subscript 0, denotes the concentration initially ( t = 0) and on the surface ( x = 0). In this case the concentration at time t and distance x from the surface, is given by the relation: € [ Sd ] t , x = [ Sd ]0, 0 1 − erf x 2 Dt
The error function, erf, is a common function occurring often in statistics, just as the sine or cosine functions appear in geometry. Like all functions, it delivers a number depending on the value of the argument. Many calculators calculate the value of the error function. If you do not have such a calculator the tabulated value for the error function are given in Table 2 below. Using this function we can calculate the concentration of a diffusing species at a given time, if we know the corresponding diffusion constant.
Try using this function to solve the problems posted on the web site and
The error function
- Table included at the end of these notes.
- 0 0.0000 0.85 0. z erf (z) z erf (z)
- 0.025 0.0282 0.90 0.
- 0.05 0.0564 0.95 0.
- 0.10 0.1125 1.0 0.
- 0.15 0.1680 1.1 0.
- 0.20 0.2227 1.2 0.
- 0.25 0.2763 1.3 0.
- 0.30 0.3286 1.4 0.
- 0.35 0.3794 1.5 0.
- 0.40 0.4234 1.6 0.
- 0.45 0.4755 1.7 0.
- 0.50 0.5205 1.8 0.
- 0.55 0.5633 1.9 0.
- 0.60 0.6039 2.0 0.
- 0.65 0.6420 2.2 0.
- 0.70 0.6778 2.4 0.
