PS3

Chem 240 Problem Set 3 Due Oct. 10 1. Transformation of variables and the Albery model of electrocatalysis. The Albery model is for electrons crossing an electrode’s surface to reduce a substrate. To do so the electron has to overcome a barrier of energy EA, and since there are many sites on the electrode surface then there is a normalized Gaussian distribution of barrier energies described as:

∫ P(EA) ∙ ∂EA = ∫ 1

√2π ∙ σEA

e

−(EA−EA 0 )

2

2σEA 2

∙ ∂EA (1)

The Arrhenius equation describes the timescale 𝜏 of a process as a function of

the barrier EA via: 1

𝜏 = 𝐴 ∙ 𝑒

− EA

𝑘𝐵∙𝑇 (2)

Try to substitute 𝜏 for into eq. 1 using the transformations of variables formula (with Jacobian) shown here:

∫ P(EA) ∙ ∂EA → ∫ P(g(𝜏)) ∙ | 𝜕EA

𝜕𝜏 | 𝜕𝜏

(where EA = g(𝜏) and is derived from eq. 2) and show that the resulting normalized distribution is as follows:

∫ kBT

√2π ∙ σEA ∙ τ

e

−(kBT∙ln(τ)+kBT∙ln(A)−EA 0 )

2

2σEA 2

∙ ∂τ

Note: Eq. 3 above is the lognormal distribution that is commonly expressed as:

∫ 1

√2π ∙ σ𝜏 ∙ τ e

−(ln(τ)−μ)2

2σ𝜏 2

∙ ∂τ (3)

where σ𝜏 = σEA

kBT and μ =

EA 0

kBT − ln(A) (10 pts)

2. A 3-point stencil for derivatives by MATLAB. Let’s say we wanted to develop a 3-point stencil for calculating a derivative:

𝑓(𝑥)′ = 𝑎 ∙ 𝑓(𝑥 − ℎ) + 𝑏 ∙ 𝑓(𝑥) + 𝑐 ∙ 𝑓(𝑥 + ℎ)

and a double derivative: 𝑓(𝑥)′′ = 𝑎 ∙ 𝑓(𝑥 − ℎ) + 𝑏 ∙ 𝑓(𝑥) + 𝑐 ∙ 𝑓(𝑥 + ℎ)

where the various points are defined as:

𝑎 ∙ 𝑓(𝑥 − ℎ) = 𝑎 ∙ 𝑓(𝑥) − 𝑎 ∙ ℎ

2 ∙ 𝑓(𝑥)′ + 𝑎 ∙

ℎ2

2 ∙ 𝑓(𝑥)′′

𝑏 ∙ 𝑓(𝑥) = 𝑏 ∙ 𝑓(𝑥)

𝑐 ∙ 𝑓(𝑥 + ℎ) = 𝑐 ∙ 𝑓(𝑥) + 𝑐 ∙ ℎ

2 ∙ 𝑓(𝑥)′ + 𝑐 ∙

ℎ2

2 ∙ 𝑓(𝑥)′′

Consequently, one would find that, for the derivative 𝑓(𝑥)′ it must be true that: 𝑎 + 𝑏 + 𝑐 = 0

−𝑎 ∙ ℎ

2 + 𝑐 ∙

ℎ

2 = 1

𝑎 ∙ ℎ2

2 + 𝑐 ∙

ℎ2

2 = 0

You can solve this using the following Matlab code: syms a b c h;

eqn1 = a + b + c == 0;

eqn2 = -a*h/2+c*h/2 == 1;

eqn3 = a*h^2/2 + c*h^2/2 == 0;

[A,B] = equationsToMatrix([eqn1, eqn2, eqn3], [a, b, c]);

der = linsolve(A,B)

der =

-1/h

0

1/h

Thus: 𝑓(𝑥)′ = − 1

ℎ 𝑓(𝑥 − ℎ) +

1

ℎ 𝑓(𝑥 + ℎ)

Please determine the coefficients a,b, and c for the double derivative 𝑓(𝑥)′′ using Matlab by modifying the code above. (10 pts)

3. Write a MATALB code to take the double derivative of this dataset. Submit your code and plot the function with its double derivative. Note that the data are clearly a sine wave- what is the double derivative of a sine? (10 pts) If you don’t recall, here is the code to make a nice plot: >> plot(stencil(:,1),stencil(:,2),'b');

>> xlabel('time'); ylabel('sine(time)'); >> set(gca,'TickDir','out');

4. Here we use Mathematica to study the Gaussian distribution:

P(x) = 1

√2πσ2 e

−(x−a)2

2σ2

a. Show that the distribution is properly normalized over the range −∞ ≤ x ≤ ∞.

b. Calculate the average value ⟨x⟩.

c. Also calculate ⟨x2⟩.

d. Now calculate the variance ⟨x2⟩ − ⟨x⟩2. (10 pts)

5. A Poisson distribution is used to describe things like the number of drops

hitting you per minute in a light rain. Please use Mathematica to show that the Poisson probability distribution:

λke−λ

k!

is properly normalized; you will use two different methods:

a. Try to demonstrate this way:

∫ λke−λ

k!

∞

0

∂k

b. Before you go too crazy, you should know that integration in pt. a won’t work.

Try summing the distribution instead. Why does this approach reveal proper normalization while integration (pt. a) failed? Note, you may have to do some googling to see how to do a summation in Mathematica.

c. Now that you know how to properly “work” the Poisson distribution, please

calculate the average value ⟨k⟩.

d. Also calculate ⟨k2⟩, and the variance ⟨k2⟩ − ⟨k⟩2. (10 pts)

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