CH-5010: Chemical Reactor Theory
July – Nov. 2023
Problem Sheet – 3
Monday, 28th August 2023
Problem 1: Pre-Read
Please read the material from Text-Book
All the examples we will do in this problem sheet are from Davis and Davis, Chapter 5.
Please pre-read Pages 144 to 171, especially the first twelve pages (until page 156).
Link to Chapter 5: https://authors.library.caltech.edu/25070/6/FundChemReaxEngCh5.pdf
Recap of Derivation in the Week of 21st August
We have already done the Langmuir adsorption isotherm for:
A + ∗ ⇌ A∗
where we showed that at equilibrium,
𝐶"∗ =
𝐾" 𝐶"
1 + 𝐾" 𝐶"
Thereafter we considered the following dissociative adsorption:
A# + 2 ∗ ⇌ 2A∗
(1)
For this case, the rate of forward reaction is:
𝑟$ = 𝑘$ 𝐶"" 𝐶∗#
and the rate of backward reaction is:
𝑟%$ = 𝑘%$ 𝐶"#∗
Equilibrium: When the reaction is in equilibrium: 𝑟$ − 𝑟%$ = 0, which leads to: 𝐶"∗ = /𝐾" 𝐶" 𝐶∗
Let us consider instead that pseudo-steady-state is invoked on A*.
Since there was some confusion among students, let’s look at another (non-catalytic) reaction:
𝐴 → 2𝐵
Clearly, in this reaction, 𝑅& = 2𝑟 = 2𝑘𝐶" , where 2 is the stoichiometric coefficient.
Likewise, for the reaction 2𝐵 → Products, we have studied that 𝑅& = 2𝑘𝐶&# .
Replace B with A* and we will see that the forward reaction in (1) is similar to the red text
and the reverse reaction is similar to the blue text. Thus, for A* involved in reactions (1) and (–1):
𝑑
𝐶 ∗ = 2𝑘$ 𝐶"" 𝐶∗# − 2𝑘%$ 𝐶"#∗
𝑑𝑡 "
Only for this example, PSSA on A* leads to the same expression: 𝐶"∗ = /𝐾" 𝐶" 𝐶∗
From this point, we can derive the isotherm for dissociative adsorption of A* at equilibrium:
𝐶"∗ =
/𝐾" 𝐶"
1 + /𝐾" 𝐶"
Γ,
or
𝜃"∗ =
/𝐾" 𝐶"
1 + /𝐾" 𝐶"
CH-5010: Chemical Reactor Theory
CH-5010: Chemical Reactor Theory
July – Nov. 2023
Problem 2: Reaction involving single reactant
Several isomerization reactions and those involving deposition in semiconductor industry
follow a mechanism where an overall reaction may be written as: 𝐴 → Products.
Part 2.A.: When A is the only MARI (most abundant reactive intermediate)
Recall that we derived Langmuir adsorption for the following reaction in partial equilibrium:
𝐴 + ∗ ⇌ 𝐴∗
1. Use Partial Equilibrium assumption to express 𝐶"∗ in terms of 𝐶" and 𝐶∗
2. Use overall site balance to obtain 𝐶∗ and 𝐶"∗ in terms of measurable quantity (𝐶" )
3. If the reaction giving products:
𝐴∗ → Products + ∗
is rate limiting, then show that the net reaction rate for 𝐴 → Products is given by:
𝑟'() =
𝑘'() 𝐶"
1 + 𝐾" 𝐶"
Part 2.B. Thought Question
If the reaction that gives products consists of two steps:
𝐴∗ → 𝑃∗
𝑃∗ ⇌ 𝑃 + ∗
4. If 𝐶*∗ is negligible our analysis remains exactly the same as Part 2.A. Please discuss why.
Part 2.C. Adsorption with multiple MARI
Let us now consider a problem like the one in Part 2.B, but where significant amount of P* is
also adsorbed. Thus, the reactions involved are:
𝑘$
𝐴 + ∗ ⇌ 𝐴∗
𝑘#$
𝑘#
𝐴∗ DE 𝑃∗
𝑘+
𝑃∗ ⇌ 𝑃 + ∗
𝑘#%
5. Use the partial equilibrium assumption for the first and third reactions.
6. Use the site balance to obtain 𝐶∗ in terms of measurable concentrations of gases.
7. Hence obtain the net rate of reaction as:
𝑟'() =
𝑘'() 𝐶"
1 + 𝐾" 𝐶" + 𝐾* 𝐶*
Also express how (i) 𝐾, are related to 𝐾" and 𝐾* , and (ii) how 𝑘'() is related to 𝑘- and 𝐾- .
CH-5010: Chemical Reactor Theory
CH-5010: Chemical Reactor Theory
July – Nov. 2023
Problem 3: More realistic case of reversible rate limiting step
In Part 2.C, rate limiting step (reaction 2) was assumed to be irreversible for simplicity.
8. Please update your analysis if the rate-limiting reaction was reversible (all else remains same)
𝑘#
𝐴 ⇌ 𝑃∗
𝑘#"
∗
Hint-1: In Part 2.C, you considered the net reaction rate was: 𝑟'() = 𝑟# . Now, the net reaction rate
may instead be considered as: 𝑟'() = (𝑟# − 𝑟%# ).
Hint-2: The net reaction gets the following form:
𝑟'() =
𝑘'() 𝐶"
𝐶*
F1 −
G
1 + 𝐾" 𝐶" + 𝐾* 𝐶*
𝐾(. 𝐶"
(How is 𝐾(. for the net reaction related to 𝐾$ , 𝐾# , 𝐾+ ?
Problem 4: Analysis if another step was rate-limiting
Please repeat the analysis if the first reaction was rate-limiting:
𝑘$
𝐴 + ∗ ⇌ 𝐴∗
𝑘#$
𝑘#
𝐴 ⇌ 𝑃∗
𝑘#"
∗
𝑘+
𝑃 ⇌ 𝑃+∗
𝑘#%
∗
Rate limiting
Partial equil.
Partial equil.
CH-5010: Chemical Reactor Theory