CHBE 590: Homework 2
Please complete the following assignment by uploading your files to Canvas by February 19th,
2025 at 11:59 PM. Submissions received any additional day after this deadline will receive a 10%
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1) Consider the Ostwald process of oxidizing ammonia for the ultimate formation of nitric acid.
The initial step involves the conversion of ammonia to nitric oxide over a platinum-rhodium
catalyst. You may assume Langmuir adsorption and mean field approximations in all questions
below.
4NH3 + 5O2 ↔ 4NO + 6 H2O
a) Suppose the reaction occurs at 850°C and 10 bar of pressure with a feed of 10 vol%
ammonia, 89.74 vol% air, 0.25 vol% water, and 0.01 vol% NO. Assume the system behaves
as an ideal gas and that no reaction has occurred. What is the chemical affinity of the total
reaction? What is the reversibility coefficient of the total reaction (ztot)? What is ztot after
the reaction reaches equilibrium?
b) Propose a series of elementary reactions involving sequential ammonia oxidation,
terminated by nitric oxide desorption to complete the net stoichiometry. You may assume
oxygen adsorbs dissociatively. Assume that OH* primarily forms and is consumed by rapid
reactions of gas phase H2O with surface oxygen (H2O + O* + * ⟷ 2OH*). Assume H2O*
and H* never form on the surface. Assume that ammonia sequentially oxidizes through
nitrogen-containing species that react primarily with monoatomic surface oxygen species
(O*).
c) Derive the complete De Donder relationship for
i) All surface intermediates in this system
ii) The general coverage expression (𝜃∗ )
iii) All forward rates of NO formation, accounting for coverages (you may express
coverages generically in terms of 𝜃∗ )
d) Consider the following scenarios and state (1) the approximate values of the reversibility
(zj) of all steps, (2) the simplified functional form of the total rate of reaction derived from
the appropriate De Donder relation from 1.c.iii, (3) a qualitative reaction coordinate
diagram of chemical potential, and (4) the apparent Gibbs Free Energy barrier and
difference in Gibbs Free Energy for the net reaction written with the appropriate chemical
potentials.
i)
Case 1: Assume the adsorption of O2 is rate-determining, and all other steps are fast.
Assume that the surface is mostly unoccupied (* MARI).
ii) Case 2: Assume the oxidation of monoatomic nitrogen is rate-determining (i.e., N* +
O* ⟶ NO* + *) and that all proceeding steps are quasi-equilibrated. Assume that the
surface is mostly unoccupied (* MARI)
iii) Case 3: Assume the oxidation of monoatomic nitrogen is rate-determining (i.e., N* +
O* ⟶ NO* + *) and that all proceeding steps are quasi-equilibrated. Assume that the
surface is mostly occupied by oxygen (O* MARI)
e) Consider Case 2 from part 1.d.ii, but relax the assumption of a * MARI. Qualitatively plot
the rate of NO formation as a function of the pressure of oxygen and ammonia.
f) Suppose we relax the mean-field assumption. Qualitatively, how might estimated rates of
reaction change? What methods would allow us to estimate this difference quantitatively?
Qualitatively draw a cartoon of the surface of a catalyst for a mean field assumption versus
a more realistic surface accounting for surface interactions of adsorbates.
g) Suppose we relax the Langmuir assumption and account for energetically repulsive
interactions of surface oxygen and nitrogen-derived species (i.e., Temkin assumption).
Qualitatively, how might the plots in part e change?
h) Why is the Ostwald process important to society?
2) Consider any generic reaction with reactant and product species j that occur with n elementary
steps. Suppose these reactions are completely linear (no branched steps) and the nth reaction is
the rate determining steps while all preceding steps are quasi-equilibrated. Prove generally that
the degree of rate control always sums to 1. What is the degree of rate control of the nth
reaction?
3) Consider the hydrogen evolution reaction over a palladium catalyst. You may assume
Langmuir adsorption and mean field approximations in all questions below.
2H3O+ + 2e– ↔ H2
You may assume that this reaction follows Heyrovsky-Volmer kinetics with the following
elementary steps:
H3O+ + e– + * ↔ H2O + H* (1)
H3O+ + e– + H* ↔ H2O + H2 + *(2)
a) Derive the coverage of H* and * by accounting for the voltage-driven adsorption of
protons on the catalyst surface as a function of potential (E) by applying the pseudosteady-state hypothesis.
b) Solve for the net turnover rate of H2 formation from the solution in (a). You may use
generic expressions for coverage in the form of θ𝐻 ∗ and θ∗ .
c) Suppose that the voltage is sufficiently negative that both steps are irreversible toward the
formation of H2, as shown below. Rederive the expressions for the coverage of H* and
the rate of H2 formation.
H3O+ + e– + * ⟶ H2O + H* (1)
H3O+ + e– + H* → H2O + H2 + *(2)
d) Given the definition of the total rate in (c), how does the rate simplify for
i) Case 1: the surface is empty (* MARI)?
ii) Case 2: the surface is saturated by surface hydrogen (H* MARI)?
e) Consider the case of irreversible hydrogen evolution with a H* saturated surface, as
described in 3.d.ii. Evaluate the apparent Tafel slope, apparent rate order with respect to
proton activity, and apparent activation barrier.
f) Draw a hypothetical reaction coordinate for case 3.d.ii. assuming (𝐸 − 𝐸10 ) = (𝐸 − 𝐸20 ) =
1
0 ),
(𝐸 − 𝐸𝑡𝑜𝑡
in which that apparent Gibbs free energy barrier is labeled. Also label the
2
Gibbs free energy differences of reaction for the first and second steps as well as the total
reaction.
0
i) Case 1: The reaction is at equilibrium (𝐸 = 𝐸𝑡𝑜𝑡
)
0
ii) Case 2: The reaction is at highly irreversible reducing potentials ( 𝐸 ≪ 𝐸𝑡𝑜𝑡
)