1 Stoichiometry

1.1 Definitions

I.) Basics: N usually denotes the number of species/components in a reaction, whereas M denotes the number of reactions in a reaction system. We use the variables i and j as follows

i ∈ [1, … N]
j ∈ [1, … M]

II.) Element species matrix:

\underline{B} contains the number of atoms of an element per molecule of a species. You have to set it up as follows for the four species O_{2}, CO_{2}, H_{O} and C_{2}H_{5}OH:

\underline{B} = ( \begin{matrix}& O_{2} & CO_{2} & H_{2}O & C_{2}H_{5}OH \\ C & β_{C,O_{2}} & β_{C,CO_{2}} & β_{C,H_{2}O} & β_{C,C_{2}H_{5}OH} \\ O & β_{O,O_{2}} & β_{O,CO_{2}} & β_{O,H_{2}O} & β_{O,C_{2}H_{5}OH} \\ H & β_{H,O_{2}} & β_{H,CO_{2}} & β_{H,H_{2}O} & β_{H,C_{2}H_{5}OH} \end{matrix} ) = ( \begin{matrix}& O_{2} & CO_{2} & H_{2}O & C_{2}H_{5}OH \\ C & 0 & 1 & 0 & 2 \\ O & 2 & 2 & 1 & 1 \\ H & 0 & 0 & 2 & 6 \end{matrix} )

III.) Amount of substance:

\underline{Δn_{i}} = ( Δn_{1} ; ⋮ ; Δn_{N} ) = ( \begin{matrix}Δn_{O_{2}} \\ Δn_{CO_{2}} \\ Δn_{H_{2}O} \\ Δn_{C_{2}H_{5}OH} \end{matrix} )

IV.) Reactions need to be in balance: The overall amount of elements is constant even though the amount of substance of particular species may change:

\underline{0} = \underline{B} · \underline{Δn}

⇝ ( \begin{matrix}0 \\ 0 \\ 0 \\ 0 \end{matrix} ) = ( \begin{matrix}& O_{2} & CO_{2} & H_{2}O & C_{2}H_{5}OH \\ C & β_{C,O_{2}} & β_{C,CO_{2}} & β_{C,H_{2}O} & β_{C,C_{2}H_{5}OH} \\ O & β_{O,O_{2}} & β_{O,CO_{2}} & β_{O,H_{2}O} & β_{O,C_{2}H_{5}OH} \\ H & β_{H,O_{2}} & β_{H,CO_{2}} & β_{H,H_{2}O} & β_{H,C_{2}H_{5}OH} \end{matrix} ) · ( \begin{matrix}Δn_{O_{2}} \\ Δn_{CO_{2}} \\ Δn_{H_{2}O} \\ Δn_{C_{2}H_{5}OH} \end{matrix} )

One will usually search for the Δn_{i} while \underline{B} is known. To do this: First one should apply the Gauss elimination and then develop the linear equation system from it. The matrix form actually is already an equation system and it is possible to use it directly to get the needed values for the unknown Δn_{i}. Please note, that the key components/species will be given (or in a laboratory measured).

V.) Extent of reaction (dt.: Reaktionslaufzahl): The extent of reaction denotes the amount of substance which has already been consumed or produced. The used symbol is the small greek letter Xi (ξ; the capital letter Xi looks like this: Ξ). The extent of reaction remains the same throughout one reaction!

ξ = \frac{n_{i} − n_{i0}}{ν_{i}} = \frac{Δn_{i}}{ν_{i}} = \frac{Δn_{1}}{ν_{1}} = ⋯ \frac{Δn_{N}}{ν_{N}}

VI.) Conversion (dt.: Umsatzgrad): The conversion denotes the percentage of amount of substance which has already reacted. It is specific to a single species but can be transferred to be meaningful for other species, too (see Stoichiometric Table).

X_{i} = \frac{n_{i0} − n_{i}}{n_{i0}}

1.2 Changes in amount of substance and key components

Basic equation: \underline{B} · \underline{Δn} = \underline{0} Approach:

I.) Set up the element species matrix \underline{B}. II.) Determine the rank of \underline{B}.

⇝ Number of key components: N_{kc} = N − Rg(\underline{B}) III.) Measure Δn of key components.

⇝ Δn of key components are known (or given). IV.) Plug in Δn of key components into the basic equation. V.) Solve the linear equation system (in form of a matrix) to get the Δn for non-key components.

1.3 Extent of reaction and key reactions

Basic equation: \underline{N} · \underline{ξ} = \underline{Δn} Approach:

I.) Set up the matrix of stoichiometric coefficients, \underline{N}. II.) Determine the rank of \underline{N}.

⇝ Number of key reactions: N_{krc} = Rg(\underline{N}) III.) Linearly independent rows describe the key reactions stoichiometrically (i.e. form the key reactions by

taking the ν’s from the rearranged matrix - rearranging was done using the Gauss elimination). IV.) Delete ξ_{j} of the linearly dependent non-key reactions (i.e. cancel out the rows for the linearly dependent

non-key reactions in the ξ-matrix/vector). V.) Delete ν_{ij} and Δn_{i} for the non-key reactions. (i.e. remove the respective rows from these matrices/vectors). VI.) Solve the equations for ξ_{j} of key reactions.

2 Thermodynamics

2.1 Important definitions

I.) Enthalpy change on reaction ΔH_{R} is the molar heat of formation of the products ΔH_{f,i} minus the molar heat of formation of educts. ⇝ The heat tonality can be exo- (ΔH_{R} < 0; i.e. heat is produced by the reaction), endo- (ΔH_{R} > 0; i.e. the reaction consumes energy) or diathermic (i.e. neutral).

II.) Hess’s law of heat summation: ΔH_{R} is independent of the reaction path. (This allows to calculate the heat of reaction for any reaction, although it might not be possible to realize it experimentally!)

III.) Le Chatelier’s principle: A system in chemical equilibrium shifts in response to any imposed change of factors governing the equilibrium. Note: Catalysts \underline{never} have any influence on the equilibrium. They accellerate or inhibit a reaction, but they never shift the equilibrium!

2.2 Important formulae

I.) Enthalpy change on reaction / heat of reaction

ΔH_{R}^{0} = ∑_{i=1}^{N} ν_{i} · ΔH_{f,i}^{0} this holds for STP

ΔH_{R}(T) = ΔH_{R}^{0} + ∫_{T^{(0)}}^{T} Δc_{p}(T)dT

Δc_{p}(T) = ∑_{i=1}^{N} ν_{i} · c_{pi}(T)

Often the formula to calculate Δc_{pi} appears similar to this

Δc_{pi}(T) = A + BT + CT^{2} + DT^{3},

where A, B, C and D are coefficients one can find in standard tables.

II.) Gibbs free energy

ΔG_{R}^{0} = ∑_{i=1}^{N} ν_{i} · ΔG_{f,i}^{0} this holds for STP
= −RT · ln K_{p}^{0}
= −RT · ln K_{a}^{0}
ln K_{p}(T) = ln K_{p}^{0} + ∫_{T^{(0)}}^{T} \frac{ΔH_{R}^{0}}{RT^{2}} dT

ln K_{a}(T) = ln K_{a}^{0} + ∫_{T^{(0)}}^{T} \frac{ΔH_{R}^{0}}{RT^{2}} dT
K_{a}^{0} ∼ K_{p}^{0}

III.) Equilibria

1.) K_{c} ⟶ concentration equilibrium

2.) K_{a} ⟶ activity equilibrium

3.) K_{p} ⟶ pressure equilibrium 4.) K_{x} ⟶ conversion equilibrium

K_{c} = ∏_{i=1}^{N} c_{i}^{ν_{i}} = ∏_{i=1}^{N} (\frac{c_{i}}{c^{(0)}} )^{ν_{i}} (”normalized form”)
K_{a} = ∏_{i=1}^{N} a_{i}^{ν_{i}}
K_{p} = ∏_{i=1}^{N} p_{i}^{ν_{i}} = K_{c} · (\frac{c^{(0)} · RT}{p^{(0)}} )^{∑_{i=1}^{N} ν_{i}} = ∏_{i=1}^{N} (\frac{p_{i}}{p^{(0)}} )^{ν_{i}} (”normalized form”)
K_{x} = ∏_{i=1}^{N} x_{i}^{ν_{i}} = K_{p} · (\frac{p}{p^{(0)}} )^{− ∑_{i=1}^{N} ν_{i}} (where p is the overall pressure)

The equilibria values relate as follows:

K_{a} = (p^{(0)})^{− ∑_{i=1}^{N} ν_{i}} · K_{p} = (\frac{p}{p^{(0)}} )^{∑_{i=1}^{N} ν_{i}} · K_{x} = c^{− ∑_{i=1}^{N} ν_{i}} · K_{c}

2.3 Explanations

I.) STP means: Standard Temperature and Pressure II.) Standard temperature: 25^{0}C = 298K III.) Standard pressure: p^{(0)} = 1, 013bar = 101, 3kP a = 1, 013 · 10^{5}P a IV.) Standard concentration: c^{(0)} = 1\frac{mol}{l} V.) Important: ΔG_{f,i}^{0} = ΔH_{f,i}^{0} = 0 for elements!

3 Microkinetics (Reaction kinetics)

3.1 Description of kinetics

I.) Reaction rate (N being the number of species, M being the number of reactions and i=1,..,N; M=1,..,M):

r_{j} = \frac{1}{ν_{i,j}} · \frac{1}{V} · \frac{dn_{i}}{dt} = \frac{1}{ν_{i,j}} · \frac{dc_{i}}{dt} (because V = const. and c = \frac{n}{V})

II.) Power law:

r_{j} = k_{j} · ∏_{i=1}^{N} c_{i}^{κ_{i,j}}  \underbrace{with κ_{i,j} = \{ \begin{matrix}|ν_{i,j}| & for educts \\ 0 & for products \end{matrix} \underline{or} 0  for educts |ν_{i,j}| for products}_{for \underline{elementary reactions} only!}

III.) Arrhenius’ law (dependency on temperature):

k_{j}(T) = k_{∞,j} · e^{− \frac{E_{a,j}}{RT}}

IV.) Net rate of changes in amount of substance:

R_{i} = \frac{dc_{i}}{dt} = ∑_{j=1}^{M} ν_{i,j}r_{j}

For ideal gases we have c_{i} = \frac{p_{i}}{R·T}, hence:

R_{i} = \frac{dc_{i}}{dt} = \frac{1}{RT} · \frac{dp_{i}}{dt}

3.2 Determination of kinetics

Measure the values for concentration at different points in time: c(t); c(t + Δt)

I.) Differential method 1.) Calculate approximation values for r:

r = \frac{1}{ν_{i}} · \frac{c_{i}}{dt} ≈ \frac{1}{ν_{i}} · \underbrace{\frac{c_{i}(t + Δt) − c_{i}(t)}{Δt}}_{similar to the definition of the differential}

2.) Use the power law (r = k · ∏_{i=1}^{N} c_{i}^{κ_{i}}) 3.) Find the logarithms:

log r = log k + ∑_{i=1}^{N}(κ_{i} log c_{i})

⇒ You get a linear set of equations, κ_{i} can be estimated by solving it. ⇒ Plot log r versus log c_{i} to determine k.

II.) Integral method \underline{Example:} A_{1} ⟶ A_{2} 1.) Use the power law and set it equal to the differential form:

r = k · ∏_{i=1}^{N} c_{i}^{k_{i}} = k · c_{1}^{κ_{1}} ⟺ r = \frac{1}{ν_{i}} · \frac{dc_{i}}{dt} = \frac{1}{−1} · \frac{dc_{1}}{dt} = −\frac{dc_{1}}{dt} ⇝ k · c_{1}^{κ_{1}} = −\frac{dc_{1}}{dt}

2.) Solve the differential equation by separating the variables

\frac{dc_{1}}{c_{1}^{κ_{1}}} = −kdt ⇝ c_{1} = c_{10} · e^{−kt}

3.) Find the logarithms

ln c_{1} = ln c_{10} − kt

⇒ Plot ln c_{1} versus t to determine k. In general for κ_{1} ≠ 1 it holds (\frac{1}{c_{1}} )^{κ_{1}−1} − (\frac{1}{c_{10}} )^{κ_{1}−1} = (κ_{1} − 1)k_{κ_{1}} · t

3.3 Special reactions

General approach:

I.) Get the number of species and of reactions.

II.) Set up the differential equations R_{i} = \frac{dc_{i}}{dt} = ∑_{j=1}^{M} ν_{i,j} · r_{j}

III.) Substitute r_{j} by the power law. IV.) Make any necessary assumptions. V.) Solve the system of differential equations.

1.) Reversible reactions

A_{1} + A_{2} r_{+} ; ⇀ ↽ ; r_{−} A_{3} + A_{4}

Considering equilibrium: r_{+} = r_{−} 2.) Multiple concurrent reactions / parallel reactions (multiple reaction pathes)

A_{1} A_{2} ; ↗^{1} ; ↘_{2} ; A_{3}

3.) Consecutive reactions

A_{1} ⟶^{1} A_{2} ⟶^{2} A_{3}

4.) Homogeneous catalyzed reactions

i. General reaction equation:

A_{1} + A_{2} ⇀ ↽_{2}^{1} A_{12} →^{3} A_{2} + A_{3}

Where A_{2} is the catalyst and A_{12} is the activated complex. ii. Set up the differential equations for R_{i} using the power law for r_{j}. iii. Assumptions:

* The total amount of catalyst is constant: c_{2} + c_{12} = c_{20}

* Reaction (3) is the rate determination step ⇝ r_{1} = r_{2} or \frac{dc_{12}}{dt} = 0

4 Reactor design

4.1 Stoichiometric table for batch systems

ν_{1}A_{1} + ν_{2}A_{2} + ν_{3}A_{3} + ν_{4}A_{4} = 0  with k=1
Species n_{i0} Δn_{i} = −\frac{ν_{i}}{ν_{k}} n_{k0}X_{k}  n_{i} = n_{i0} + Δn_{i}
A_{1}  n_{10}  −n_{10}X_{1}  n_{1} = n_{10} − n_{10}X_{1}
A_{2}  n_{20}  −\frac{ν_{2}}{ν_{1}} n_{10}X_{1}  n_{2} = n_{20} − \frac{ν_{2}}{ν_{1}} n_{10}X_{1}
A_{3}  n_{30}  −\frac{ν_{3}}{ν_{1}} n_{10}X_{1}  n_{3} = n_{30} − \frac{ν_{3}}{ν_{1}} n_{10}X_{1}
A_{4}  n_{40}  −\frac{ν_{4}}{ν_{1}} n_{10}X_{1}  n_{4} = n_{40} − \frac{ν_{4}}{ν_{1}} n_{10}X_{1}
Inert  n_{I0}  −  n_{I} = n_{I0}
Sum n_{sum0}  −  nsum = n_{sum0} − (1 + \frac{ν_{2}}{ν_{1}} + \frac{ν_{3}}{ν_{1}} + \frac{ν_{4}}{ν_{1}} )n_{10}X_{1}

4.2 Stoichiometric table for flow systems

ν_{1}A_{1} + ν_{2}A_{2} + ν_{3}A_{3} + ν_{4}A_{4} = 0  with k=1
Species F_{i0} ΔF_{i} = −\frac{ν_{i}}{ν_{k}} F_{k0}X_{k}  F_{i} = F_{i0} + ΔF_{i}
A_{1}  F_{10}  −F_{10}X_{1}  F_{1} = F_{10} − F_{10}X_{1}
A_{2}  F_{20}  −\frac{ν_{2}}{ν_{1}} F_{10}X_{1}  F_{2} = F_{20} − \frac{ν_{2}}{ν_{1}} F_{10}X_{1}
A_{3}  F_{30}  −\frac{ν_{3}}{ν_{1}} F_{10}X_{1}  F_{3} = F_{30} − \frac{ν_{3}}{ν_{1}} F_{10}X_{1}
A_{4}  F_{40}  −\frac{ν_{4}}{ν_{1}} F_{10}X_{1}  F_{4} = F_{40} − \frac{ν_{4}}{ν_{1}} F_{10}X_{1}
Inert  F_{I0}  −  F_{I} = F_{I0}
Sum F_{sum0}  −  Fsum = F_{sum0} − (1 + \frac{ν_{2}}{ν_{1}} + \frac{ν_{3}}{ν_{1}} + \frac{ν_{4}}{ν_{1}} )F_{10}X_{1}

4.3 Design equations

I.) Batch Reactor:

\overbrace{input}^{=0} = \overbrace{output}^{=0} +disappearance + accumulation

II.) Continuous Stirred Tank Reactor:

input = output + disappearance + \overbrace{accumulation}^{=0}

III.) Plug Flow Reactor: input = output + disappearance + \overbrace{accumulation}^{=0}

Batch Reactor (BR) n_{10} · dX_{1} = r · V · dt t = n_{10} · ∫_{0}^{X_{1}} \frac{dX_{1}}{rV}

Continuous Stirred Tank Reactor (CSTR) F^{10} · X^{1} = r · V τ = n^{10} · ^{\frac{dX_{1}}{r}}

Plug Flow Reactor (PFR)  F_{10} · dX_{1} = r · dV  τ = c_{10} · ∫_{0}^{X_{1}} \frac{dX_{1}}{r}

Where t is the time that the mixture is in the batch reactor, τ is the space time. A space time of 5min denotes that every five minutes one reactor volume of feed is being treated by the reactor. τ = \frac{1}{s}, where s is the space velocity (in min^{−1}) and denotes how many reactor volumes of feed are being treated during a given time.

Note: Volume of feed (V) is not necessarily the same as volume of reactor(V_{R}). But we only consider volume of feed and how much time is needed to treat one reactor volume of feed. ⇝ V = ϵ_{R} · V_{R} (with ϵ_{R} being the ratio of filling volume by reactor volume).

References

[1] Baerns Manfred, Hofmann Hanns, Renken Albert. Chemische Reaktionstechnik. Georg Thieme Verlag Stuttgart - New York.

German and available in the BTU library

In my opinion the best book since it explains everything from the very scratch. BUT it also has a drawback: there is much information we simply don’t need!

[2] Levenspiel Octave.

Chemical Reaction Engineering. John Wiley & Sons, Inc..

English and available in the BTU library

Very useful for Microkinetics and Reactor Design parts.

[3] Fogler H. Scott.

Elements of Chemical Reaction Engineering. Prentice Hall.

English and available in the BTU library

This is partially useful for Reactor Design but not at all for the rest. Description of different reactors and the origin of their design equations good explained.

[4] Bechmann Wolfgang, Schmidt Joachim.

Einstieg in die Physikalische Chemie für Nebenfächler . B.G. Teubner Stuttgart - Leipzig.

German and available in the BTU library

Microkinetics.

[5] Müller-Erlwein Erwin.

Chemische Reaktionstechnik. B.G. Teubner Stuttgart - Leipzig.

German

Stoichiometry, Thermodynamics and partially Microkinetics and Reactor Design.

[6] Butt John B..

Reaction Kinetics and Reactor Design. Prentice Hall.

English and available in the BTU library

Partially useful for Microkinetics and Reactor Design parts.

[7] Atkins Peter William.

The Elements of Physical Chemistry. Oxford University Press.

English and available in the BTU library

Thermodynamics and partially Microkinetics.

[8] Kopka Helmut.

L^{A}TEX Einführung. Band 1 Addison-Wesley.

German

[9] Dalheimer Mathias Kalle.

L^{A}TEX kurz & gut. O’Reilly.

German