Understanding Chemical Kinetics

Chemical Kinetics

• Reaction rate is the change in concentration of reactants or products with respect to time.
• The formula for the rate of a reaction, $aA + bB \rightarrow cC + dD$, is: $Rate = -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$.
• The rate is always positive and the stoichiometric coefficients normalize the rate for each reactant and product.
• Reactants have a negative sign as they are consumed, while products have a positive sign as they are produced.

Factors Affecting Reaction Rate

• Concentration: Increased concentration generally increases the reaction rate.
• Temperature: Higher temperatures usually speed up reactions by increasing molecular collision frequency and energy.
• Surface Area: Increasing surface area increases the reaction rate for reactions involving solids.
• Catalyst: Catalysts accelerate reactions by lowering the activation energy.
• Pressure: Increased pressure increases the rate of gaseous reactions by increasing concentration.
• Light: Certain reactions are initiated or accelerated by light.

Rate Law

• The rate law relates reaction rate to reactant concentrations, determined experimentally
• For a reaction $aA + bB \rightarrow cC + dD$, it is expressed $Rate = k[A]^m[B]^n$.
• $k$ stands for the rate constant; $m$ and $n$ represent reaction orders with respect to A and B.
• $m + n$ is the overall reaction order

Determining Reaction Order

• Method of Initial Rates: The initial rate of the reaction gets measured with varying starting reactant concentrations. Reaction order is determined by measuring how the rate fluctuates with concentration.
• Graphical Method: The concentration of a reactant gets plotted against time. A straight line plot will reveal the reaction order.
  • Zero order: $[A]$ vs t is linear
  • First order: $ln[A]$ vs t is linear
  • Second order: $1/[A]$ vs t is linear

Common Rate Laws

• Zero-Order Reactions: Rate is independent of reactant concentration.
  • $Rate = k$
  • $[A]_t = -kt + [A]_0$
  • $t_{1/2} = \frac{[A]_0}{2k}$
• First-Order Reactions: Rate is directly proportional to the concentration of one reactant.
  • $Rate = k[A]$
  • $ln[A]_t = -kt + ln[A]_0$
  • $t_{1/2} = \frac{0.693}{k}$
• Second-Order Reactions: Rate is proportional to the square of one reactant's concentration or the product of two reactants' concentrations.
  • $Rate = k[A]^2$
  • $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • $t_{1/2} = \frac{1}{k[A]_0}$
• $[A]_t$ is the concentration of $A$ at time $t$, $[A]0$ is the initial concentration, and $t{1/2}$ is the half-life.

Collision Theory

• Collision: Reactant molecules must collide for a reaction to occur.
• Energy: Colliding molecules must possess a minimum activation energy ($E_a$).
• Orientation: Molecules must collide with proper orientation.

Arrhenius Equation

• The Arrhenius equation relates the rate constant to temperature, activation energy, and a frequency factor: $k = Ae^{-\frac{E_a}{RT}}$
• $k$ is the rate constant, $A$ is the frequency factor, $E_a$ is the activation energy, $R$ is the gas constant ($8.314 J/(mol \cdot K)$), and $T$ is temperature in Kelvin.

Determining Activation Energy

• By plotting $ln(k)$ versus $\frac{1}{T}$, a straight line is obtained with a slope of $-\frac{E_a}{R}$
• The activation energy $E_a$ can be determined from this slope.
• $ln(k) = ln(A) - \frac{E_a}{RT}$ or $ln(k) = -\frac{E_a}{R}(\frac{1}{T}) + ln(A)$

Reaction Mechanisms

• A reaction mechanism is a step-by-step sequence of elementary reactions for an overall chemical change.

Elementary Steps

• Elementary steps are single-step reactions. Molecularity is the number of reactant molecules:
  • Unimolecular: one molecule decomposes or rearranges.
  • Bimolecular: two molecules collide and react.
  • Termolecular: three molecules collide and react.

Rate-Determining Step

• The slowest step in a reaction mechanism, determining the overall reaction rate.

Intermediates

• Species produced in one step and consumed in a subsequent step, not in the overall balanced equation.

Validating a Reaction Mechanism

• Elementary steps must add up to the overall balanced equation.
• The mechanism must align with the experimentally determined rate law.

Example

• $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$
• Proposed Mechanism:
  • $NO(g) + NO(g) \rightleftharpoons N_2O_2(g)$ (fast, equilibrium)
  • $N_2O_2(g) + O_2(g) \rightarrow 2NO_2(g)$ (slow, rate-determining)
• Overall Reaction: $2NO(g) + O_2(g) \rightarrow 2NO_2(g)$
• Since $N_2O_2$ is an intermediate $[N_2O_2] = K[NO]^2$ from $K = \frac{[N_2O_2]}{[NO]^2}$
• Rate Law is $Rate = k[N_2O_2][O_2]$ which simplifies to $Rate = kK[NO]^2[O_2] = k'[NO]^2[O_2]$

Algèbre linéaire

Definições

• Espace vectoriel: Conjunto $E$ com adição e multiplicação por escalar que satisfaz associatividade, comutatividade, elemento neutro, inverso, distributividade e elemento neutro para multiplicação escalar.
• Sous-espace vectoriel: Subconjunto $F$ de $E$ não vazio, fechado para adição e multiplicação por escalar.
• Combinação linéaire: Expressão $\lambda_1 v_1 + \lambda_2 v_2 +... + \lambda_n v_n$, onde os $v_i$ são vetores e os $\lambda_i$ são escalares.

Famílias de vetores

• Espaço engendrado ("Span"): Conjunto $Vect(v_1, v_2,..., v_n)$ de todas as combinações lineares de $v_1, v_2,..., v_n$.
• Família livre: Conjunto de vetores $(v_1, v_2,..., v_n)$ cuja única combinação linear nula é com todos os coeficientes nulos.
• Família geradora: Conjunto de vetores $(v_1, v_2,..., v_n)$ que pode expressar qualquer vetor do espaço como combinação linear.
• Base: Família que é simultaneamente livre e geradora.
• Dimension: Número de vetores em uma base.

Aplicações lineares

• Aplicação linear: Função $f: E \rightarrow F$ entre espaços vetoriais tal que $f(u+v) = f(u) + f(v)$ e $f(\lambda \cdot u) = \lambda \cdot f(u)$.
• Noyau (Kernel): Conjunto $Ker(f) = {u \in E | f(u) = 0_F}$ de vetores de $E$ que são mapeados para o vetor nulo de $F$.
• Image (Imagem): Conjunto $Im(f) = {v \in F | \exists u \in E, f(u) = v}$ de vetores de $F$ que são a imagem de pelo menos um vetor de $E$.

Théoème du rang (Teorema do posto)

• Relação entre dimensões: $dim(E) = dim(Ker(f)) + dim(Im(f))$.

Chapter 5: Distributed Coordination

• Need for distributed coordination involves:

  • Leader election
  • Mutual exclusion
  • Commitment

• Impossibility results, even with crash failures arise from:

  • FLP (Fischer, Lynch, Paterson)
  • CAP (Brewer)

Agreement

• Binary consensus: All processes each propose a bit

  • Non-faulty processes must decide on a single bit
  • All non-faulty processes must decide on the same bit
  • If all processes propose the same bit, non-faulty processes must decide on that bit

• k-agreement: All processes propose a value

  • Non-faulty processes must decide on a single value
  • All non-faulty processes must decide on the same value
  • The value decided must be one of the proposed values

Failure Models

• Crash failure
• Byzantine failure (arbitrary or malicious) where:
  • A faulty process can do anything

Impossibility of Agreement

• FLP Impossibility Result [FLP85] states that in an asynchronous system, consensus cannot be guaranteed within a bounded time if there's even a single crash failure.
• Sketch of Proof:
  • Assume an algorithm A that solves binary consensus.
  • A must have at least one critical state.
  • A state from which, depending on the next message, either a 0 or 1 outcome is decided.
  • Delay m results in system decides 1
  • Deliver m, the system decides 0, which results in a Contradiction.

Paxos

• It's a family of protocols for solving consensus in distributed systems, despite failures.

• Based on the state machine approach

  • If all processes propose the same sequence of commands, then all non-faulty processes will execute the same sequence of commands.
  • Replicated state machines are used to achieve fault tolerance.

• Basic Paxos assumptions:

  • Single proposer
  • Multiple acceptors
  • Multiple learners

• Illustrated using 3 Diagrams that portray it:

  • First, the Client sends a command which the Proposer delivers to the Acceptors, and with mutual consensus they deliver the outcome to the Learners who notify the Client.
  • Two-Phase Communication protocol used: Phase 1 with Preparing and Promising, and Phase 2 with Acepting and Learning.
  • Each process has a state machine enabling fault tolerance through consistent replicated executions.

Algorithmes de tri

Tri par insertion

• Principe: Insérer un élément dans une partie déjà triée du tableau.
• Algorithme (Pseudocode):

fonction tri_insertion(tableau T)
  n ← taille(T)
  pour i de 1 à n - 1 faire
    x ← T[i]
    j ← i
    tant que j > 0 et T[j - 1] > x faire
      T[j] ← T[j - 1]
      j ← j - 1
    fin tant que
    T[j] ← x
  fin pour
fin fonction

• Complexité: $O(n^2)$

Tri fusion

• Principe: Diviser le tableau en deux, trier chaque partie, fusionner les deux parties triées.
• Algorithme (Pseudocode):

fonction tri_fusion(tableau T)
  si taille(T) ≤ 1 alors
    retourner T
  sinon
    m ← taille(T) / 2
    T1 ← tri_fusion(T[0..m - 1])
    T2 ← tri_fusion(T[m..taille(T) - 1])
    retourner fusion(T1, T2)
  fin si
fin fonction

fonction fusion(tableau T1, tableau T2)
  n1 ← taille(T1)
  n2 ← taille(T2)
  T ← tableau de taille n1 + n2
  i ← 0
  j ← 0
  k ← 0
  tant que i < n1 et j < n2 faire
    si T1[i] < T2[j] alors
      T[k] ← T1[i]
      i ← i + 1
    sinon
      T[k] ← T2[j]
      j ← j + 1
    fin si
    k ← k + 1
  fin tant que
  tant que i < n1 faire
    T[k] ← T1[i]
    i ← i + 1
    k ← k + 1
  fin tant que
  tant que j < n2 faire
    T[k] ← T2[j]
    j ← j + 1
    k ← k + 1
  fin tant que
  retourner T
fin fonction

• Complexité: $O(n \log n)$

Tri rapide (Quick Sort)

• Principe: Choisir un pivot, partitionner le tableau, trier récursivement chaque partie.
• Algorithme (Pseudocode):

fonction tri_rapide(tableau T, entier debut, entier fin)
  si debut < fin alors
    pivot ← partitionner(T, debut, fin)
    tri_rapide(T, debut, pivot - 1)
    tri_rapide(T, pivot + 1, fin)
  fin si
fin fonction

fonction partitionner(tableau T, entier debut, entier fin)
  pivot ← T[fin]
  i ← debut
  pour j de debut à fin - 1 faire
    si T[j] ≤ pivot alors
      echanger(T[i], T[j])
      i ← i + 1
    fin si
  fin pour
  echanger(T[i], T[fin])
  retourner i
fin fonction

fonction echanger(tableau T, entier i, entier j)
  tmp ← T[i]
  T[i] ← T[j]
  T[j] ← tmp
fin fonction

• Complexité:
• Meilleur cas et cas moyen : $O(n \log n)$
• Pire cas : $O(n^2)$

Comparaison des algorithmes

Lógica proposicional

• Lógica proposicional is a system for representing and manipulating propositions, which can be true or false, but not both.

Elementos

• Variáveis proposicionais (Propositional variables): represent simple propositions ( p, q, r, etc.)
• Conectivos lógicos (Logical connectives): operators to create complex propositions.
• Negação ($\neg$): reverses truthness.
• Conjunção ($\land$): true if both true.
• Disjunção ($\lor$): true if at least one is true.
• Implicação ($\rightarrow$): only false if first true and second false.
• Bicondicional ($\leftrightarrow$): true if both proposition have the same truth value.
• Parênteses use to grouping to avoid ambiguities.

Tabelas verdade (Truth tables)

• The truth table definitions for values in proposition are shown below.

Leis (Laws)

• Comutatividade
• $p \land q \equiv q \land p$
• $p \lor q \equiv q \lor p$
• Associatividade
• $(p \land q) \land r \equiv p \land (q \land r)$
• $(p \lor q) \lor r \equiv p \lor (q \lor r)$
• Distributividade
• $p \land (q \lor r) \equiv (p \land q) \lor (p \land r)$
• $p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)$
• Idempotência
• $p \land p \equiv p$
• $p \lor p \equiv p$
• Negação
• $\neg (\neg p) \equiv p$
• $\neg (p \land q) \equiv \neg p \lor \neg q$ (De Morgan)
• $\neg (p \lor q) \equiv \neg p \land \neg q$ (De Morgan)
• Identidade
• $p \land V \equiv p$
• $p \lor F \equiv p$
• Dominância
• $p \land F \equiv F$
• $p \lor V \equiv V$
• Absorção
• $p \land (p \lor q) \equiv p$
• $p \lor (p \land q) \equiv p$

Inferência lógica (Logical inference)

• Process to define new proposition from existing propositions. Valid when the conclusion is true with true premises.
• Modus ponens: if $p \rightarrow q$ and $p$ are true, then $q$ is true.
• Modus tollens: if $p \rightarrow q$ is true and $q$ is false, then $p$ is false.
• Silogismo hipotético: if $p \rightarrow q$ and $q \rightarrow r$ are true, then $p \rightarrow r$ is true.
• Silogismo disjuntivo: if $p \lor q$ is true and $p$ is false, then $q$ is true.

Aplicações (Applications)

• Aplicações to verify program correctness, represent knowledge for AI, digital circuit design, math for theorem proving and analisysing arguments in philisophy.

Bernoulli's Principle

• For nonviscous, incompressible fluid, speed increases with decreasing pressure or potential energy.

• Bernoulli's equation mathematical expression is : $P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2$

Variables:

• If height is constant, then speed increases with decreasing pressure.
• If speed is constant, then pressure increases with decreasing height.

Lecture 11: Bioreactors

Continuous Culture

• Conditions change in batch culture, In continuous culture fresh medium is added and old fermentation broth is removed contitnuously.
• In start-up of continuous culture, conditions and volume inside the bioreactor are constant
• Two types used:
• Chemostat
• Turbidostat

Chemostat

• Constant fow rate.
• Incoming medium's limiting nutrient concentration defines cell density.
• Dilution rate ($D$) determines organism growth rate via $D = F/V$, where $F$ is the flow rate and $V$ is the volume of the bioreactor.
• At steady state, $\mu = D$
• $\mu$ is the specific growth rate.
• Cells washout if $D$ is too high
• Cells grow slowly or die if $D$ is too low.

Chemostat

• Use Monod kinetics to calculate bioreactor's limiting nutrient concentration
• $ \mu = \mu_{max} \frac{S}{K_s + S} $ yields $ S = \frac{K_s D}{\mu_{max} - D} $
• $S$ is limiting nutrient concentration, $K_s$ is saturation constant, and $\mu_{max}$ is the maximum specific growth rate.

Turbidostat

• Turbidity (cell density) is kept constant adjusting rate.
• Sensor adjusts flow rate accordingly.
• Not Limiting nutrient
• Works at high cell density
• More complex and costly.

Bioreactor configurations:

• Batch
• Fed-Batch
• Continuous
• Chemostat
• Turbidostat
• Plug flow reactor
• Fluidized bed reactor

Plug Flow Reactor

• A long, cylindrical reactor
• No mixing occurs
• Similar to a continuous batch reactor

Fluidized Bed Reactor

• Cells are immobilized on solid particles that are suspended.
• High mass transfer rates.
• Suitable for product inhibited reactions

Bioreactor Design

• Design depends on application depending on organism, product, scale, and also the equipment cost.

Bioreactor Operation

• Bioreactor operation parameters to control are temperature, pH, dissolved oxygen, agitation rate, nutrient feed rate, and product removal rate.

Bioreactor Control

• Computer controls bioreactors by monitoring parameters and adjusting them to reach optima conditions.
• Also to be used for data collection and report generation.

Algorithmes de tri

####Tri par Insertion

• Consider the first $i$ items $T,...,T[i]$ of the table which are already organized, and insert the element $T[i+1]$ in its place among the first $i$ elements.

Algoritme

procedure tri_insertion(tableau T, entier n)
  donnée : tableau T de n éléments
  pour i de 2 à n faire
    x ← T[i]
    j ← i - 1
    tant que j > 0 et T[j] > x faire
      T[j+1] ← T[j]
      j ← j - 1
    fin tant que
    T[j+1] ← x
  fin pour
fin procedure

Complexité

• Worst case The table is already ordered inversely such that the calculation is:

$\qquad \sum_{i=2}^{n} (i-1) + i = \sum_{i=2}^{n} (2i-1) = 2 \frac{n(n+1)}{2} - 2 - (n-1) = n^2 + n - 2 - n + 1 = n^2 - 1$

• Complexité of : $O(n^2)$.

• Best case table is already organized: $O(n)$

• Average case: complexité $O(n^2)$.

Avantages

• Implementation of concept is simple.
• Efficient for smaller tables.
• Accommodated into tables already almost organized.
• Stable, where the order of items are conserved.
• In place: Does not create use for more tables.

Inconvénients

• Inefficient against more larger tables
• Quadratic complex

Tri par Sélection

• A selection sort searches for the smallest element in the table and if they're badly organized, swap one with the first element. Then continue from the rest and search again from smaller one.

Algoritme

procedure tri_selection(tableau T, entier n)
  donnée : tableau T de n éléments
  pour i de 1 à n-1 faire
    min ← i
    pour j de i+1 à n faire
      si T[j] < T[min] alors
        min ← j
      fin si
    fin pour
    si min ≠ i alors
      echanger(T[i], T[min])
    fin si
  fin pour
fin procedure

Complexité

• Worst cast, the complexity of this operation is:

$\qquad \sum_{i=1}^{n-1} (n-i) = \sum_{i=1}^{n-1} i = \frac{n(n-1)}{2} = \frac{n^2 - n}{2}$

$O(n^2)$.

• Best case The operation is, O(n^2)

• Average case the complexity is, $O(n^2)$.

Avantages

• Simple
• Doesn't not create tables, in place.
• Minimal swaps

Inconvénients

• Inefficient for larger tables
• Quadratic complex Non stable and doesn't follow the order of the selected items.

Tri a bulles

• A "Bubble sort" travels around the table comparing and the organized swaps done based on the mal-aligned sorted items. The operations get repeated until no more swaps left.

Algoritme

procedure tri_bulles(tableau T, entier n)
  donnée : tableau T de n éléments
  répéter
    echange ← faux
    pour i de 1 à n-1 faire
      si T[i] > T[i+1] alors
        echanger(T[i], T[i+1])
        echange ← vrai
      fin si
    fin pour
  tant que echange
fin procedure

Complexité

• Worst case, for a unorganized tabler, operations for each loop is the following is: - $n-i$ where it equals O(n^2);

• Best case is is,

echange T a faux

With such, there's just 
  $ n-1$ for a $ O(n) time.

• Average: $O(n^2).

Avantages

• Simple

Inconvénients.

• Iffenficetn tables Quadratic Complex; Non stable doesn't follow elements set.

Tri Rapide (The quicksort).

Chooses and partitions the array. The two subarrays of the table which are recursive.

procedure tri_rapide(tableau T, entier debut, entier fin)
  donnée : tableau T, entier debut, entier fin
  si debut < fin alors
    pivot ← partitionner(T, debut, fin)
    tri_rapide(T, debut, pivot-1)
    tri_rapide(T, pivot+1, fin)
  fin si
fin procedure

procedure partitionner(tableau T, entier debut, entier fin)
  donnée : tableau T, entier debut, entier fin
  pivot ← T[fin]
  i ← debut
  pour j de debut à fin-1 faire
    si T[j] ≤ pivot alors
      echanger(T[i], T[j])
      i ← i + 1
    fin si
  fin pour
  echanger(T[i], T[fin])
  retourner i
fin procedure

The complexity is, for the following:

Best case. O(n log n): Averagee case . O(n log n). And : In all O(n^2);

Avantages

• Efacicnet n larget tables, -in En palce where

Inconvénients

-Complex:

TrI Fusion

• Divide the table into a sub-table that has the same values