• Chemical kinetics involves reaction rates, rate laws, temperature dependence, catalysis, and reaction mechanisms.

Reaction Rate

• It is the change in reactant or product concentration over time.
• For $aA + bB \rightarrow cC + dD$, Rate $= -\frac{1}{a} \frac{\Delta[A]}{\Delta t} = -\frac{1}{b} \frac{\Delta[B]}{\Delta t} = \frac{1}{c} \frac{\Delta[C]}{\Delta t} = \frac{1}{d} \frac{\Delta[D]}{\Delta t}$
• $[A]$, $[B]$, $[C]$, $[D]$ represent reactant and product concentrations.
• $a$, $b$, $c$, $d$ are stoichiometric coefficients.
• $\Delta t$ is the change in time.

Rate Law

• The rate law equation relates reaction rate to reactant concentrations raised to specific powers.
• General form: Rate $= k[A]^m[B]^n$.
• $k$ is the rate constant.
• $m$ and $n$ are reaction orders with respect to A and B.
• Overall reaction order: $m + n$.
• Reaction orders are determined experimentally.

Common Rate Laws

• Zero-Order: Rate = k, rate is independent of reactant concentration.
• First-Order: Rate = k[A], rate is directly proportional to reactant concentration.
• Second-Order: Rate = k[A]^2 or Rate = k[A][B], rate is proportional to the square of one reactant's concentration or the product of two reactant concentrations.

Temperature Dependence via Arrhenius Equation

• The Arrhenius Equation links the rate constant k to temperature T: $k = Ae^{-\frac{E_a}{RT}}$.
• A is the pre-exponential factor.
• $E_a$ is the activation energy.
• R is the gas constant, equal to 8.314 J/(mol⋅K).
• T is absolute temperature in Kelvin.

Activation Energy

• $E_a$ represents the minimum energy for a reaction to occur.
• Higher $E_a$ results in a slower reaction rate.

Catalysis

• A catalyst speeds up a chemical reaction without being consumed.
• Homogeneous catalysis: catalyst is in the same phase as reactants.
• Heterogeneous catalysis: catalyst is in a different phase from reactants.

Reaction Mechanisms

• A reaction mechanism includes a series of elementary steps from reactants to products.
• Elementary step: A single step in a reaction mechanism, the rate law can be deduced from stoichiometry.
• Rate-determining step: the slowest step in a reaction mechanism, determines the overall reaction rate.

Algorithmus von Floyd-Warshall

Ziel

• Findet die kürzesten Pfade zwischen allen Knotenpaaren in einem Graphen.

Eingabe

• Graph $G = (V, E)$ mit Kantengewichten $w: E \rightarrow \mathbb{R}$.
• Adjazenzmatrix $D = (d_{ij})$ mit $d_{ij} = w(i, j)$ , $d_{ij} = \infty$, $d_{ij} = 0$.

Ausgabe

• Distanzmatrix $D^* = (d_{ij}^)$, wobei $d_{ij}^$ die Länge des kürzesten Pfades von Knoten $i$ zu Knoten $j$ ist.

Algorithmus

• For k in 1 to |V|: For i in 1 to |V|: For j in 1 to |V|: D[i][j] := min(D[i][j], D[i][k] + D[k][j])

Eigenschaften

• Korrektheit: Finds shortest paths if no negative cycles exist.
• Laufzeit: (\Theta(|V|^3))
• Speicherplatz: (\Theta(|V|^2))

Beispiel

• Graphen: (1)->(2)->(3), (3)->(1).
• Adjazenzmatrix: $D = \begin{bmatrix} 0 & 2 & \infty \ \infty & 0 & 3 \ 4 & \infty & 0 \end{bmatrix}$
• Distanzmatrix: $D^* = \begin{bmatrix} 0 & 2 & 5 \ 7 & 0 & 3 \ 4 & 6 & 0 \end{bmatrix}$

Anmerkungen

• Works with negative weights if no negative cycles.
• Negative cycles are found if $D[i][i] < 0$.

Lecture 24: Photodiodes

Light as a Source of Energy

• Electron-hole pairs are generated when light shines on a semiconductor material.
• Electron-hole pairs recombine without a field
• Separation of electron-hole pairs by a potential barrier creates a voltage.

Device & Circuit Operation

• A photodiode collects light-generated carriers efficiently.
• Light-generated carriers are collected to produce current under reverse bias.
• Operated in reverse bias, current increases linearly with light intensity.

Photodiode Structure

• Basic Structure: p-n junction, light shone on intrinsic region, light is absorbed creating electron-hole pairs.
• Improved Structure: p-i-n photodiode increases absorption volume.

Current Gain

• Responsivity ((\Re)): Ratio of photocurrent ((I_{ph})) to incident optical power ((P_{opt})): (\Re = \frac{I_{ph}}{P_{opt}}).
• Quantum Efficiency ((\eta)): Number of electron-hole pairs per incident photon: (\eta = \frac{I_{ph}/q}{P_{opt}/hv})
• Avalanche Gain (M): Due to impact ionization, where (M = \frac{I_M}{I_{ph}}).

Key Performance Parameters

• Includes Responsivity, Speed, Dark Current, Noise, Capacitance.

Advantages & Disadvantages

• Advantages: High sensitivity, fast response, low noise, small size, low cost.
• Disadvantages: Temperature dependence, dark current, capacitance, fragility.

Applications

• Light detection, measurement, optical communication, medical imaging, bar code scanners, optical storage, remote control.

Guía de inicio rápido de Markdown

• Markdown: Texto plano con sintaxis de formato ligero.

Sintaxis de Markdown

Encabezados

• From # Encabezado 1 to ###### Encabezado 6.

Énfasis

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Listas

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Enlaces

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Imágenes

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Citas

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Código

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Tablas

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Líneas horizontales

• ---.

Fórmulas matemáticas

• En línea: (x=y+2); Bloque: [f(x) = ...].

Combinación de elementos

• Combina to create richer documents.

Chemical Kinetics Formulae

Reaction Rate

$aA + bB \rightarrow cC + dD$

• Rate $= -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt}$

Rate Law

• Rate $\propto [A]^x [B]^y$
• Rate $= k[A]^x [B]^y$
• x + y = overall order of reaction

Zero Order Reaction

• Rate $= k[A]^0 = k$
• $[A] = [A]_0 - kt$
• $t_{1/2} = \frac{[A]_0}{2k}$

First Order Reaction

• Rate $= k[A]^1 = k[A]$
• $\ln[A] = \ln[A]_0 - kt$
• $t_{1/2} = \frac{0.693}{k}$

Second Order Reaction

• Rate $= k[A]^2$
• $\frac{1}{[A]} = \frac{1}{[A]_0} + kt$
• $t_{1/2} = \frac{1}{k[A]_0}$

Arrhenius Equation

• $k = A e^{-E_a/RT}$
• $\ln(\frac{k_2}{k_1}) = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$
• Ea = activation energy, R = 8.314 J/mol·K, A =frequency factor

Productos notables

Binomio al cuadrado

• ((a+b)^2 = a^2 + 2ab + b^2)
• ((a-b)^2 = a^2 - 2ab + b^2)

Binomios conjugados

• ((a+b)(a-b) = a^2 - b^2)

Binomio con término común

• ((x+a)(x+b) = x^2 + (a+b)x + ab)

Binomio al cubo

• ((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3)
• ((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3)

Factorización

Factor común

• (ab + ac = a(b+c))

Diferencia de cuadrados

• (a^2 - b^2 = (a+b)(a-b))

Trinomio cuadrado perfecto

• (a^2 + 2ab + b^2 = (a+b)^2)
• (a^2 - 2ab + b^2 = (a-b)^2)

Trinomio de la forma (x^2 + bx + c)

• (x^2 + bx + c = (x+p)(x+q)), where (p+q = b) and (pq = c)

Suma o diferencia de cubos

• (a^3 + b^3 = (a+b)(a^2 - ab + b^2))
• (a^3 - b^3 = (a-b)(a^2 + ab + b^2))

Markdown Quick Start Guide

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Static Equilibrium Study Notes

Objectives

• Study the conditions for static equilibrium.
• Verify that the vector sum of forces is zero.
• Verify that the sum of the torques is zero.

Key Definitions

• Static equilibrium means the object is at rest and remains at rest with zero translational and rotational acceleration.

Translational Equilibrium

• Vector sum of the forces must be zero: (\sum \overrightarrow{F} = 0)

Rotational Equilibrium

• Sum of the torques must be zero: (\sum \tau = 0.)
• Torque Formula: (\tau = rF\sin\theta )
  • r = distance from axis of rotation to the point where the force is applied.
  • F = magnitude of the force.
  • θ = angle between the force vector and the vector from the axis of rotation to the point where the force is applied.

Procedure

• Part 1 covers translational equilibrium.
• Part 2 covers rotational equilibrium.

Data Analysis

Translational Equilibrium:

• Magnitude of the force using: (F = \sqrt{F_x^2 + F_y^2}).
• Direction formula: (\theta = \arctan(\frac{F_y}{F_x})).

Rotational Equilibrium:

• Torque formula: (\tau = rF\sin\theta).

Questions:

• Difference between static and dynamic?
• Conditions for static equilibrium?
• Torque calculation and units?
• Possible sources of error in each experiment?