Chemical Kinetics: Rates and Reaction Order

Chemical Kinetics

• Reaction rate is the change in concentration of a reactant or product over time.

Rate Law

• For a general reaction $aA + bB \rightarrow cC + dD$, the rate law is expressed as $rate = k[A]^m[B]^n$.
• $k$ is the rate constant in the rate law.
• $[A]$ and $[B]$ represent reactant concentrations in the rate law.
• $m$ and $n$ in the rate law are the reaction orders with respect to A and B.

Order of Reaction

• The order of a reaction is calculated by summing the exponents in the rate law: $order = m + n$.

Factors Affecting Reaction Rates

• Increased temperature generally leads to increased reaction rate.

Temperature and Arrhenius Equation

• The Arrhenius equation, $k = Ae^{-E_a/RT}$, describes the relationship between temperature and reaction rate.
• $k$ is the rate constant in the Arrhenius equation.
• $A$ is the pre-exponential factor in the Arrhenius equation.
• $E_a$ is the activation energy in the Arrhenius equation.
• $R$ is the gas constant in the Arrhenius equation.
• $T$ is the absolute temperature in the Arrhenius equation.

Catalyst

• A catalyst increases the rate of a reaction without being consumed in the process.
• Catalysts function by lowering the activation energy of the reaction.

Concentration

• An increased concentration of reactants typically increases the reaction rate.

Surface Area

• For reactions involving solids, a larger surface area generally results in an increased reaction rate.

Reaction Mechanisms

• A reaction mechanism is a sequence of elementary steps that describe the overall reaction.

Rate-Determining Step

• The rate-determining step is the slowest step in the reaction mechanism.
• The rate of the overall reaction depends on the rate of the rate-determining step.

Types of Reactions

First-Order Reactions

• For a first-order reaction, $rate = k[A]$.
• The integrated rate law for a first-order reaction is $ln[A]_t - ln[A]_0 = -kt$.
• $[A]_t$ is the concentration of A at time t.
• $[A]_0$ is the initial concentration of A.
• $k$ is the rate constant of the reaction.

Second-Order Reactions

• For a second-order reaction, $rate = k[A]^2$.
• The integrated rate law for a second-order reaction is $\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt$.
• $[A]_t$ is the concentration of A at time t.
• $[A]_0$ is the initial concentration of A.
• $k$ is the rate constant of the reaction.

Introducción

• Python's clear syntax facilitates easy learning and coding.
• Python is versatile and used widely in web development, data science, AI, and automation.
• Python has multiplatform functionality and works on Windows, macOS, and Linux.

Instalación de Python

• Visit python.org to download the latest version based on your operating system.
• Windows requires running the installer and marking "Add Python to PATH" to execute from the command line.
• Linux installations requires using the distribution's package manager.
• To verify the installation, open the command line and use the command: python3--version.

Entorno de Desarrollo

• Commonly used IDEs are Visual Studio Code and PyCharm.
• VSCode is lightweight and supports extensions and Python.
• PyCharm is a complete IDE with many features for Python.
• Jupyter Notebooks are great for interactive learning and data science.

Variables y Tipos de Datos

• Variables store data, such as strings (str), integers (int), floats (float), and Booleans (bool).

Tipos de Datos

• Str is text strings.
• Int is integer values.
• Float is decimals values.
• Bool is a statement is either true or false.
• print() prints string outputs to the console

Operadores

• Python features: (+,-,*,/,**,//,%)
  • + represents adding a value
  • - represents subtracting a value
  • * represents multiplying a value
  • / represents dividing a value
  • // represents dividing and rounding down to the nearest integer
  • % represents the modulo

Comparación

• Different conditional operators are available
• == represents checking for if the value is equal to
• != represents checking if the value is NOT equal to
• > represents checking if the value is greater than
• < represents checking if the value is lesser than
• >= represents checking if the value is greater than or equal to
• <= represents checking if the value is lesser than or equal to

Lógicos

• Different logic comparisons are available
• and evaluates true if both statements are true
• or evaluates true if at least one statement is true
• not reverses the results, returns false if results is true

Estructuras de Control

• Conditional and loop statements allow different control flows
• Loops such as the for and while loops are available
• for loops can iterate through 0 to one less than the value given
• while loops will only continue if the value set is still true

Estructuras de Datos

• Lists can be created with square brackets and support several methods
• Tuples, are ordered and unchangeable, and may contain similar or different data types, and are represented using parenthesis
• Dictionaries are similar to hashmaps, represented with braces, where the key corresponds to the value

Funciones y Módulos

• Functions allow different types of statements to be reused over and over

Archivos

• Files for reading and writing can be opened in read mode "r", write mode "w", and append mode "a"

Programación Objetos

• Classes define objects whereas instances are specific instances of the calls
• Classes contain attributes and methods
• Inheritance allows object to call functions from their superclass

Herencia

• Inheritance is the ability to create new classes based on existing classes, inheriting attributes and methods.

Reglas de Inferencia

• Rules of inference are schematic methods for constructing valid arguments, deriving true conclusions from true premises.

Modus Ponens (MP)

• MP's form involves a conditional statement "If P then Q," asserting P, and concluding Q.
• P → Q, P, therefore Q

Modus Tollens (MT)

• MT's form starts with "If P then Q," denies Q, and concludes not P.
• p → Q, ¬Q, therefore ¬P

Silogismo Hipotético (HS)

• HS chains conditionals: "If P then Q," "If Q then R," leading to "If P then R."
• P → Q, Q → R, therefore P → R

Silogismo Disyuntivo (DS)

• DS provides options: "Either P or Q," denies P, and concludes Q.
• P ∨ Q, ¬P, therefore Q

Adición (Ad)

• Ad introduces possibilities: from P, infer "P or Q."
• p, therefore P ∨ Q

Simplificación (Simp)

• Simp extracts components: from "P and Q," one can infer P.
• P ∧ Q, therefore P

Conjunción (Conj)

• Conj Joins statements: From P and Q, derive "P and Q."
• p, Q, therefore P ∧ Q

Doble Negación (DN)

• DN asserts equivalence: P is equivalent to "not not P."
• P ≡ ¬¬P

Ley de De Morgan (DM)

• DM illustrates negation: "Not (P and Q)" is equivalent to "Not P or Not Q," and vice versa for OR
• ¬(P ∧ Q) ≡ ¬P ∨ ¬Q and ¬(P ∨ Q) ≡ ¬P ∧ ¬Q

Conmutación (Conm)

• Comm rearranges statements: P OR Q is the same as Q OR P; P AND Q is the same as Q AND P.
• (P ∨ Q) ≡ (Q ∨ P) and (P ∧ Q) ≡ (Q ∧ P)

Asociación (Asoc)

• Asoc Groups without changing meaning: The way ANDs or the way ORs are grouped does not change the result.
• ((P ∨ Q) ∨ R) ≡ (P ∨ (Q ∨ R)) and ((P ∧ Q) ∧ R) ≡ (P ∧ (Q ∧ R))

Distribución (Dist)

• Dist Combines AND and OR: P AND (Q OR R) is (P AND Q) OR (P AND R); P OR (Q AND R) is (P OR Q) AND (P OR R).
• (P ∧ (Q ∨ R)) ≡ ((P ∧ Q) ∨ (P ∧ R)) and (P ∨ (Q ∧ R)) ≡ ((P ∨ Q) ∧ (P ∨ R)).

Implicación Material (IM)

• IM Converts conditionals: "If P then Q" is "Not P or Q."
• (P → Q) ≡ (¬P ∨ Q)

Equivalencia Material (EM)

• EM Equates equivalencies: "P if and only if Q" is "(If P then Q) and (If Q then P)".
• (P ↔ Q) ≡ ((P → Q) ∧ (Q → P)) and (P ↔ Q) ≡ ((P ∧ Q) ∨ (¬P ∧ ¬Q))

Exportación (Exp)

• Exp Rearranges conditions: (If P and Q, then R) is (If P, then (If Q, then R)).
• ((P ∧ Q) → R) ≡ (P → (Q → R))

Reglas de Inferencia Cuantificacional

• Rules for quantified statements involve instantiation and generalization.

Instanciación Universal (IU)

• From "all x have property P," one instance 'a' also has property P.
• ∀x P(x), therefore P(a)

Generalización Universal (UG)

• If an unrelated 'a' has property P, one can generalize that all x have property P.
• P(a), therefore ∀x P(x)

Instanciación Existencial (IE)

• From "there exists an x with property P," pick one 'c' that has this property.
• ∃x P(x), therefore P(c)

Generalización Existencial (EG)

• If 'a' has property P, then there is an entity x that has property P.
• P(a), therefore ∃x P(x)

Physics

Definitions

• Kinematics describes object motion without considering motion's causes, based on position, displacement, velocity, and acceleration.

Equations and Motion

• Constant velocity ($v = \frac{\Delta x}{\Delta t}$) and constant acceleration ($v = v_0 + at$, $x = x_0 + v_0t + \frac{1}{2}at^2$, $v^2 = v_0^2 + 2a\Delta x$) equations relate motion variables.

Dynamics

• Dynamics links motion to forces, governed by Newton's laws.

Newton's Laws of Motion

• First law affirms inertia.
• Second law says $F = ma$.
• Third law notes action-reaction pairs.

Common Forces

• Gravity ($F_g = mg$).
• Normal force counteracts gravity.
• Friction opposes motion where static friction ($F_s \le \mu_s N$) prevents it, and kinetic friction ($F_k = \mu_k N$) slows it.
• Tension pulls via ropes and strings.
• Spring force ($F = -kx$) follows Hooke's law.

Work and Energy

• Work is energy transfer ($W = Fd\cos\theta$).
• Kinetic energy is due to motion ($KE = \frac{1}{2}mv^2$).
• Potential energy refers to gravity ($U_g = mgh$) and springs ($U_s = \frac{1}{2}kx^2$).
• Energy Conservation says $KE_i + U_i = KE_f + U_f$.

Momentum and Collisions

• Momentum is mass times velocity ($p = mv$).
• Impulse changes momentum ($J = \Delta p = F\Delta t$).
• Conservation of momentum affirms $p_i = p_f$.
• Collisions are either perfectly elastic or inelastic.

Rotational Motion

• Rotation uses angular position, displacement, and velocity ($\omega = \frac{\Delta \theta}{\Delta t}$)

Angular Acceleration

• Constant is $\alpha = \frac{\Delta \omega}{\Delta t}$.
• Torque rotates ($F = qvB\sin\theta$).
• Inertia and angular momentum describe resistance to and quantity of rotational action.

Rotational Equations

• Rotation can be expressed with equations mirroring linear ones.
• Newton's second law describes rotation where $\tau = I\alpha$.
• Angular momentum is is conserved in closed system.

Waves and Optics

Types of Waves

• Includes transverse (perpendicular displacement), longitudinal (parallel), and electromagnetic (charged particle acceleration) waves.

Wave Properties

• Amplitude is maximum displacement, wavelength is crest-to-crest distance, frequency is waves per second, period is wave duration, and speed is $v = f\lambda$.

Interference and Diffraction

• Overlapping waves superpose, causing constructive or destructive interference, and diffract around obstructions.

Optics

• Reflection bounces light and refraction bends it with lenses focusing, mirrors reflecting.

Thermodynamics

• Thermodynamics includes temperature, heat, internal energy, and Entropy

Laws of Thermodynamics

• Zeroth law sets thermal equilibrium, first law equates energy change to heat and work, second law states entropy increase, and third law describes entropy at absolute zero.

Heat Transfer

• Heat transfers through conduction (direct contact), convection (fluid movement), and radiation (electromagnetic waves).

Thermodynamic Processes

• Processes are isobaric (constant pressure), isochoric (constant volume), isothermal (constant temperature), adiabatic (no heat exchange).

Electricity and Magnetism

• Charge interacts via electric fields ($E = \frac{F}{q}$) following Coulomb's law ($F = k\frac{q_1q_2}{r^2}$).

Electric Potential and Capacitance

• Potential $(V = \frac{U}{q})$ drives charge storage in capacitors ($C = \frac{Q}{V}$).

Electric Current and Resistance

• Current is passing charge (I=ΔQ/Δt ), opposed by resistance(R=V/I), per Ohm's law ( V=IR ).

Magnetic Field and Magnetic Force

• Magnetic Influence is exerted via magnetic fields and forces: $F=qvBsin\theta$

Electromagnetic Induction

• Field Change induces Voltage as described by Faraday's +Lenz's Law.

Modern Physics

Quantum Mechanics

• Quantum mechanics involves wave-particle duality and the uncertainty principle.

Nuclear Physics

• Nuclei decay creating radiation, reactions with fission/fusion.

Relativity

• Space & time bend creating time dilation, and mass can become pure energy: $E=mc^2$.

Graphics Output

• Coordinate systems are essential for displaying data graphically. R creates scatterplots with plot(x, y), automatically setting scales.

High-Level Plotting Functions

• R's high-level plotting (plot(), hist(), boxplot()) can create complete graphics.
• Customize titles, x/y labels with arguments like main, xlab, ylab.

Low-Level Plotting Commands

• R uses low-level commands (points(), lines(), text()) to add to existing plots, supporting customization.

The par() Function

• The par() function controls persistent global graphics parameters.

Graphics Device Drives

• Graphics device drivers allow the specifying of graphic output (pdf(), png(), jpeg()).

Example: Scatterplot of eruption versus waiting times

plot(faithful$eruptions, faithful$waiting,
     main="Eruption vs. Waiting Time",
     xlab="Eruption Time (minutes)",
     ylab="Waiting Time (minutes)")

• This R code generates a scatterplot of the Old Faithful dataset, labeling eruption duration and waiting time.

Algorithmic Game Theory

• Players, strategies, and payoffs are involved in game theory.

Key Questions in AGT

• Concerns include computational complexity, mechanism design, strategic learning, and outcome efficiency

Key elements

• Players: Decision-makers in the game.
• Strategies: Possible actions for each player.
• Payoffs: Outcomes for players based on combined strategies.

The Prisoner's Dilemma

• A classic example illustrates game dynamics.

Selfish Routing

• Nodes/edges compromise a network.
• Each edge $e$ possesses a cost function $l_e(x)$ which depends on the edge's flow $x$.
• $A$ amount of total flow $r$ is routed from a source $s$ to a destination $t$.

Selfish Behavior

• Each flow decides the path that reduces its own cost.
• As such, a Nash equilibrium exists when no flow can have improves.

Shannon Channel Coding Theorem

Braess's Paradox

• Adding an edge can hinder the cost of the Nash equilibrium

Lecture 18: The Hydrogen Atom

.1 Schrödinger Equation in Spherical Coordinates

.1.1. Laplacian in spherical coordinates

• $\nabla^2 = \dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2\dfrac{\partial}{\partial r}\right) + \dfrac{1}{r^2\sin\theta}\dfrac{\partial}{\partial \theta}\left(\sin\theta\dfrac{\partial}{\partial \theta}\right) + \dfrac{1}{r^2\sin^2\theta}\dfrac{\partial^2}{\partial \phi^2}.$

.1.2 Schrödinger equation for the Hydrogen atom

• $-\dfrac{\hbar^2}{2\mu}\nabla^2\psi(r,\theta,\phi) - \dfrac{e^2}{r}\psi(r,\theta,\phi) = E\psi(r,\theta,\phi)$

.2 Separation of Variables

• $\psi(r,\theta,\phi) = R(r)\Theta(\theta)\Phi(\phi)$

.2.1 Φ equation

• $\dfrac{d^2\Phi}{d\phi^2} = -m^2\Phi(\phi)$, where $m = 0, \pm 1, \pm 2,...$ Solution is $\Phi_m(\phi) = A e^{im\phi}$

.2.2 Θ equation

• $\dfrac{1}{\sin\theta}\dfrac{d}{d\theta}\left(\sin\theta\dfrac{d\Theta(\theta)}{d\theta}\right) - \dfrac{m^2}{\sin^2\theta}\Theta(\theta) = -l(l+1)\Theta(\theta)$, where $l = |m|, |m|+1,...$

.2.3 Radial equation

• $\dfrac{1}{r^2}\dfrac{d}{dr}\left(r^2\dfrac{dR(r)}{dr}\right) + \left[\dfrac{2\mu}{\hbar^2}\left(E + \dfrac{e^2}{r}\right) - \dfrac{l(l+1)}{r^2}\right]R(r) = 0$

Channel Capacity

• Channel capacity is the maximum rate of reliable communication over a noisy channel.

Definition: Channel Capacity

• $C = \max_{p(x)} I(X; Y)$, where $p(x)$ is the input distribution and $I(X; Y)$ is the mutual information between the input $X$ and the output $Y$.
• $C$ is the maximum number of bits per channel use that can be reliably transmitted over the channel.

Properties of Channel Capacity

• $C \geq 0$ since $I(X; Y) \geq 0$
• $C \leq \min {\log |X|, \log |Y|}$
• $C$ is a property of the channel alone, not of any particular use of the channel.

Oscillations

Definition

• In SHM, the restoring force is proportional to the displacement.

Characteristics

• The amplitude (A) measures max separation and equilibrium
• The period (T) determines time per cycle
• The $f = 1/T$, determine the oscillations per unit
• $ \omega = 2\pi/T$ equation provides equation for angular frequency

Oscillation Equations

• Displacement: equation $ x(t) = Acos(ωt+ϕ)$ defines position using phase shift
• At time t equation for:
  • $\omega*isin(\omega t+\phi)$ defines velocity
  • $\omega^2 * x(t)$ defines accelerations

Energy in SHM

• SHM (oscillations) possess kinetic and potential energies.

Kinetic Equation

• The term, $0.5kA * cos^2 * (\omega t+ \phi))$, defines the value for potential Equation $0.5mA^2 \omega sin^2(\omega t+ \phi)$ defines the value for Kinetic Summation of both values are $0.5kA^2$.

Equations of Motion

• Equation theta(max) define relationship between values in motion. Angular frequency omega= sqrt(g/L) and T defines equations and is equal $2(PI) sqrt(L/g)$.

Forced Oscillations and Resonance

• External application with periodic force makes waves with external periodic motion

• $m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega t)$ defines amplitude within forced movement.