Chemical Kinetics: Reaction Rates and Rate Laws

Chemical Kinetics

• Chemical reaction rate is the change in reactant or product concentration with time.
• Rate law shows the relationship between reaction rate and reactant concentrations.
• For a reaction like $aA + bB \rightarrow cC + dD$, the rate law is $rate = k[A]^m[B]^n$.
• k is the rate constant, [A] and [B] are reactant concentrations, m and n are reaction orders for A and B respectively.
• Reaction order is the sum of the exponents in the rate law (Overall order = m + n).

Units of Rate Constant (k)

• The units of k depend on the reaction's overall order.
• Zero Order: $mol \cdot L^{-1} \cdot s^{-1}$
• First Order: $s^{-1}$
• Second Order: $L \cdot mol^{-1} \cdot s^{-1}$
• Third Order: $L^2 \cdot mol^{-2} \cdot s^{-1}$
• n Order: $L^{n-1} \cdot mol^{1-n} \cdot s^{-1}$

Integrated Rate Laws

• Zero Order:
  • Rate = k
  • $[A]_t = -kt + [A]_0$
  • $t_{1/2} = \frac{[A]_0}{2k}$
• First Order:
  • Rate = $k[A]$
  • $ln[A]_t = -kt + ln[A]_0$
  • $t_{1/2} = \frac{0.693}{k}$
• Second Order:
  • Rate = $k[A]^2$
  • $\frac{1}{[A]_t} = kt + \frac{1}{[A]_0}$
  • $t_{1/2} = \frac{1}{k[A]_0}$
• $[A]_t$ is concentration of A at time t, $[A]0$ is initial concentration of A, and $t{1/2}$ is half-life.

Activation Energy

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

Determining Ea

• Taking the natural logarithm of the Arrhenius equation:$lnk = \frac{-E_a}{R} (\frac{1}{T}) + lnA$
• Plotting ln k versus $\frac{1}{T}$ provides a straight line with slope $\frac{-E_a}{R}$ $ln\frac{k_2}{k_1} = \frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2})$

Reaction Mechanisms

• A series of elementary steps depicting the path of a reaction.
• The rate-determining step is the slowest step, governing the overall reaction rate.

Catalysis

• A catalyst speeds up a reaction without being consumed.
• Catalysts function by decreasing a reaction's activation energy.

Cardiovascular System

Blood Vessels

• Arteries: Carry blood away from the heart.
  • Walls have 3 layers: Tunica intima (endothelium), Tunica media (smooth muscle/elastic fibers controlled by sympathetic nervous system for vasoconstriction/vasodilation), and Tunica externa (collagen fibers).
• Capillaries: Smallest vessels comprised of thin tunica intima, facilitating gas, nutrient, waste, and hormone exchange between blood and interstitial fluid.
• Veins: Carry blood to the heart, with thinner walls than arteries, use valves to prevent backflow due to lower blood pressure.

Circulatory Routes

• Pulmonary Circuit: Right ventricle $\rightarrow$ Pulmonary trunk $\rightarrow$ Pulmonary arteries $\rightarrow$ Lungs $\rightarrow$ Pulmonary veins $\rightarrow$ Left atrium.
• Systemic Circuit: Left ventricle $\rightarrow$ Aorta $\rightarrow$ Systemic arteries $\rightarrow$ Capillaries in tissues $\rightarrow$ Systemic veins $\rightarrow$ Superior vena cava/inferior vena cava $\rightarrow$ Right atrium.

Blood Pressure

• Force per unit area exerted by blood on vessel walls.
• Measured via systolic (ventricular contraction) and diastolic (ventricular relaxation) pressures in mm Hg (e.g., 120/80 mm Hg).
• Affected by cardiac output, peripheral resistance (blood viscosity, vessel length/diameter), and blood volume.

Lab 4 Report

Introduction

• Filters pass signals of certain frequencies and block others and can be:
  • Low-pass (pass low-frequency signals/block high-frequency)
  • High-pass (pass high-frequency signals/block low-frequency)
  • Band-pass (pass signals in a specific frequency range)
  • Band-stop (block signals in a specific frequency range).

Design and Simulation

• First-Order Low-Pass Filter
  • Cutoff frequency: $f_c = \frac{1}{2\pi RC}$
  • For $f_c = 2kHz$ and $C = 0.1 µF$, $R \approx 795 \Omega$
  • Circuit design verified via simulation from 1Hz to 100kHz.
• First-Order High-Pass Filter
  • Cutoff frequency: $f_c = \frac{1}{2\pi RC}$
  • For $f_c = 2kHz$ and $C = 0.1 µF$, $R \approx 795 \Omega$
  • Circuit design verified via simulation from 1Hz to 100kHz.
• Second-Order Band-Pass Filter
  • Parameters include $f_c = 2kHz$, $Q = 5$, $G = 1$.
  • $f_c = \sqrt{f_1 f_2}$, $Q = \frac{f_c}{BW} = \frac{f_c}{f_c2 - f_c1}$ , $G = \frac{R_2}{2R_1}$
  • For $C_1 = C_2 = 0.1 µF$, $R_1 \approx 3978.87 \Omega$ and $R_2 \approx 7957.74 \Omega$ with gain.
  • Circuit design verified via simulation from 100Hz to 10kHz.

Experimental Results

• Filter circuits are constructed, and their frequency response is measured by varying input signal frequency and measuring output voltage.
• Gain is calculated by: $Gain(dB) = 20 \log_{10} \left( \frac{V_{out}}{V_{in}} \right)$

Discussion

• Experimental results from the constructed filters are analyzed and compared with simulation results to identify discrepancies.

Lecture 1 (Jan 8, 2013): What is Quantum Mechanics?

• Quantum mechanics is a theory of "small things" where action is on the order of Planck's constant ($h = 6.626 \times 10^{-34} Js = 4.136 \times 10^{-15} eVs$) involving atoms, molecules, and electrons in solids.
• Classical mechanics fails to explain experiments like black body radiation, photoelectric effect, atomic spectra.
• Quantum mechanics involves superposition, quantum entanglement, and the measurement problem.

Course Outline

• Part 1: Basic Quantum Mechanics
  • Mathematical tools: Linear vector spaces, Operators, Hilbert space
  • Postulates: States, Observables, Measurement, Time evolution
  • Examples: Particle in a box, Harmonic oscillator, Hydrogen atom
• Part 2: Advanced Topics
  • Quantum entanglement
  • Quantum information
  • Quantum computing

Mathematical Tools

• Linear Vector Spaces: Vectors that can be added and multiplied by scalars.
  • Addition: If $|\alpha \rangle$ and $|\beta \rangle$ are vectors, then $|\alpha \rangle + |\beta \rangle$ is also a vector.
  • Scalar multiplication: If $|\alpha \rangle$ is a vector and $c$ is a scalar, then $c|\alpha \rangle$ is also a vector.
• Examples: $n$-tuples of complex numbers ($\mathbb{C}^n$), functions f(x) defined on the interval [a, b]

Inner Product

• A function that takes two vectors and returns a scalar; follows conjugate symmetry, positivity, and linearity ($ \langle \alpha | \beta \rangle = \langle \beta | \alpha \rangle^*$, $\langle \alpha | \alpha \rangle \geq 0$, $\langle \alpha | c_1 |\beta_1 \rangle + c_2 |\beta_2 \rangle \rangle = c_1 \langle \alpha | \beta_1 \rangle + c_2 \langle \alpha | \beta_2 \rangle$).
• Examples: $\langle \alpha | \beta \rangle = \sum_{i=1}^{n} \alpha_i^* \beta_i$ for $\mathbb{C}^n$; $\langle f | g \rangle = \int_{a}^{b} f^*(x) g(x) dx$ for functions f(x) and g(x).

Algorithmic Game Theory

• Game theory is a framework for analyzing strategic interactions between rational decision-makers, to understand and predict outcomes where entity's actions affect others.
• Key Concepts: Players, Strategies, Payoffs, Rationality, Equilibrium.
• Types: Cooperative vs Non-cooperative, Symmetric vs Asymmetric, Perfect vs Imperfect Information, Zero-sum vs Non-zero-sum, Simultaneous vs Sequential.

Normal-Form Games

• A representation specified by players, strategies, and payoffs.
• Defined as: $(N, A, u)$ $N$: finite set of players, where indexed by $i$
  $A = A_1 \times \dots \times A_n$: a set of actions, where $A_i$ is the set of actions available to player $i$
  $u = (u_1, \dots, u_n)$: function for player $i$, $u_i : A \rightarrow \mathbb{R}$
• Example: Prisoner's Dilemma.

Strategies

• Pure Strategy: Selecting an action deterministically.

• Mixed Strategy: Assigning a probability distribution over actions, expressed as: $s_i \in [0, 1]^{A_i} \text{ and } \sum_{a_i \in A_i} s_i(a_i) = 1$

• Utility of player $i$:
  $u_i(s) = \sum_{a \in A} u_i(a) \prod_{j \in N} s_j(a_j)$

Dominant Strategies

• Strategy $s_i \in S_i$ strictly dominates $s'i \in S_i$ if:
  $u_i(s_i, s{-i}) > u_i(s'i, s{-i})$

• Strategy $s_i \in S_i$ weakly dominates $s'i \in S_i$ if:
  $u_i(s_i, s{-i}) \ge u_i(s'i, s{-i})$
  and if there exists a strategy profile $s'{-i} \in S{-i}$ such that:
  $u_i(s_i, s'_{-i}) > u_i(s'i, s'{-i})$

• Iterated Elimination of Dominated Strategies is a concept where players remove strictly dominated strategies until there are none to remove.

Nash Equilibrium

• Strategy profile is equilibrium if for all players, $s_i$ is the best response.
• $u_i(s_i, s_{-i}) \ge u_i(s'i, s{-i}) \quad \forall s'_i \in S_i$
• Nash's Theorem (1950): asserts every game with a finite amount of player has at least one Nash Equilibrium

Correlated Equilibrium

• $p(a)$ is a correlated equilibrium with action profiles $ a_i, a'_i \in A_i$

$$ \sum_{a_{-i} \in A_{-i}} p(a_i, a_{-i}) u_i(a_i, a_{-i}) \ge \sum_{a_{-i} \in A_{-i}} p(a_i, a_{-i}) u_i(a'i, a{-i}) $$

Extensive-Form Games

• Representation specifying: players, moves, information, and payoffs defined by $(\mathcal{N}, \mathcal{A}, H, P, f_c, \mathcal{I}, u)$
• Where $s_i : \mathcal{I}_i \rightarrow \mathcal{A}$ assigned to each information set.
• Backwards induction is a solution concept that works backward in perfect information.

Reglas de la Derivada (Rules of Differentiation - Spanish Notes)

Regla de la Potencia (Power Rule)

• If $f(x) = x^n$, then $f'(x) = nx^{n-1}$
  • Example: $f(x) = x^3$, then $f'(x) = 3x^2$ or $f(x) = \sqrt{x} = x^{1/2}$, then $f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$

Regla de la Constante (Constant Rule)

• If $f(x) = c$, c is a constance, then $f'(x) = 0$
  • Example: If $f(x) = 5$, then $f'(x) = 0$

Regla del Múltiplo Constante (Constant Multiple Rule)

• If $f(x) = c \cdot g(x)$, where c is a constant, then $f'(x) = c \cdot g'(x)$
  • Example: if $f(x) = 3x^2$, then $f'(x) = 3(2x) = 6x$

Regla de la Suma/Resta (Sum/Difference Rule)

• If $f(x) = u(x) \pm v(x)$, then $f'(x) = u'(x) \pm v'(x)$
  • Example: If $f(x) = x^3 + 4x^2$, then $f'(x) = 3x^2 + 8x$

Regla del Producto (Product Rule)

• If $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$ - Example: $f(x) = x^2 \sin(x)$, then $f'(x) = 2x\sin(x) + x^2\cos(x)$

Regla del Cociente (Quotient Rule)

• If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$ - Example: $f(x) = \frac{x^2}{x+1}$, then $f'(x) = \frac{2x(x+1) - x^2(1)}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}$

Regla de la Cadena (Chain Rule)

• If $f(x) = u(v(x))$, then $f'(x) = u'(v(x)) \cdot v'(x)$
  • Example: $f(x) = \sin(x^2)$, then $f'(x) = \cos(x^2) \cdot 2x$ or $f(x) = (2x + 1)^3$, then $f'(x) = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2$

Derivadas de Funciones Trigonométricas (Derivatives of Trig Functions)

• $f(x) = \sin x$ | $f'(x) = \cos x$
• $f(x) = \cos x$ | $f'(x) = -\sin x$
• $f(x) = \tan x$ | $f'(x) = \sec^2 x$
• $f(x) = \csc x$ | $f'(x) = -\csc x \cot x$
• $f(x) = \sec x$ | $f'(x) = \sec x \tan x$
• $f(x) = \cot x$ | $f'(x) = -\csc^2 x$

Derivadas de Funciones Exponenciales y Logarítmicas (Derivatives of Logarithmic and Exponential Functions)

• $f(x) = e^x$ | $f'(x) = e^x$
• $f(x) = a^x$ | $f'(x) = a^x \ln a$
• $f(x) = \ln x$ | $f'(x) = \frac{1}{x}$
• $f(x) = \log_a x$| $f'(x) = \frac{1}{x \ln a}$

Plan de estudios de maestría en matemáticas 2024-2026 (Masters in Mathematics Study Plan)

• Designed to provide advanced training to prepare students for careers in academia, industry, and public service.

Objetivos (Objectives)

• General Objective: train qualified math professionals capable of original research, applying math in complex problems, and contributing to scientific/technological growth.
• Specific Objectives: provide a theoretical/methodological foundation, develop analytical problem-solving, foster high quality research, enable effective communication.

Perfil de Ingreso (Entry Profile)

• Bachelor's degree in mathematics or related fields.
• Solid knowledge of calculation, algebra, equations, probability.
• Reading comprehension in English.
• Basic programming skills.
• Interest in research

Perfil de Egreso (Graduate Profile)

• Deep knowledge and application of tools
• Conduct of original research
• Effective communication and collaboration

Estructura Curricular (Curricular Structure)

• Course is structured around two years (four semesters)
• Consists of 36 credits, 24 optative credits, 6 for complementary activities (workshops, seminars), and a 30 credit thesis.

Plan de Estudios Detallado (Detailed Study Plan)

• Primer Semestre (First Semester):
  • Real Analysis (9 credits)
  • Modern Algebra (9 credits)
  • Research Seminar I (3 credits)
• Segundo Semestre (Second Semester):
  • Complex Analysis (9 credits)
  • General Topology (9 credits)
  • Research Seminar II (3 credits)
• Tercer Semestre (Third Semester):
  • Elective Courses (12 credits)
  • Thesis Progress I (3 credits)
• Cuarto Semestre (Fourth Semester):
  • Elective Courses (12 credits)
  • Thesis Progress II (3 credits)

Cursos Optativos (Elective Courses)

• Students must pick among areas: Algebra, Analysis, Geometry, Topology, Differential Equations, Applied Mathematics, Statistics.

Trabajo de Tesis (Thesis Work)

• Thesis consists in original research supervised by faculty.

Requisitos de Graduación (Graduation Requirements)

• Pass all courses and have a minimum 8.0 grade
• 2. Complete Thesis
• 3. Demonstrate in English (level B2)