Phase Equilibria and Diagrams

Phase Equilibria

• At equilibrium, a system's free energy is at its minimum for given temperature, pressure, and composition.
• A closed system will spontaneously change toward its equilibrium state.

Phases

• A phase is a homogeneous portion of a system with uniform physical and chemical characteristics.
• A phase can exist in solid, liquid, or gaseous state.

Phase Diagrams

• Phase diagrams represent relationships between phases in a system at equilibrium.
• These diagrams are also known as equilibrium or constitutional diagrams.

Equilibrium and Metastability

• Equilibrium represents the state of minimum free energy.
• Metastable states exist when free energy is slightly higher than equilibrium; they can persist for extended durations.

Cooling Curves

• Cooling curves illustrate temperature changes over time as a material cools.
• They aid in determining a material's phase diagram.

Solubility Limit

• This is the maximum solute concentration dissolving in a solvent to form a solid solution.
• Increases with temperature.

Components and Phases

• Components are the independent chemical species within a system.
• A phase is a homogeneous portion of the system characterized by uniform physical and chemical properties.

Gibbs Phase Rule

• The Gibbs Phase Rule is expressed as P + F = C + N
• P = number of phases present
• F = number of degrees of freedom
• C = number of components
• N = number of noncompositional variables (T, P)
• For constant pressure, the formula simplifies to P + F = C + 1

Binary Phase Diagrams

• Binary phase diagrams are for systems with two components.
• They define phases at varying temperatures and compositions.
• They define phase compositions at different temperatures and compositions.
• They show phase fractions at different temperatures and compositions

Isomorphous Systems

• These systems feature complete solubility of one component in another.
• The alpha ($\alpha$) phase field spans from 0 to 100 wt%.
• Examples of isomorphous systems include Cu-Ni, Ag-Au, and Ge-Si.

Capítulo 2. Álgebra de Boole

2.1 Introducción

• George Boole (1815-1864) developed Boolean algebra.
• Boolean algebra is a deductive mathematical system focused on zero and one (false and true).
• A binary operator accepts a pair of inputs and yields a single Boolean value.
• The Boolean equivalent of arithmetic addition (+) is OR.
• The Boolean equivalent of arithmetic multiplication (.) is AND.
• The Boolean equivalent of complementation (-) is NOT.
• Boolean values are represented by capital letters such as A, B, C.

2.2 Postulados fundamentales del álgebra booleana

• Boolean algebra is governed by fundamental postulates:
  • Closure: Boolean algebra is closed under AND, OR, and NOT operators.
  • Identity element: Identity elements exist for AND (1) and OR (0) such that A + 0 = A and A * 1 = A.
  • Commutativity: AND and OR operators are commutative: A + B = B + A and A * B = B * A.
  • Distributivity: AND and OR are distributive over each other: A * (B + C) = (A * B) + (A * C) and A + (B * C) = (A + B) * (A + C).
  • Complement element: For every Boolean value A, a complement exists like A + A' = 1 and A * A' = 0.
  • Idempotence law: A + A = A and A * A = A.
  • Absorption law: A + (A * B) = A and A * (A + B) = A.
  • DeMorgan's laws: (A + B)' = A' * B' and (A * B)' = A' + B'.

Lab 3: Spectral Analysis of Signals

Objective

• Spectral analysis on signals is performed using tools like Matlab.
• Concepts of Fourier series, Fourier transform, power spectral density, and time-frequency analysis get reinforced.

Preliminaries

• Read Signals and Systems, by Haykin & Van Veen, (2nd ed.) Ch. 4.
• Matlab functions used include: fft, abs, angle, imag, real, linspace, plot, soundsc, axis, xlabel, ylabel, title, stem, fftshift, ifftshift, spectrogram.

Procedure

Fourier Series

• A periodic square wave $x(t)$ with period $T_0 = 1$ is considered, defined as: $$ x(t) = \begin{cases} 1, & -\frac{T_1}{2} < t < \frac{T_1}{2} \\ 0, & \frac{T_1}{2} < t < T_0 - \frac{T_1}{2} \end{cases} $$ where $T_1 = \frac{T_0}{4} = 0.25$.
• Analytically determine the Fourier series coefficients $c_k$ for $x(t)$.
• Sampling $x(t)$ with sampling period $T_s = \frac{T_0}{200}$ creates a discrete-time signal $x[n]$.
• One period ($0 \le n \le 199$) of $x[n]$ is generated in Matlab.
• The Discrete Fourier Series (DFS) coefficients, $X[k]$, of $x[n]$ are computed with fft.
• Plots of the magnitude $|X[k]|$ and phase $\angle X[k]$ of DFS coefficients are shown for $0 \le k \le 199$.
• It is important to understand how DFS coefficients $X[k]$ relate to Fourier series coefficients $c_k$.
• The sample period for the discrete-time signal $x[n]$ is changed to $T_s = \frac{T_0}{2}$.
• Generating one period ($0 \le n \le 1$) of the new $x[n]$ in Matlab happens with the new sampling period.
• The Discrete Fourier Series coefficients, $X[k]$, for the new $x[n]$ are again computed using the fft command in Matlab.
• The magnitude $|X[k]|$ and phase $\angle X[k]$ are plotted for $0 \le k \le 1$.
• Sampling $x(t)$ using a sampling period of $T_s = \frac{T_0}{200}$ yields a discrete-time signal $x[n]$.
• Generating five periods ($0 \le n \le 999$) of $x[n]$ in Matlab is next.
• Again, the Discrete Fourier Series coefficients, $X[k]$, for $x[n]$ are computed using the fft command in Matlab.
• Magnitudes $|X[k]|$ and phases $\angle X[k]$ of the DFS coefficients are plotted for $0 \le k \le 999$.

Fourier Transform

• Consider the continuous-time signal $x(t) = e^{-|t|}$.
• Analytically determine the Fourier transform $X(f)$ for $x(t)$.
• Sampling $x(t)$ with sampling period $T_s = 0.05$ creates a discrete-time signal $x[n]$.
• The discrete-time signal $x[n]$ is generated in Matlab, ranging from $-10 \le t \le 10$.
• The Discrete-Time Fourier Transform (DTFT), $X(e^{j\omega})$, of $x[n]$ is computed via the fft command in Matlab.
• Plots of the magnitude $|X(e^{j\omega})|$ and phase $\angle X(e^{j\omega})$ of the DTFT are generated for $-\pi \le \omega \le \pi$.
• The focus is how $X(e^{j\omega})$ relates to the Fourier transform $X(f)$.
• Parts are repeated using sampling periods of $T_s = 0.5$ and $T_s = 1.0$.
• Consider the continuous-time signal $x(t) = \cos(2\pi f_0 t)u(t)$, with $f_0 = 100$ Hz where $u(t)$ is the unit step function.
• Creating a discrete-time signal $x[n]$ is the next step by sampling $x(t)$ with a sampling frequency $F_s = 10$ kHz.
• Generating $1$ second of discrete-time signal $x[n]$ in Matlab follows.
• The DTFT, $X(e^{j\omega})$, of $x[n]$ is computed using the fft command.
• The magnitude $|X(e^{j\omega})|$ of the DTFT is plotted for $0 \le \omega \le \pi$.
• Changing the length of signal leads to different results.

Power Spectral Density

• Generating $10$ seconds of zero-mean white Gaussian noise $w[n]$ in Matlab with variance $\sigma^2 = 1$.
• A sampling frequency of $F_s = 10$ kHz is used.
• Computed and plot the power spectral density (PSD), $S_w(e^{j\omega})$, of the white Gaussian noise $w[n]$.
• The relationship $x[n] = ax[n-1] + w[n]$ is considered with the first-order AR process where $a = 0.9$.
• Generating $10$ seconds of the AR process $x[n]$ in Matlab is the next step, with a sampling frequency of $F_s = 10$ kHz.
• It is key to compute and plot the PSD, $S_x(e^{j\omega})$, of the AR process $x[n]$.
• How the PSD changes when varying parameter $a$.

Spectrogram

• Creating a discrete-time signal $x[n]$ consisting of three tones with frequencies $f_1 = 100$ Hz, $f_2 = 200$ Hz, and $f_3 = 300$ Hz.
• Each tone should have a duration of $1$ second.
• With a sampling frequency of $F_s = 1$ kHz.
• Computing and displaying the spectrogram of $x[n]$ using the spectrogram command.
• Window length and overlap factor experimentation
• Creating a discrete-time signal $x[n]$ consisting of a chirp signal, with a frequency increasing linearly from $0$ Hz to $500$ Hz over $5$ seconds.
• The sampling frequency is set to $F_s = 1$ kHz.
• Computing and displaying the spectrogram of $x[n]$ using Matlab's spectrogram command.
• Experimentation with different window lengths and overlap factors.

Matrizen

Definition

• A matrix A is a rectangular array of numbers arranged in rows and columns.
• $a_{ij}$ is the element in the i-th row and j-th column.
• An m x n matrix has m rows and n columns.
• An n x n matrix is called a square matrix.

Spezielle Matrizen

• Nullmatrix: All elements are zero.
• Identity matrix $I_n$: Square matrix with ones on the main diagonal and zeros elsewhere.
• Diagonal matrix: Square matrix with all elements outside the main diagonal being zero.
• Transpose matrix $A^T$: Rows and columns of $A$ are interchanged.

Matrixoperationen

• Matrix Addition and Subraction: Two matrices A and B can be added / subtracted only if they have the same dimensions. The addition / subraction is performed element-wise
  • $(A + B){ij} = a{ij} + b_{ij}$
  • $(A - B){ij} = a{ij} - b_{ij}$
• Scalar mutliplication: A matrix A can be scaled with a scalar c, by multipyling each element of A with c.
  • $(cA){ij} = c \cdot a{ij}$
• Matrix Multiplication: Given to matrices and $A (m \times n)$ and $B (n \times p)$, they are multiplied to the resulting matrix $C(m \times p)$ with:
  • $c_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj}$

Eigenschaften der Matrixmultiplikation

• Associativity: $(AB)C = A(BC)$.
• Distributivity: $A(B + C) = AB + AC$ and $(A + B)C = AC + BC$.
• Non commutative: In general $AB \neq BA$.

Lineare Gleichungssysteme

• A linear system of equations can be represented in matrix form: $Ax = b$
• A is the coefficient matrix.
• x is the vector of unknowns.
• b is the vector of constants.

Inverse Matrix

• The inverse matrix $A^{-1}$ of a matrix A exists, such as $AA^{-1} = A^{-1}A = I_n$.

Determinante

• The determinant is a function that associates a number to a square matrix. (Ausschuss über die Eigenschaften der Matrix und ist wichtig für die Lösung linearer Gleichungssysteme.)
• 2 x 2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is $\det(A) = ad - bc$.
• Larger matrices can be recursively determined of with other methods.

Rang einer Matrix

• Matrix Rank is the maximum number of its linear independent rows and columns.

Chemical Kinetics

Overview

• Chemical kinetics, or reaction kinetics, examines the rates of chemical reactions.
• It studies how experimental conditions affect reaction speeds.
• It provides insights into reaction mechanisms and transition states.
• Mathematical models are constructed to describe reaction characteristics.

Factors Affecting Reaction Rates

• Reaction rate often rises with increased reactant concentrations due to more frequent collisions.
• Reaction rate is determined by contact area when reactants are in different phases.
• Higher reaction temperatures increase the rate by increasing collision energy.
• Catalysis accelerate reactions without being consumed, usually by lowering activation energy.

Reaction Rate Expression

• For $aA + bB \rightarrow cC + dD$: $$ -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt} = \frac{1}{d} \frac{d[D]}{dt} $$ where $[X]$ denotes concentration; a, b, c, d are stoichiometric coefficients; t is time.

Rate Laws

• Rate laws link reaction rate with reactant concentrations or pressures.
• Typically has the form: rate = $k[A]^m[B]^n$
  • k = rate constant
  • m, n = reaction orders with respect to A and B.
  • m + n = is the overall reaction order.

Reaction Order Types

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

Activation Energy

• Activation energy ($E_a$) is the minimum energy needed for a reaction.
• Can use $k = Ae^{-\frac{E_a}{RT}}$ to describe the activation energy
  • k: rate constant
  • A: frequency factor
  • Ea: activation energy
  • R: gas constant 8.314J/(mol·k)
  • T: temperature(K)

Reaction Mechanisms

• Reaction mechanisms detail elementary reaction steps in an overall chemical change.
• These mechanisms are theoretical constructs that describe what takes place during each step.

Resumen de la Ley de Charles

• Charles's law is an experimental gas law that looks at how gasses expand under heating with

For a fixed mass of an ideal gas at a constant pressure, the volume is directly proportional to the absolute temperature.

• V1 / T1 = V2 / T2
• V1: Initial
• V2: Final
• T1: aboslute initial
• T2: absolute final

Antecedentes

• The law is named for Jacques Charles who found the law in the 1780s

Plan général

• Overview of building layout.
• Scale: Typically 1:50 or 1:100.
• Shows space relationships.
• Indicates walls, windows, doors.

Plan d'étage

• Building floor drawing.
• Scale: Typically 1:50.
• Shows room layouts, walls, windows, doors, stairs.
• Indicates furniture & appliances.

Élévation

• Exterior building view.
• Scale: Typically 1:50 or 1:100.
• Shows building appearance from outside.
• Indicates construction materials.

Coupe

• Vertical building cut drawing.
• Scale: Typically 1:50.
• Shows inter-floor relationships.
• Indicates building structure.

Détail

• Larger scale drawing of part of building.
• Scale: Typically 1:5, 1:10, or 1:20.
• Shows construction of specific building parts.
• Indicates construction materials.

Numéro de port

• The chart below lists the door number along with the width, height and thickness of the doors | Numéro de porte | Largeur (MW) | Hauteur (MH) | Épaisseur | | :-------------: | :----------: | :----------: | :-------: | | D1 | 900 mm | 2100 mm | 40 mm | | D2 | 850 mm | 2100 mm | 40 mm | | D3 | 800 mm | 2100 mm | 40 mm | | D4 | 750 mm | 2100 mm | 40 mm | | D5 | 700 mm | 2100 mm | 40 mm | | D6 | 650 mm | 2100 mm | 40 mm | | WD1 | 1800 mm | 2100 mm | 40 mm | | WD2 | 1500 mm | 2100 mm | 40 mm | | WD3 | 1200 mm | 2100 mm | 40 mm |