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What the Chemical Principles book is designed for

• Designed for students with at least one introductory chemistry course
• Covers stoichiometry, atomic theory, chemical bonding, states of matter, and more

Topics Covered

• Thermochemistry
• Chemical Equilibrium
• Acids and Bases
• Electrochemistry
• Chemical Kinetics
• Nuclear Chemistry
• Organic Chemistry
• Biochemistry

Goals

• To help students think like chemists, not just memorize facts.
• Develop problem-solving using chemical principles

Prerequisites

• Complete at least one introductory chemistry course
• Comfortable with basic algebra,

How to Succeed in the Course

• Read the textbook for a solid foundation of chemical principles
• Attend class, for the opportunity to ask questions
• Homework problems helps with applying concepts learned
• Seek help when needed from instructors or tutors

Matrices Defined

• A matrix is a rectangular array of real numbers
• Denoted by a capital letter, like $A$
• A matrix $A$ with $m$ rows and $n$ columns is an $m \times n$ matrix
• $a_{ij}$: element in the $i$-th row and $j$-th column

Types of Matrices

Square Matrix

• Number of rows equals number of columns, $m=n$

Row Matrix

• Single row, $m = 1$

Column Matrix

• Single column, $n = 1$

Null Matrix

• All elements are zero

Diagonal Matrix

• Square matrix with all non-diagonal elements zero

Identity Matrix

• Diagonal matrix with all diagonal elements equal to 1.
• Example: $I_3$

Upper Triangular Matrix

• Square matrix with all elements below the main diagonal zero

Lower Triangular Matrix

• Square matrix with all elements above the main diagonal zero

Matrix Operations

Addition and Subtraction

• Only matrices of the same format can be added or subtracted
• $A + B = [a_{ij} + b_{ij}]$
• $A - B = [a_{ij} - b_{ij}]$

Multiplication by a Scalar

• Each element of the matrix is multiplied by the scalar
• $kA = [ka_{ij}]$

Matrix Multiplication

• For $AB$ multiplication to be possible, the number of columns in $A$ must equal the number of rows in $B$
• If $A$ is $m \times n$ and $B$ is $n \times p$, then $AB$ is $m \times p$
• $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$

Transposition

• The transpose of matrix $A$, denoted $A^T$, is obtained by swapping rows and columns.

Inverse of a Matrix

• Inverse of a square matrix $A$, denoted $A^{-1}$, such that $AA^{-1} = A^{-1}A = I$
• A matrix with an inverse is invertible or regular, otherwise, it is singular

Vectors

Component Addition Method:

• Useful for vector manipulation
• Step 1: List knowns and unknowns
• Step 2: Sketch the vectors
• Step 3: Find x and y components
• Step 4: Add x components to find x resultant
• Step 5: Add y components to find y resultant
• Step 6: Find magnitude of resultant vector
• Step 7: Find direction of resultant vector
• Step 8: Report answer with units

Kinematics

Useful Equations:

• $\bar{v} = \frac{\Delta x}{\Delta t}$ : Average velocity formula
• $\bar{a} = \frac{\Delta v}{\Delta t}$: Average acceleration formula
• $v = v_0 + at$ : Final speed calculation with initial speed, acceleration, and time
• $x = x_0 + v_0t + \frac{1}{2}at^2$ : Position with initial position, initial speed, time, and acceleration
• $v^2 = v_0^2 + 2a\Delta x$ : Final speed formula with change in position and acceleration
• Note* $x = \text {position}, v = \text{speed}, a = \text {acceleration}, t=\text{time}$

Problem Solving:

• Step 1: Read the problem carefully
• Step 2: Draw a diagram
• Step 3: Choose a coordinate system
• Step 4: List knowns and unknowns
• Step 5: Choose an equation
• Step 6: Estimate the answer
• Step 7: Solve the equation
• Step 8: Check the answer
• Step 9: Report answer with units

Dynamics

Useful Equations:

• $\Sigma \overrightarrow{F} = m\overrightarrow{a}$: Sum of forces equals mass times acceleration
• $F_g = mg$: Force of gravity formula
• $F_{fr} = \mu F_N$: Force of friction euqation

Problem Solving:

• Step 1: Read the problem carefully
• Step 2: Draw a diagram
• Step 3: Choose a coordinate system
• Step 4: List knowns and unknowns
• Step 5: Draw a free body diagram
• Step 6: Apply Newton's 2nd Law
• Step 7: Estimate the answer
• Step 8: Solve the equation
• Step 9: Check the answer
• Step 10: Report answer with units

Chemical Kinetics

• Chemical kinetics studies the rates of chemical reactions

Reaction Rates

• Reaction rate is the change in concentration of reactants or products per unit time
• rate = $\frac{\Delta [A]}{\Delta t}$
• $[A]$ represents concentration of reactant or product
• $\Delta t$ is the change in time

Stoichiometry and Rate

• Rate must be quantified with a specified component
• Rate is related to the coefficients in the stochiometry

Rate Positivity

• The rate is always positive

Instantaneous Rate

• Rate at a particular moment is called the "instantaneous rate" denoted by;
• $rate = \lim_{\Delta t \to 0} \frac{\Delta [A]}{\Delta t} = \frac{d[A]}{dt}$
• Found by determining the slope of the curve

Rate Law

• Expresses the connection between the rate of a reaction and the rate constant along with the concentrations of the reactants raised to certain powers

Reaction Order

• Sum of the orders is the overall order of the reaction
• $rate = k[A]^x[B]^y$, overall order $= x + y$

Units of Rate Constants

• Units varies based on the overall reation order

Determining Rate Laws

• Method of initial rates determines rate law
• Initial rate is at the beginning of the reaction, done experimentally to determine an order with respect to that reactant

Integrated Rate Laws

First-Order Reactions

• Rate depends on one reactant to the first power
• $A \rightarrow products$, $rate = -\frac{d[A]}{dt} = k[A]$

Half-Life

• Time for the concentration of a reactant to decrease to half of its initial value

Second-Order Reactions

• Rate depends on on reactant to the second power, or two reactants to the first power
• $A \rightarrow products$, $rate = -\frac{d[A]}{dt} = k[A]^2$

Half-Life

• Depends on the initial concentration of the reactant in this kind of reaction

Zero-Order Reactions

• Rate is independent of the reactant concentration
• $A \rightarrow products$, $rate = -\frac{d[A]}{dt} = k[A]^0 = k$

Summary of Kinetic Equations

• Equations are separated based on rate law, integration, linearity, etc to deduce rate laws efficiently

Collision Theory

• Reacting species need to collide in order for a reaction to occur
• Not all collisions result in a reaction, must collide in the correcct orientation and with correct energy

Activation Energy

• Minimum energy required for a reacion to occur

Arrhenius Equation

• $k = A e^{-E_a/RT}$, $k$ is the rate constant, $A$ the frequency factor, $E_a$ is activation energy, $R$ is the gas constant (8.314 J/mol·K) and $T$ is the absolute temperature in Kelvin

Catalysts

• Lowers the activation energy of a reaction, but is not consumed
• Speeds up a reaction

Enzymes

• Biological catalysts that are very specific, using an active site
• Follow the Michaelis-Menten mechanism

Diagram Description

• A reaction catalyzed takes less amount of energy compared to a uncatalyzed raction

Statics

Definitions

• Scalars are are simply positive or negative numbers like mass, volume, length for example
• Quantities with magnitude, direction and sense are called Vectors

Vector Operations

• Multiplying by a positive scalar increases the magnitude of the vector, multiplying by a negative scalar reverses it
• Vector addition follows the parallelogram law such that $\overrightarrow{R} = \overrightarrow{A} + \overrightarrow{B}$

Vector Subtraction

• Subtracting vector $\overrightarrow{B}$ from vector $\overrightarrow{A}$ is defined as $\overrightarrow{R} = \overrightarrow{A} - \overrightarrow{B} = \overrightarrow{A} + (-\overrightarrow{B})$

Vector Addition of Forces

• Replacing forces with a resultant force, we can use $\overrightarrow{F_R} = \overrightarrow{F_1} + \overrightarrow{F_2}$

Analysis Procedure

• First find the components of the vectors to work with
• Add x and y components seperately
• Then compute magnitude with $F_R = \sqrt{F_{Rx}^2 + F_{Ry}^2}$
• Compute direction with $\theta = tan^{-1}(\frac{F_{Ry}}{F_{Rx}})$

Cartesian Vectors

• Using the axes (x,y,z) we can designate directions to develop vector algerbra
• Cartesian unit vectors, designated by the direction of their axes i, j, k
• dimensionless vectors of unit magnitude

Vector Cartesian Respresentation

$\overrightarrow{A} = A_x\overrightarrow{i} + A_y\overrightarrow{j} + A_z\overrightarrow{k}$

Magnitude of a Cartesian Vector

• From ther Pythagorean therom, we have can calculate the magnitude of cartesian vectors with $A = \sqrt{A_x^2 + A_y^2 + A_z^2} $

Direction of a Cartesian Vector

• The orientation of vector is defined by the coordinate direction angles α, β, and γ
• The cosines of $\alpha$, $\beta$, and $\gamma$ are known as the direction cosines of vector

Addition of Cartesian Vectors

• Results are conveniently found if the forces are resolved into Cartesian components.
• Exmaple: Forces $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$ can be added as $\overrightarrow{F_R} = \overrightarrow{F_1} + \overrightarrow{F_2} = (F_{1x} + F_{2x})\overrightarrow{i} + (F_{1y} + F_{2y})\overrightarrow{j} + (F_{1z} + F_{2z})\overrightarrow{k}$

Position Vectors

• A position vector $\overrightarrow{r}$ locates a point in space relative to another point
• Vector from point A to point B is: $\overrightarrow{r} = (x_B - x_A)\overrightarrow{i} + (y_B - y_A)\overrightarrow{j} + (z_B - z_A)\overrightarrow{k}$
• Written as length using: $r = \sqrt{x^2 + y^2 + z^2}$
• The direction of $\overrightarrow{r}$ is defined by the coordinate direction angles $\alpha$, $\beta$, and $\gamma$

Force Vector Directed Along a Line

• $\overrightarrow{F} =F\overrightarrow{u} = F(\frac{\overrightarrow{r}}{r})$

Dot Product

The dot product of vectors with the format:

$\overrightarrow{A} \cdot \overrightarrow{B} = A B cos\theta$

• Where $\theta$ is the angle between the tails of the two vectors.

Dot Product Laws

• Commutative law: $\overrightarrow{A} \cdot \overrightarrow{B} = \overrightarrow{B} \cdot \overrightarrow{A}$, we can switch the order of the vectors in question.
• Multiplication by a scalar: $a(\overrightarrow{A} \cdot \overrightarrow{B}) = (a\overrightarrow{A}) \cdot \overrightarrow{B} = \overrightarrow{A} \cdot (a\overrightarrow{B}) = (\overrightarrow{A} \cdot \overrightarrow{B})a$.
• Distributive law: $\overrightarrow{A} \cdot (\overrightarrow{B} + \overrightarrow{C}) = (\overrightarrow{A} \cdot \overrightarrow{B}) + (\overrightarrow{A} \cdot \overrightarrow{C})$.
• Note: the dot product distributes as expected.

High-Frequency Trading (HFT)

• Making money is the primary use-case, and motivation for computers to assist.

Computers assist with;

• Speed
• Volume
• Complexity

History of use of Computers in Trading

• Late 1990's: Specialists begin automating.
• 2001: Decimalization
• 2000's: Colocation and direct feeds: allows for fast volume processing with high levels of automated access.
• 2007: Flash Crash

How do HFT's Make Money?

• Capture bid-ask spread
• Capture short term directional changes

Market Making

• Goal Capture Bid-Ask Spread by doing the following
• Method: place Limit Orders on both sides

Directional Strategies

• Goal: Predict Short term pricr movements
• Methods statistical Arbitrage Event Arbitrage

Consequences

• Liquidity Provision
• Increased Volatility
• Fairness

Order Book

• Limit order specifies price and quantity, market order specifies quantity only
• Components of the order books consists of different orders depending on which side

Sides

• Bid Sides: where only buy orders appear in descending order
• Ask Sides: where only sell orders are displayed ascending

Order Book Dynamics

Orders that arrive can be immediately executed market orders or limit orders that can be then added to the book

Mathematical Modelling

• Kyle Model for, quantifying impacts of informed trading on prices

Preventative Maintenance Plan for an Elevator

• Goal: Guarantee User Safety, Prolong Equipment Life, Reduce Failure Risks
• Maintenance is important for optimizating the elevators performanc and to also meet regulatory requirements.

Activities Table

• Inspections: Weekly Visual General review
• Monthly Verification of Light/Indicators
• Trimester lubrication of guideways/cables and verification of safety breaks.
• Yearly overhauls and tests

Important Note

• Record all activities that have been performed, and all incidents that have been detected. If there are any faults, contact a specialized service.

Tension, Compression, and Shear

Stress

• Force acting per unit area, equation $\sigma = \frac{F}{A}$ where $\sigma$=stress in $Pa$or $psi$, and $F$=Force,and $A$ =Area

Normal Stress

• Stress component perpendicular to the surface, either tensile or compressive

Shear Stress

• Stress component parallel to surface, expressed mathematically as $\tau = \frac{V}{A}$

Strain

• Deformation of a material caused by stress, expressed change in length divded by orginal length $\epsilon = \frac{\Delta L}{L_0}$ where all measures are in same unit

Shear Strain

• Change in angle between two lines that were orginally perpendicular with format: $\gamma = \frac{\Delta x}{L}$

Hooke's Law

• Hooke's Law states that stress is proportional to strain for elastic materials, usually following the eqution: $\sigma = E\epsilon$

Material properties

• Stress is related to strain

Shear properties

• Modulus is ratio of shear stress to shear strain
• Using relation $\tau = G\gamma$

Axial Deformation

The axial deformation ($\delta$) of a member subjected to axial loading is: $\delta = \frac{PL}{AE}$

Thermal Stress

Thermal stress is usually stress caused by changes in temperature, normally by finding the stress using the relation $\sigma_T = E\alpha \Delta T$

Causality Intro

• Causality helps show relations where cause produces an effect
• Understanding these connections help with prediction and explanation

Association vs. Causation

• Association does not imply causation
• Association shows that evnets co-exsist, causation confirms one events produces the other

Notation

• $Y_i$: Outcome for individual i.
• $D_i$: Treatment for individual i (1 if treated, 0 if not).
• $Y_{1i}$: Potential outcome if individual i is treated.
• $Y_{0i}$: Potential outcome if individual i is not treated.

Causal Effect

• The causal effect of D on Y for individual i is: $\qquad \tau_i = Y_{1i} - Y_{0i}$