Atomic Radius and Ionization

Atomic Radius

• Atomic radius is defined as one-half the distance between the nuclei of identical bonded atoms.

Trends in Atomic Radii

Group Trend

• Atomic size increases from top to bottom within a group in the periodic table.
• This increase is due to the addition of energy levels as you move down a group.

Period Trend

• Atomic size decreases from left to right across a period in the periodic table.
• This decrease results from increasing nuclear charge and an increased attraction to the nucleus.

Radius of Atoms

Metallic Radius

• Metallic radius is defined as half the distance between nuclei in a solid metal.

Covalent Atomic Radius

• Covalent atomic radius is defined as half the distance between nuclei in a molecule.

Ions

• An ion is an atom or group of atoms with a positive or negative charge.
• Ionization is any process that leads to the formation of an ion.

Ion Sizes

Cation

• A cation is a positive ion formed through the loss of one or more electrons.
• Cation formation leads to a decrease in radius compared to the neutral atom.

Anion

• An anion is a negative ion formed through the gain of one or more electrons.
• Anion formation causes an increase in radius compared to the neutral atom.

Linear Algebra Summaries

Vectors

Definition

• A vector is a matrix of n × 1.
• $$\mathbf{v} = \begin{bmatrix} v_1 \ v_2 \ \vdots \ v_n \end{bmatrix}$$
• ℝ^n denotes the set of all vectors with n real-valued entries.

Vector Sum

• $$\begin{bmatrix} u_1 \ u_2 \end{bmatrix} + \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = \begin{bmatrix} u_1 + v_1 \ u_2 + v_2 \end{bmatrix}$$

Scalar Multiplication

• $$c \begin{bmatrix} v_1 \ v_2 \end{bmatrix} = \begin{bmatrix} cv_1 \ cv_2 \end{bmatrix}$$

Linear Combination

• Given vectors v₁, v₂, ..., v^n in ℝ^n and scalars c₁, c₂, ..., c^n, then $$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_n\mathbf{v}_n$$ is known as a linear combination of v₁, v₂, ..., v^n with weights c₁, c₂, ..., c^n.

Dot Product (Scalar Product)

• $$\mathbf{u} = \begin{bmatrix} u_1 \ u_2 \end{bmatrix}, \mathbf{v} = \begin{bmatrix} v_1 \ v_2 \end{bmatrix}$$
• $$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2$$

Length (Norm) of a Vector

• $$| \mathbf{v} | = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2} = \sqrt{\mathbf{v} \cdot \mathbf{v}}$$

Distance between u and v

• $$\text{dist}(\mathbf{u}, \mathbf{v}) = | \mathbf{u} - \mathbf{v} |$$

Unit Vector

• $$\mathbf{u} = \frac{1}{| \mathbf{v} |} \mathbf{v}$$

Zero Vector

• $$\mathbf{0} = \begin{bmatrix} 0 \ 0 \end{bmatrix}$$

Orthogonal Vectors

• u and v are orthogonal if u ⋅ v = 0.

Angle between Two Vectors

• $$\mathbf{u} \cdot \mathbf{v} = | \mathbf{u} | | \mathbf{v} | \cos \theta$$
• $$\cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{| \mathbf{u} | | \mathbf{v} |}$$

Projection of a Vector onto Another

• The projection of y onto u is represented as: $$\text{proj}_{\mathbf{u}} \mathbf{y} = \frac{\mathbf{y} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}} \mathbf{u}$$

Matrices

Definition

• A matrix is a rectangular array of numbers.
• $$\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}$$

Matrix Sum

• $$\mathbf{A} + \mathbf{B} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}$$

Scalar Multiplication

$$\mathbf{A} + \mathbf{B} = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}$$

Matrix Multiplication

• If A is an m × n matrix and B is an n × p matrix, then the product AB is an m × p matrix.
• (\mathbf{AB}){ij} = a{i1}b*{1j} + a*{i2}b*{2j} + \dots + a*{in}b*{nj}

Transpose of a Matrix

• The transpose of a matrix A is denoted by Aᵀ and it is obtained by interchanging the rows and columns of A.
• $$\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix} \implies \mathbf{A}^T = \begin{bmatrix} a_{11} & a_{21} \ a_{12} & a_{22} \end{bmatrix}$$

Identity Matrix

• The identity matrix I is a square matrix with 1s on the main diagonal and 0s in the rest of the entries.
• $$\mathbf{I} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$$

Inverse of a Matrix

• The inverse of a matrix A is denoted by A⁻¹ and it satisfies A A⁻¹ = A⁻¹ A = I.
• $$\mathbf{A} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \implies \mathbf{A}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}$$

Determinant of a Matrix

• The determinant of a matrix A is denoted by det(A) or |A|.
• $$\mathbf{A} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \implies \det(\mathbf{A}) = ad - bc$$

Rank of a Matrix

• The rank of a matrix A is the number of linearly independent columns of A.

Null Space of a Matrix

• The null space of a matrix A is the set of all vectors x such that A * x = 0.

Linear Transformations

Definition

• A linear transformation is a function T: ℝ^n → ℝ^m that satisfies:
• T(u + v) = T(u) + T(v)
• T(cu) = cT(u)

Transformation Matrix

• Every linear transformation T: ℝ^n → ℝ^m can be represented by a matrix A of m × n such that T(x) = A * x.

Chemical Kinetics

• Chemical kinetics, also known as reaction kinetics, studies reaction rates and chemical processes.
• Investigations include the influence of experimental conditions on reaction speed, information about reaction mechanisms and transition states, and developing mathematical models for reaction characteristics.

Factors affecting reaction rates

Reactant concentration

• Reaction rate is often proportional to reactant concentration; increasing reactant concentration usually increases reaction rate.

Temperature

• Temperature greatly affects reaction rate; higher temperatures increase thermal energy in molecules, almost always increasing reaction rate.

Catalysts

• Catalysts speed up reactions by lowering activation energy or changing the reaction mechanism.
• Catalysts are not consumed during the reaction and do not appear in the overall chemical equation.

Pressure

• Increased pressure in gaseous reactions increases collisions between reactants, thereby increasing the reaction rate.

Rate Law

• Rate law, or rate equation, is a mathematical expression relating reaction rate to the molar concentrations of reactants raised to certain powers.
• For the reaction: aA + bB → cC + dD; Rate ∝ [A]^x [B]^y; Rate = k[A]^x [B]^y
  • [A] and [B] represent concentrations of reactants A and B
  • x and y are reaction orders with respect to A and B
  • k is the rate constant (specific reaction rate constant)

Order of a reaction

• The order of a reaction is the sum of the powers to which concentration terms are raised in the rate law (Order = x + y).

Molecularity

• Molecularity is the number of reacting species (atoms, ions, or molecules) that collide simultaneously in an elementary reaction to bring about a chemical reaction.

Difference between order and molecularity of a reaction

Collision Theory of Chemical Reactions

• Collision theory explains reaction occurrence and rate differences based on molecular collisions.

Activation energy

• Activation energy is the minimum energy required for a reaction to occur and to break bonds.

Orientation factor

• Orientation factor is the fraction of collisions with correct orientation for reaction, also known as the steric factor.

Executive Summary

What is the New Deep Learning Economy?

• The Deep Learning Economy (DLE) views intelligence as the ability to rapidly and efficiently acquire complex skills.
• The DLE focuses on the importance of working with data and computational infrastructure.
• Key Principles:
  • Talent is widely distributed, but opportunity is not.
  • AI enables mass adaptation and radical personalization.
  • Deliberate practice driven by rapid feedback is essential for complex skill acquisition.
  • Deep learning requires an expanding ecosystem of educational institutions, companies, and other organizations.

3 Problems That the DLE Is Solving

• Skills gaps
• Inequality
• Productivity

4 Opportunities That the DLE Is Opening

• New labor markets
• Increased productivity
• Innovation
• Economic growth

5 Predictions About the Future of Work

• Deep learning will become more important as AI continues to develop.
• The future of work will be more fluid and flexible.
• Demand for workers with cognitive skills will increase.
• The importance of soft skills will increase.
• The emphasis on lifelong learning will grow.

6 Steps You Can Take to Prepare for the Deep Learning Economy

• Focus on developing complex skills.
• Learn how to learn.
• Build a strong network.
• Be adaptable.
• Embrace lifelong learning.
• Develop your soft skills.

Bayes' Theorem

Definition

• Bayes' Theorem describes the probability of an event based on prior knowledge of conditions related to the event.
• Expressed mathematically: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
  • $P(A|B)$: Conditional probability of A given B is true.
  • $P(B|A)$: Conditional probability of B given A is true.
  • $P(A)$ and $P(B)$: Probabilities of A and B occurring independently (marginal probability).

Deduction

• Can be deduced from conditional probability definitions:
  • $P(A|B) = \frac{P(A \cap B)}{P(B)}$
  • $P(B|A) = \frac{P(B \cap A)}{P(A)}$
• Since $P(A \cap B) = P(B \cap A)$, rewrite equations as: $P(A|B)P(B) = P(B|A)P(A)$
• Divide by $P(B)$ to get Bayes' Theorem: $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Example

• Tech company with two chip factories:
  • Factory A: 60% of chips, 4% defect rate
  • Factory B: 40% of chips, 6% defect rate
• If a defective chip is found, find the probability it came from Factory A.
• Let:
  • $A$: Chip produced in Factory A
  • $B$: Chip is defective
• Known:
  • $P(A) = 0.60$
  • $P(B|A) = 0.04$
  • $P(\neg A) = 0.40$
  • $P(B|\neg A) = 0.06$
• Want to find $P(A|B)$.
• Calculate $P(B)$ using total probability law:
  • $P(B) = P(B|A)P(A) + P(B|\neg A)P(\neg A)$
  • $P(B) = (0.04 \times 0.60) + (0.06 \times 0.40) = 0.048$
• Apply Bayes' Theorem:
  • $P(A|B) = \frac{P(B|A)P(A)}{P(B)} = \frac{0.04 \times 0.60}{0.048} = 0.5$
• Probability that a defective chip was produced in Factory A is 50%.