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Astronomy

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Celestial Coordinates
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Celestial Navigation
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Distance Units
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Location of North and South Celestial Poles

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve

Cosmology

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Penrose Diagrams
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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Friedmann-Robertson-Walker Equations
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes
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Olbers' Paradox
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Planck Units
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Stephen Hawking's Last Paper
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Stephen Hawking's PhD Thesis
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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Stress-Energy-Momentum Tensor
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Tensors
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The Area Metric
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Hamiltonian Formulation
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras
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Heron's Formula
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Index Notation (Tensors and Matrices)
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Related Rates
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Sets, Groups, Modules, Rings and Vector Spaces
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Similar Matrices and Diagonalization
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Symmetric Groups
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Tangent and Normal Line
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Taylor and Maclaurin Series
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The Essential Mathematics of Lie Groups
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The Integers Modulo n Under + and x
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Entangled States
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Heisenberg Uncertainty Principle
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Ladder Operators
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Multi Electron Wavefunctions
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Pauli Exclusion Principle
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Pauli Spin Matrices
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Photoelectric Effect
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Position and Momentum States
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Probability Current
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Schrodinger Equation for Hydrogen Atom
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Schrodinger Wave Equation
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Schrodinger Wave Equation (continued)
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Qubit
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids
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Fermi-Dirac Statistics
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1
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Spinors - Part 2
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The Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

test

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test

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: January 26, 2018

Zeroth Law of Thermodynamics ---------------------------- Most natural processes are not reversible. i.e. Thot -> Tcold, Tcold -/-> Thot First law of Thermodynamics --------------------------- ΔU = Q - W U = Internal energy Q = Heat added/subtracted to/from the system W = Work done on or by the system Second law of Thermodynamics ---------------------------- It is impossible to extract an amount of heat QH from a hot reservoir and use it all to do work W. Some amount of heat must be exhausted to a cold reservoir. Work Done by a Gas ------------------ Consider a piston with area A travelling a distance d: W = Fd = (F/A)Ad = PΔV = the area under the P versus V curve. = ∫PdV Now PV = nRT so P = nRT/V. Therefore, W = ∫PdV = nRT∫dV/V = nRTln(Vf/Vi) = (N/NA)RTln(Vf/Vi) = NkBTln(Vf/Vi) If the gas is expanding, ΔV is positive and constitutes work being done by the gas. Since ΔU = Q - W this means that internal energy of the gas must decrease. Conversely, if the gas is being compressed, the ΔV term is negative. This constitutes work being done on the gas. In this case ΔU = Q - (-W) = Q + W so the internal energy of the gas inreases. Isothermal, Adiabatic, Isobaric and Isochoric Processes ------------------------------------------------------ Isothermal: Isothermal processes take place at constant temperature. In order for this to happen the system is assumed to be connected to a heat reservoir that supplies/absorbs heat to maintain the temperature. Changes are assumed to take place slowly so that equilibrium with the reservoir is maintained at all times. We can summarize Isothermal processes by saying: ΔT = 0 and Q ≠ 0 Since ΔU is proportional to T, this implies that ΔU = 0. Thus, 0 = Q - W ∴ Q = W Thus, all the heat added to the system is used to do work. Adiabatic: Adiabatic processes take place at non-constant temperature. The system is insulated and not connected to a heat reservoir so there is no heat flow to keep the temperature constant. Changes are assumed to take place rapidly. We can summarize Adiabatic processes by saying: Q = 0 and ΔT ≠ 0 Correspondingly, ΔU = -W Isobaric: Isobaric processes take place at constant pressure. Isochoric (aka isovolumetric) processes: Isochronic processes take place at constant volume. Since there is no ΔV involved there is no work done. Heat Capacities: U is a funtion of T and V. dU = (∂U/∂T)V dT + (∂U/∂V)T dV For constant V, dV = 0. Therefore, dU = (∂U/∂T)V dT = CV dT where CV is the heat capacity at constant V Define Enthalpy, H = U + PV H is a funtion of T and P. dH = (∂H/∂T)P dT + (∂H/∂P)T dP For constant P, dp = 0. Therefore, dH = (∂H/∂T)P dT = CP dT where CP is the heat capacity at constant P Now, (∂H/∂T)P = (∂U/∂T)P + (∂PV/∂T)P (∂H/∂T)P = (∂U/∂T)P + (PdV/dT)P Now PV = nRT so PdV/dT = R if n = 1 CP = (∂U/∂T)P + R Now, using the Chain Rule, we can write: (∂U/∂T)P = (∂U/∂T)V + (∂U/∂T)P(∂V/∂T)P (∂U/∂T)P = (∂U/∂T)V since there is no change in U when T is constant. and so, CP = CV + R Boyle's Law for Adiabatic process: dU = dQ - PdV = dQ - (nRT/V)dV For adiabatic process dQ = 0, therefore: dU = -(nRT/V)dV = CVdT CVdT/T = -nRdV/V Integrate both sides and set n= 1 to get: CVln(Tend/Tbegin) = -Rln(Vbegin/Vend) ∴ CVln(Tend/Tbegin) = Rln(Vend/Vbegin) ∴ ln(Tend/Tbegin)CV = ln(Vend/Vbegin)R ∴ (Tend/Tbegin) = (Vend/Vbegin)R/CV Now, from before: CP = CV + R ∴ R/CV = (CP - CV)/CV = γ - 1 which leads to: (Tend/Tbegin) = (Vend/Vbegin)γ - 1 Using the ideal gas equation PV = nRT we can write: (PendVend/nR)/(PbeginVbegin/nR) = (Vend/Vbegin)γ - 1 (PendVend)/(PbeginVbegin) = (Vend/Vbegin)γ - 1 Pend/Pbegin = (Vend/Vbegin)γ which leads to: PbeginVbeginγ = constant = PendVendγ This is Boyle's Law where the temperature is not constant. Entropy: Consider an isothermal process such that Q = W From before we can write that: Q = W = NkBTln(Vf/Vi) = NkBln(Vf/Vi) = kBln(Vf/Vi)N If we assume that the volume of the gas is proportional to the number of states in the system, the RHS looks very similar to the definition of the change in entropy derived from statistical mechanics. Thus, we can write an alternative thermodynamics equation for the change in entropy as: ΔS = S - S' = Q/T S is a state function or variable. A state function is a property of a system that is not dependent on how the system arrived at its present condition (is not time dependent). Any change in a state function is only dependent on the initial and final conditions. P, V, and T are other examples of state functions whereas heat transfers, work transfers and rates of chemical reactions are not. Carnot Cycle ------------ A to B - The gas expands and does work on the surroundings. The temperature of the gas does not change during the process, and so U remains constant. The gas expansion is due to the absorption of heat energy, Qhot. The entropy change, ΔS = Qhot/Thot B to C - The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy, U. The loss of U causes it to cool to the "cold" temperature, Tcold. ΔS = 0. C to D - The surroundings do work on the gas, causing an amount of heat energy, Qcold, to flow out of the gas to the low temperature reservoir. Again, the temperature does not change so U remains constant at its new level. ΔS = -Qcold/Tcold D to A - During this step, the surroundings do work on the gas, increasing its internal energy and compressing it, causing the temperature to rise to Thot. ΔS = 0. The Carnot cycle is the most efficient heat engine cycle allowed by physical laws. When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used. In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy. Consider the adiabatic processes, Q = 0 ΔU = -PΔV ∴ ΔU + PΔV =0 (3/2)nRΔT + nRTΔV/V = 0 (3/2)dt/T + dV/V = 0 (3/2)∫dt/T + ∫dV/V = 0 Tcold C [(3/2)lnT] + [lnV] = 0 Thot B ln(Tcold/Thot)3/2 + ln(Vc/VB) = 0 (Tcold/Thot)3/2(VC/VB) = 1 Take the reciprocal to get: (Thot/Tcold)3/2(VB/VC) = 1 For D to A we get: (Thot/Tcold)3/2(VA/VD) = 1 Thus, we can say, VB/VA = VC/VD What about the entropy? Consider the isothermals (ΔU = 0 so Q = W) ΔS = Qhot/Thot + 0 + Qcol/Tcold + 0 = Whot/Thot + Wcol/Tcold = (nRThot/Thot)∫dV/V + (nRTcold/Tcold)∫dV/V = nR{∫dV/V + ∫dV/V} = nR{ln(VB/VA) + ln(VD/VC)} = nR{ln[(VB/VA)(VD/VC)]} = nR{ln[(VB/VA)/(VC/VD)]} = nRln(1) from before = 0 The efficiency, E is: W = Work done during a cycle = Qhot - Qcold (ΔU = 0) E = W/Qhot = (Qhot - Qcold)/Qhot = 1 - Qcold/Qhot The overall change in internal energy around the cycle is 0. Therefore, ΔU = 0 and Q = PdV We can write from before that: Qhot = nRThotln(VB/VA) and Qcold = -nRTcoldln(VD/VC) = nRTcoldln(VC/VD) ∴ E = 1 - nRTcoldln(VC/VD)/nRThotln(VB/VA) = 1 - Tcoldln(VC/VD)/Thotln(VB/VA) But from before we have shown that VB/VA = VC/VD. Thus, = 1 - Tcold/Thot If the efficiency is less than 100% we get: W/Qhot ≤ (1 - Tcold/Thot) ∴ W ≤ Qhot(1 - Tcold/Thot) (Qhot - Qcold) ≤ Qhot(1 - Tcold/Thot) Qcold ≥ QhotTcold/Thot Qcold/Tcold ≥ Qhot/Thot Scold ≥ Shot ... The 2nd Law of Thermodynamics This implies that the entropy at the end of the process is greater than that at the beginning. The more efficient the process, the less the increase in entropy will be. In other words, entropy provides a measure of the amount of thermal energy that cannot be used to do work. The change in entropy will only be 0 for a reversible process (100% efficiency). The connection between reversibility and efficiency can be thought of as follows: Consider an upright cylinder and piston in a vacuum. | W | |=====| | | | | ----- The gas in the cylinder is compressed by a weight, W. If W is suddenly removed, the molecules in the gas will not instantaneously rearrange themselves and there will be a period of non-equilibrium in P, V and T. If, on the other hand, W is removed slowly in very small increments, we can consider the system to be in constant equilibrium. Now, if there is friction between the piston and the cylinder, any movement of the piston will generate heat. Removing increments of W and adding increments of W to return to the original state both result in the generation of heat. Thus, returning both the system and its surroundings to their original state simultaneously cannot be achieved. In summary, a reversible process has to take place slowly and be frictionless. An implication of the fact that entropy can never decrease is that at the time of the 'Big Bang', the universe had low entropy and a high degree of order. Since then, however, the entropy has increased (and continues to increase) with time. It is worthwhile noting that systems that are not isolated may decrease in entropy, provided they increase the entropy of their environment by at least that same amount. An example is the freezing of water. The entropy of the water decreases as it freezes, but the heat extracted increases the entropy of the environment. The Heat Pump ------------- The heat engine cycle described can be run in the opposite direction to become the refrigeration cycle. The cycle remains exactly the same except that the directions of any heat and work flows are reversed. Heat is absorbed from the low temperature reservoir and dumped into the high temperature reservoir with work input required to accomplish all this. The P-V diagram is the same except that the directions of the processes are reversed. In the case of an a/c system, the cold reservoir is the inside of the building and the outdoors is the hot reservoir. In the case of a heat pump, the cold reservoir is the outdoors and the inside of the building is the cold reservoir. In both cases the work input is the electrical power. - The compressor compresses the refrigerant and raises its pressure and temperature. Latent heat is also released as the gas condenses to a liquid (the boiling point is raised). This corresponds to CB of the Carnot cycle. - While passing through the condenser, the heated refrigerant yields part of its heat to the hot surroundings with a lower temperature. This corresponds to BA of the Carnot cycle. - The expansion phase reduces the pressure of the fluid, and consequently its temperature. Latent heat is also absorbed as the liquid evaporates to a gas (the boling point is lowered). This corresponds to AD of the Carnot cycle. - When it passes through the evaporator, as its temperature is lower than the cold surroundings, the fluid picks up heat and the cycle repeats. This corresponds to DC of the Carnot cycle. Clausius Statement of the 2nd Law --------------------------------- | Thot | ----------- | Q | ----------- | Tcold | Q flowing from hot to cold: ΔS = -Q/Thot + Q/Tcold Since Thot > Tcold ΔS > 0 Q flowing from cold to hot: ΔS < 0 ... not allowed. Spontaneous Action ------------------ Consider a gas in a container with a partition. After the partition is removed we get: -------------- ------------ | gas | vac | | gas | | | | | | -------------- ------------ We always get the situation of left proceeding to right but never the other way around in which the gas collects in the left side of the container. Why is this? For the gas expanding we get: U = Q - PdV or Q = U + PdV Now, ΔS ≥ Q/T ∴ Q ≤ TΔS. Thus, U + PdV ≤ TΔS or U + PdV - TΔS ≤ 0 This formula is satisfied for the left to right process but can never be satisfied in the reverse direction. A Little More About Enthalpy ---------------------------- From before we had: H = U + PV ∴ ΔH = ΔU + Δ(PV) = Q - W + Δ(PV) = Q - PdV + Δ(PV) For constant P (isobaric) Δ(PV) = PΔV ∴ ΔH = Qp ... the heat added at constant P. Enthalpy is useful in chemistry because most reactions occur at constant P