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Mathematics for Chemists: Differential Equations Guide

Explore ordinary and partial differential equations in chemistry, from kinetics to thermodynamics. Learn to solve ODEs and PDEs with practical examples and classroom exercises. Understand chemical kinetics and the principles of superposition for linear equations.

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Mathematics for Chemists: Differential Equations Guide

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  1. 化學數學（一） The Mathematics for Chemists (I) (Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)Department of ChemistryNational Sun Yat-sen University

  2. Chapter 5 Differential Equations • Simple Ordinary Differential Equations (ODE) • Kinetics of Chemical Reactions • Partial Differential Equations (PDE) • Chemical Thermodynamics • Gamma Functions • Beta Functions • Hermite Functions • Legendre Functions • Laguerre Functions • Bessel Functions Contents Covered in Chapters 11-14

  3. Assignment • P.260: 38,40,42 • P.286: 24,27,28 • P.302: 7,9,12,15 • PP.323-324: 2, 8, 10

  4. Overview of Differential Equations (DE) DE: Equations that contains (partial) derivatives. • Ordinary DE (ODE): One variable First-order ODE, Second-order ODE, … Constant coefficient ODE, Variable coefficient ODE • Partial DE (PDE): Multi-variable

  5. Examples ODE First order: Second order: constant coefficients Second order: variable coefficients PDE

  6. Some First- and Second-order ODEs First order rate process (growth/decay) Second-order rate process Free falling of an object Classical harmonic oscillator One-dimensional Vibration of atomic bonds

  7. Solving A DE • Find the function(s) (of one or more variables) that satisfy the ODE/PDE. This step normally involves integration and/or series expansion. • Initial or boundary conditions are usually required to specify the solution. Therefore, both equations and initial/boundary conditions are equally important in solving a specific practical problem.

  8. I. First Order ODE • Examples: First order rate process (growth/decay) Second-order rate process Initial condition: y=10 when x=0

  9. Classroom Exercise Find the general and particular solutions of the following equation with the given initial condition:

  10. Solving First Order ODE Separable Equations: + initial conditions First-order linear equations:

  11. Example: Separable First-Order ODE

  12. Classroom Exercise:Separable First-Order ODE

  13. Reduction to Separable Form: Homogeneous Equations For n=0: Example:

  14. Example: Separation of a Homogeneous Equation Check:

  15. Chemical Kinetics

  16. Rate of Reaction

  17. Rate Constant and Order A  products first order 2A  products second order A + B  products second order

  18. ln[A] ln[A]0 -k 0 t A  products: first order process

  19. 1/[A] 2k 1/[A]0 0 t 2A  products: second order process

  20. A + B  products: second order process

  21. First-Order Linear Equations:The Homogeneous Case

  22. First-Order Linear Equations:The inhomogeneous Case

  23. Example: Linear Equation

  24. Classroom Exercise: Linear Equation

  25. Chemical Kinetics Example

  26. A C A C A B B B C

  27. R E L Example: Electric Circuit Three sources of electric potential drop ( drop of voltage): For constant electromotive force: E=E0 Initial condition, I(0)=0: Inductive time constant:

  28. II. Second-Order ODE: Constant Coefficients Inhomogeneous, linear, variable coefficients: Inhomogeneous, linear and constant coefficients: Homogeneous and linear, variable coefficients: Homogeneous, linear and constant coefficients:

  29. Principle of Superposition: Example Linearly independent (not related by a proportional coefficient) Particular solutions

  30. Principle of Superposition(for Homogeneous Linear DEs) The linear combination of two (particular) solutions of a homogeneous DE is also a solution of the DE.

  31. The general solution (constant coefficients) guess (characteristic equation or auxiliary equation)

  32. Example The two particular solutions being linearly independent, the general solution is

  33. Three Cases

  34. Example: Double root

  35. Example: Complex roots

  36. Classroom Exercise Find the general solution of the following ODE:

  37. Classroom Exercise Find the general solution of the following ODE:

  38. Particular Solutions Solutions with initial or boundary conditions.

  39. Boundary Conditions

  40. Example: The particle in a 1D box The microscopic entity cannot be outside of the well: Two distinct regions: well and wall Within the well, the particle is a free particle:

  41. Boundary Conditions To ensure

  42. Quantization of Energy Only some energies are allowed: Where there is constraint, there is quantization n: quantum numbers

  43. Normalization for

  44. First five normalized wavefunctions Standing wave Where there is constraint, there is quantization

  45. Orthogonality

  46. Example: The particle in a ring Choosing c2=0 (because n can take both positive and negative values) and normalizing the wavefunction:

  47. Probability distribution

  48. Orthogonality

  49. Quiz • Solve the following ODEs:

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