SymMath Applications
ABC Chemical Kinetics JCE
Jorge A. Carrazana García, Universidad de Santiago de Compostela
In this collection of worksheets, a step-by-step study of the “ABC” system is accomplished with the aid of Mathcad. Working with real mathematics notation, interactive graphs, and symbolic processing allows useful equations to be obtained, transformed and applied live, with the result of clarifying the study goals and objectives.
A Brief Introduction to the Gaussian Distribution, Sample Statistics, and the Student's t Statistic JCE
Scott Van Bramer, Widener University
This document provides students with an brief introduction to the gaussian distribution, sample statistics, and the student's t statistic. It was designed for quantitative analysis, instrumental analysis, and physical chemistry courses.
A Chemical Kinetics Application of Mathcad
Alvin F. Bopp, Southern University at New Orleans
A Mathcad simulation for a three step chemical reaction. Students are to choose the reaction conditions that will maximize profit over an eight hour period of operation of the chemical plant.
Analysis of the Vibrational Spectrum of a Linear Molecule JCE
Richard W. Schwenz, University of Northern Colorado
William F. Polik, Hope College
Sidney H. Young, University of South Alabama
This document provides a moderately interactive tutorial describing the analysis of vibrational frequencies obtained from the infrared spectrum of hydrogen chloride.
An Introduction to Mathcad
Theresa Julia Zielinski, Monmouth University
Arthur Ferguson, Worcester State College
Sidney H. Young, University of South Alabama
This document provides a tutorial for Mathcad vs 8, 2001i, and 11.
An Introduction to NMR Concepts JCE
Scott Van Bramer, Widener University
Mathcad introduction to NMR concepts. Acquisition parameters of the instrument, the basis of quadrature detection, apodization, and zero filling are discussed.
An Introduction to Statistical Mechanics JCE
Michelle M. Francl, Bryn Mawr College
An introduction to the partition function for vibrational and rotational energy by a review of Sterling's approximation and an examination of the harmonic oscillator partition function. The document can be used with experimental data or a supplied spectrum of HCl/DCl or some other diatomic gas to examine the effect of temperature on the partition function and the changes in the partition function for different molecules.
An Introduction to the Fourier Transform JCE
Scott Van Bramer, Widener University
This document simultaneously introduces students to how the Fourier transform works and how various instrument parameters affect the results of the Fourier transform. Waveforms are produced and integrated using sine waves, cosine waves, and simple functions. The fast Fourier transform is used with simulated data to introduce students to the relationship between the signal waveform and the signal frequency of a spectrum. Students are then led to the concepts of dwell time and resolution through a series of interactive exercises. The document ends with an introduction to the Fourier transform of decaying signals. Students and teachers may use IntroFourierTransform.mcd as written, or teachers may remove some graphs to increase discovery learning and class discussion.
Approximate Molecular Orbital Calculations for H2+
George M. Shalhoub, La Salle University
This document presents a solution to the H2+ ion Schrodinger equation using the Variational Method with all intermediate steps shown and many plots of functions. Many exercises focus on student learning. The document concludes with the plots of the molecular orbitals and the probability density functions.
A Summary of Statistical Thermodynamic Calculations JCE
Theresa Julia Zielinski, Monmouth University
Sidney H. Young, University of South Alabama
In this document students can explore the full set of statistical thermodynamic calculations leading to the prediction of the heat capacity at constant volume from the translational, rotational, vibrational, and electronic partition functions. An extension of the calculation of the thermodynamic properties of a molecule is made to predict the equilibrium constant of the dissociation of N2. The document concludes with the study of the NO molecule which has a low lying electronic energy level.
Blackbody Radiation
George M. Shalhoub, La Salle University
Provides students with a hands on exploration of the BlackBody radiation concept and the functions used to describe that radiation.
CP, ΔCP , ΔrH, S, ΔrS, and ΔrG as a function of T: N2, H2 and NH3
Theresa Julia Zielinski, Monmouth University
An introduction to thermodynamic calculations leading to a calculaton of the equilibrium constant for a gas phase reaction.
Circular Birefringence and Circular Dichroism Simulation JCE
Zachary Brown, University of Wisconsin-Green Bay
Ronald Starkey, University of Wisconsin-Green Bay
Fundamental principles of circular dichroism and circular birefringence are examined, including interactive graphs which permit the user to alter the phase difference or absorption between the circular components to simulate both phenomena. Advanced graphing and matrix manipulation are also used to show both the 2D and 3D perspectives, and how to create animations which can be used by various media players. This document can be used by an undergraduate audience in either an instrumental or physical chemistry, or the animations could be used as a visual aid to accompany a lecture.
Computing a Flame Temperature
Theresa Julia Zielinski, Monmouth University
Joseph H. Noggle (1936 - 1998), Monmouth University
Mathcad document that allows students to learn how to compute adiabatic flame tempertures.
Computing a Liquid-Vapor Phase Diagram
Sidney H. Young, University of South Alabama
A Mathcad worksheet focused on determining vapor-liquid phase diagrams for ideal and non-ideal binary mixtures for learning about activity and activity coeeficients.
Computing Enthalpies of Reaction
Theresa Julia Zielinski, Monmouth University
A symbolic mathematics document to provide hands on practice for computing heats of reaction and heat of reaction as a function of temperature given a constant value for Cp.The thermodynamic data required for a large variety of reactions is provided in the file ThDATA.xls.
Computing Liquid-Vapor Phase Diagrams for Non-ideal Binary Mixtures JCE
Franklin M.C. Chen, University of Wisconsin-Green Bay
A Mathcad worksheet focused on determining vapor-liquid phase diagrams for ideal and non-ideal binary mixtures for learning about activity and activity coeeficients. Van Laar theory and the Leavenberg-Marquardt algorithm are used.
Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom JCE
Mark David Ellison, Wittenberg University
Thomas S Kuntzelman, Spring Arbor University
John Tippin, Spring Arbor University
A Mathcad template that provides guided inquiry practice with angular functions used as parts of the solutions to the Hydrogen atom Schrodinger equation.
Construction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems JCE
Thomas S Kuntzelman, Spring Arbor University
A Mathcad template that provides guided inquiry practice with radial functions used as parts of the solutions to the Hydrogen atom Schrodinger equation.
Conventional Entropies and the Third Law of Thermodynamics JCE
Theresa Julia Zielinski, Monmouth University
Todd M. Hamilton, Georgetown College
After fitting a polynomial to heat capacity as a function of temperature data, students calculate the standard molar third law entropy for a solid, a liquid, and a gas.
Data Analysis (Damped Oscillations) Using the Genfit Function JCE
R. D. Poshusta, Washington State University
Shows how the Mathcad Genfit function does a general least squares fit of a model function to experimental data. Sampole data is provided. The template is easy for students to use.
Diffusion: A Brief Overview
Theresa Julia Zielinski, Monmouth University
An introduction to diffusion concepts using Fick's Laws.
Effect of Activity Coefficients on Excess Functions and Phase Equilbria JCE
Baudilio Coto, ESCET, Rey Juan Carlos University
Immaculada Suárez, ESCET, Rey Juan Carlos University
In this document students use Maple to explore the relationship between activity coefficients, excess functions, and phase equilibria. start from the activity coefficient as a known property. In this study students examine how some functions change due to activities resulting in excess functions that differ from zero and vapor-liquid diagrams show complex behavior. The model avoids complex iterative procedures. In the end the vapor-liquid equilibrium equations are solved to compute both the isobaric and isothermal binary diagrams. The results are plotted and avialable for student interpretation. Additional exercises are presents for student practice.
Energies and Wavefunctions for Several One-Dimensional Potentials JCE
Ricardo Metz, University of Massachusetts
This Mathcad module allows calculation of energies and wavefunctions for several one-dimensional potentials: Particle in a finite box, Particle in a box with a barrier, Harmonic oscillator, Morse oscillator, and a Double-minimum potential, comparison of the wavefunctions and their energies, and how barriers affect wavefunction tunneling by looking at two double minima potentials.Aa particle in a box basis set and the Variational method are used.
Enzyme Activity as a Function of pH
Paul Krause, University of Central Arkansas
A matched pair of documents providing an introduction to the role of pH in the regulation of enzyme activity.The tutorial document contains all the equations and graphs for students to use to study the role of pH in enzyme kinetics. In the EnzymeExercise twin document all quations are omitted so that students can develop these interactively. The Exercise document is also ideal for display during lecture where the ideas can be developed interactively with the class as a whole.
Evaluating Volumetric and Other Thermodynamic Properties by Means of Cubic Equations of State JCE
Baudilio Coto, ESCET, Rey Juan Carlos University
Immaculada Suárez, ESCET, Rey Juan Carlos University
In this document students use Maple to determine volumetric properties and compare values obtained from the ideal gas and the Soave (SRK) equations of state. Several aspects of the calculation as the effect of pressure, the lowest available volume, the number of roots, and the relation between volumen and compresibility factor are analyzed.
Examples of common quantum mechanical procedures and calculations carried out in Mathcad
Erica Harvey, Fairmont State University
A Mathcad template that provides students with instruction for using Mathcad to perform several basic quantum mechanical calculatons.
Exploring Digital Signals and Noise in Instrumental Analysis JCE
Augustus W. Fountain III, United States Military Academy
Mathcad document to allow the student to gain a familiarity with the concepts of signal-to-noise ratios and to explore the advantages of ensemble averaging and digital filtering analytical signals.
Exploring Exotic Kinetics: An Introduction to the Use of Numerical Methods in Chemical Kinetics JCE
Michelle M. Francl, Bryn Mawr College
An introduction to the use of numerical methods to solve systems of partial differential equations for chemical kinetics. The Runge-Kutta algorithm is used to explore two simple mechanisms as well as an auto-catalyzed system (Lotka-Volterra).An exercise for mastery based on the Gray-Scott mechanism for glycolysis is included.
Exploring Fourier Transform Techniques with Mathcad JCE
Mark Iannone, Millersville University
These four Mathcad documents illustrate different aspects of the Fourier transform. They explore the source of the Nyquist frequency and the dependence of the resolution of the spectrum on sampling. The entire set of documents contains many student exercises and hands-on activities for learning the concepts presented.
Exploring Light Amplification by Stimulated Emission in Lasers JCE
Michael A. Waxman, University of Wisconsin-Superior
This document allows the students to use a simple approach to study the time evolution of the power output of a laser and its dependence upon the amplification coefficient, the length of the active medium, and the reflectivity of the cavity mirrors.
Exploring Orthonormal Functions JCE
Theresa Julia Zielinski, Monmouth University
A Mathcad template that provides students with quided inquiry practice with orthonormal functions.
Exploring the Harmonic Oscillator Wave Function Components JCE
Theresa Julia Zielinski, Monmouth University
A Mathcad template that provides students with guided inquiry practice with harmonic oscillator wave functions.
Exploring the Morse Potential JCE
Theresa Julia Zielinski, Monmouth University
The goal of the 'Exploring the Morse Potential' document is to provide hands on practice with the Morse potential energy function and the units used for calculating this potential. In the document is a step-by-step description of how to create the Morse potential function for HCl.
Exploring Thermodynamics Using Non-traditional Systems: Elastomers and DNA JCE
Jeffrey A. Draves, Monmouth College
In this worksheet thermodynamic analysis is applied to elastomers, in particular DNA stretching to show the application of the first and second laws of thermodynamics to solids of interest to biochemists and biophysicists.
Exploring the Uncertainty Principle JCE
Franklin M.C. Chen, University of Wisconsin-Green Bay
Students uses two Gaussian functions to explore the uncertainty principle. Students build a linear combination of orthogonal particle-in-a-box eigenfunctions to represent the Gaussian functions. They discover that a Gaussian function with smaller standard deviation requires more eigenfunctions to build than that with a larger standard deviation. Students are guided to discover the relationship between position and momentum uncertainty to a given wavefunction.
Femtochemistry JCE
Mark David Ellison, Wittenberg University
Students explore a simple solution to the time-dependent Schrödinger equation in the context of understanding femtochemistryl. After a review of the time-independent harmonic oscillator model studentsstudy the process of exciting molecules with an ultrafast laser pulse and in a superposition that has time-dependent behavior.
Finding Molecular Vibrational Frequencies from HCl to SO2 JCE
Franklin M.C. Chen, University of Wisconsin-Green Bay
Theresa Julia Zielinski, Monmouth University
Students learn about finding molecular vibrational frequencies through normal mode analysis and numerical solutions for simple molecule such as HCl and more complicate molecules such as CO2 and SO2. Topics such as displacement coordinates, internal coordinates, symmetry coordinates, and their relationship are developed.
Fitting a Polynomial to CP vs. T for Ag
Theresa Julia Zielinski, Monmouth University
This document demonstrates the method of fitting a polynomial of any reasonable power to a set of data and provides an example of how to use matrix methods to solve simultaneous equations in physical chemistry.
Following the Extent of a Chemical Reaction with the Help of Maple JCE
Baudilio Coto, ESCET, Rey Juan Carlos University
Immaculada Suárez, ESCET, Rey Juan Carlos University
In this document students use Maple to explore the relationship between the extent of a reaction and the change in Gibbs free energy for that reaction. Extensions include examination of a reaction at different pressures and different temperaturescan The formation of ammonia is the chemical system used for this document..
Fourier Transforms of Molecular Vibrations JCE
W. Tandy Grubbs, Stetson University
This series of three Mathcad instructional documents permits students to explore the relationship between the time and frequency representations of vibrational motion in simple diatomic molecules. Exercises included throughout and at the end of each document allow students to practice the numerical methods and further investigate how bond anharmonicity and vibrational damping affect the frequency spectrum.
Gaussian Distributions
Kevin Lehmann, Princeton University
The author generates a set of data from a Gaussian distribution to illustrate the properties of the distribution such as the mean of a set of data, the confidence level for the mean, the chi-squared function, and the Student t Distribution The document concludes with a discussion of the mean absolute deviation and a survey of the Moments of a distribution and how the mean absolute deviation and the moments are used to characterize the shape of the distribution. Variance, skew, and kurtosis are the three moments discussed in this document.
Graphing Orbitals in Three Dimensions with Rotatable Density Plots JCE
Louis Kijewski, Monmouth University
This Mathcad document uses symmetry to simplify the evaluation of an eigenvalue problem by reducing a matrix to sparse form. The sparse matrix is then transformed to block diagonal form. Group theory is used to set up the calculations without going through the proofs of the methods. The ability to use symmetry principles to get the vibrational frequencies and other information about molecules with symmetry is an important asset to any chemist.
H2+ MO Bonding and AntiBonding Orbitals
Robert Zschocher, University of North Carolina Charlotte
Fowler S. Bush, University of North Carolina Charlotte
Jordan C. Poler, University of North Carolina Charlotte
To introduce students toMO Theory by use of theone-electron hydrogen molecule ion, H2+. The essential features of bonding of this molecule can be used as a guide to the structures of more complex molecules. Studentsfamiliarity with functions related to the linear combination of atomic orbitals andmolecular orbitals and studyhow variation in the distance between the hydrogen atoms affects the shape of the contour diagram, and the consequent bonding energy.
Heat, Work and Entropy: A Molecular Level Illustration JCE
Jeffrey A. Draves, Monmouth College
This worksheet is intended to help students understand the difference between heat and work at the molecular level and to appreciate the proper use of the term (dis)order when applied to entropy. The worksheet makes use of quantum mechanics, statistical mechanics and classical thermodynamics to illustrate these differences.
Hückel Theory I and II JCE
James M. LoBue, Georgia Southern University
These documents provide an introduction to the Heisenberg formulation of quantum mechanics.Huckel I provides an introduction to generating Huckel Theory results along with an introduction to HOMO-LUMO transition energies and the prediction of nucleophilic and electrophilic substitution. Huckel II treatsthe generation of partial charges, bond orders, and the determination of aromatic character. Since these topics have ab-initio analogs, Huckel I and II provide a "hands-on" introduction to important aspects of computational chemistry as well.
Ideal Gas Law - Intro to Mathcad
George L. Hardgrove, St. Olaf College
This Mathcad document introduces students to Mathcad and gives them practice with the software by using the Ideal Gas Law.
Illustrating the Bohr Correspondence Principle JCE
Glenn V. Lo, Nicholls State University
This document introduces the Bohr Correspondence Principle. In an incomplete student document students relate nodal features of the wavefunction to the quantum number (n), interpret the square of the wavefunction as probability density, and learn that QM can be interpreted to agree with CM in a case that is adequately described by Kinetic Molecular Theory. A completed version of the document is available for teachers.
Intermolecular Interactions: Dipole-Dipole, Dipole-Induced Dipole, and London Dispersion Forces JCE
Thomas S Kuntzelman, Spring Arbor University
Masayuki Shibata, University of Medicine and Dentistry of New Jersey
Mathcad tool that permits students to compare various intermolecular interactions.
Intermolecular Potentials and The Second Virial Coefficient JCE
Patrick L. Holt, Bellarmine University
A worksheet for exploring the relationship between intermolecular potentials and the second virial coefficient using the hard sphere, square-well, and Lennard-Jones potentials. Students can compute the second virial coefficient and explore the coefficient?s temperature dependence for several substances and use a computed B value to investigate the temperature dependence of the compression factor, Z.
Introduction to Franck-Condon Factors JCE
Theresa Julia Zielinski, Monmouth University
George M. Shalhoub, La Salle University
The background document contains a very brief introduction to Franck-Condon factors through a sequence of guided inquiry type exercises. Students use potential energy diagrams for a diatomic molecule to examine a transition from a ground electronic state to an excited electronic state including consideration of the vibrational levels of each state. The overlap of vibrational wave functions introduces Franck-Condon factors. All of the exercises in this document are done with pencil and paper as preparation for more detailed work to be done in the companion computational document The Franck-Condon Factors.
Introduction to Matrices: A Tutorial for Physical Chemists
Melissa S. Reeves, Tuskegee University
This Mathcad document provide a series of exercises that will permit students to learn fundamentals concepts about matrices, some preliminary practice with matrix algebra and the application of matrix operations to symmetry operations.
Introduction to the Basic Terms of Band Structures JCE
Gion Calzaferri, University of Berne
Stephan Glaus, University of Berne
Marc Meyer, University of Berne
In this Mathcad document you will find an explanation of the concepts and applications of band structures, translational symmetry, crystal orbitals, Bloch functions, wave vectors, the Peierls distortion, density of states (DOS), crystal orbital overlap population (COOP), and Brillouin zones.
Introduction to the Propagation of Error
Peter J. Hansen, Northwestern College
This document presents a systematic development of the basic principles of propagation of error using the area of a rectangle and the density of a substance.
Introductory Explorations of the Fourier Series JCE
Theresa Julia Zielinski, Monmouth University
A Mathcad template that provides students with quided inquiry practice using a Fourier series expansion.
Kinetics of Complex Reactions
Flick Coleman, Wellesley College
In this Mathcad document students can explore series irreversible first order chemical reactions using both the Runge-Katta and integrated equations methods. The level of detail for the Runge-Katta method makes this a useful introduction to this technique for solving differential equations.
Kinetics of Complex Reactions: Steady-State and Equilibrium Approximations
Flick Coleman, Wellesley College
Here studentsexplore the steady-state and equilibrium approximations for a two step series reaction where the first step is an equilibrium using the Runge-Katta method to solve the differential equations for this reaction type.
Learning Molecular Geometry & Symmetry Through Quantum Computations and Mathcad™ Exercises JCE
Franklin M.C. Chen, University of Wisconsin-Green Bay
A Mathcad template to help students learn about molecular geometry and symmetry calculations based on the coordinate datagenerated from quantum computation software such as HyperChem. The molecular coordinate data serve as excellent real-world materials for students to appreciate the applications of linear algebra and to refresh their algebra learning experiences about linear transformations learn about fundamental properties of quantum and symmetry operators.
Linear Least Squares Regression JCE
Sidney H. Young, University of South Alabama
Andrzej Wierzbicki, University of South Alabama
Linear least-squares regression is the workhorse mathematical tool of the physical chemistry laboratory. This Mathcad worksheet and its accompanying data files demonstrate various implicit and explicit methods for determination of slope and intercept of a regressed line.
Maple-Assisted Template for Automatic Calculation of Second Order AA'BB' NMR Spectra JCE
Mihai Scarlete, Bishops University
This paper presents a Maple template for the interpretation and simulation of second order AA'BB' NMR-spectra, starting from the core information offered in a regular, undergraduate physical chemistry course.
Maxwell Boltzman Distribution
Flick Coleman, Wellesley College
This Mathcad document provides an excellent graphical presentation of the Maxwell Boltzman distribution, integration for different limits, and differentiation on a variable to get the most probable velocity.
Maxwell distribution of Gas Molecule Velocities
George L. Hardgrove, St. Olaf College
This Mathcad document provides students with instruction and practice with the Maxwell distribution of gas velocities.
Mean Versus Median
Kevin Lehmann, Princeton University
This worksheet provides a comparison of the mean and median values for both theoretical distributions and for data sets sampled from Gaussian and Lorentzian distribution functions. The document shows that the mean value provides a moderately better estimate of the central value than the median for the case of a Gaussian. However, in the case of a Lorentzian, due to its slow fall off for large displacements from the central value, the mean is almost useless as a statistic, while the median functions quite well. The document also introduces the idea of finding the optimal estimate by using the method of maximum likelihood. This document requires Mathcad 6.0+ including upgrade through patch 'e' .
Modeling Chemical Bonds with a Mathematical Function
Theresa Julia Zielinski, Monmouth University
Mathcad documents providing a framework for exploration of basic harmonic oscillator concepts used as a model for the chemical bond. The treatment is classical Hooke's Law as used in Molecular Mechanics.
Modeling pH in Natural Waters JCE
Morten Sielemann
Niels Peder Raj Andersen
Kristian Keiding
Modelling the pH variation in rivers is an interesting exercise, not only from an environmental point of view, but also because it makes use of basic physical, chemical, biological and numerical principles obtained through undergraduate courses. The pH model is built systematically using theory and logical assumptions to include or exclude different parameters and processes. After that, a short discussion will show how the initialisation values were found. The final model will show how equilibrium problems can be solved in discreet steps for a dynamic system using the simple assumption, that equilibrium reactions are much faster than transport phenomena and biological transformation. Furthermore, the model will be used to predict the behaviour of pH when changing temperature, ionic strength, buffer capacity and biological transformation rates.
Modeling Stratospheric Ozone Kinetics, Part I: The Chapman Cycle JCE
Erica Harvey, Fairmont State University
Bob Sweeney, Fairmont State College
The four-step Chapman cycle of stratospheric ozone reactions is used to introduce numerical solutions of complex rate laws and kinetic modeling. By using this Mathcad template students apply a majority of the chemical kinetics concepts taught in standard texts (using first-, second-, and third-order rate constants, writing differential rate laws for each of the species appearing in a chain reaction mechanism, investigating temperature dependence with Arrhenius and other models, and handling photolysis rate constants).
Modeling Stratospheric Ozone Kinetics, Part II: Addition of Hydrogen, Nitrogen and Chlorine JCE
Erica Harvey, Fairmont State University
Bob Sweeney, Fairmont State College
This Mathcad document builds on the Chapman cycle considered in the Part I document to create a more realistic model of stratospheric ozone chemistry. In this document, students are asked first in a series of leading exercises to set up and solve the system of differential equations that includes the kinetics of the NOx and HOx cycles, then to investigate the effects of changes in the input values.
Non-Linear Least Squares Regression JCE
Sidney H. Young, University of South Alabama
Andrzej Wierzbicki, University of South Alabama
Nonlinear least-squares regression is often required in the physical chemistry laboratory. It is especially important for fitting functions that cannot be linearized. This template demonstrates various implicit and explicit methods for determining the parameters of the regressed curve obtained by nonlinear curve-fitting.
Nonlinear regression: Kinetics of sucrose inversion*
Theresa Julia Zielinski, Monmouth University
An introduction to nonlinear curve fitting using data using first order data. Explorqtion of the basic concepts of ploting data, writing the fitting function, estimating the fitting parameters and then finding the best fit parameters by minimizing the sum of the squares of the deviations (SSD) between the fitting function and the data.SSD from before and after obtaining the optimum parameters are compared. The document concludes withcalculation of residuals and a plot of residuals to demonstrate that points lie between. ? 2 sigma.
Numerical Solution of Second Order Differential Equations
Theresa Julia Zielinski, Monmouth University
Danny G. Miles, Jr., Mount St. Mary's University
In this document students can explore using numerical methods for solving second order differential equations. Students are expected to apply the method to the quantum mechanical oscillator and explore the properties of the resulting solutions.
Optimization of Geometries by Energy Minimization
Tracy P. Hamilton, University of Alabama at Birmingham
A Mathcad document designed to help students understand the theory underpinning energy minimization and optimization of the geometry of a molecule. Newton-Raphson Method is used. Hessian matricies are used throughout.
Orbital Graphing JCE
Mark David Ellison, Wittenberg University
This document provides three-dimensional graphs of the angular parts of hydrogen-like atomic orbitals. Linear combinations of orbitals result in the familiar px and py orbitals. The process is repeated for some of the d orbitals. The second part of the document shows linear combinations of s and p orbitals to construct hybrid sp, sp2, and sp3 orbitals.
Particle-in-a-Box Dynamics JCE
Theresa Julia Zielinski, Monmouth University
David M. Hanson, State University of New York at Stony Brook
This Mathcad document explores the time-dependent quantum mechanics for a particle-in-a-box using guided inquiry and the animation tools of the software.
Playing with Waves
Theresa Julia Zielinski, Monmouth University
A Mathcad template that provides students with quided inquiry practice to learn about wave behavior. This documents serves as a review and extends the lessons to include creation of left and right moving waves, standing waves from traveling waves and the Euler relation.
Potential Barriers and Tunneling JCE
Mark David Ellison, Wittenberg University
A document to provide students with the opportunity to develop their understanding of the behavior of particles in the presence of finite barriers. Color enhancement of the wave function and probability density plots clearly delineate the different regions and makes clear what has happened.Also Scanning Tunneling Microscopy is examined followed by applications of the tunneling concept to chemical reactions.
Powder Pattern Simulations
George L. Hardgrove, St. Olaf College
This Mathcad file provides an introduction to powder diffraction patterns of cubic crystal lattices.
Properties of the Radial Functions
George M. Shalhoub, La Salle University
This Mathcad document provides exercises exploring the radial functions of the hydrogen atom. The document concludes with Slater Rules practice exercises for heavier atoms and an application to inorganic chemistry, SF6.
Radial Distributions Functions for Hydrogen-like Orbitals
Flick Coleman, Wellesley College
This document permits students to explore some of the properties of the radial distribution functions of hydrogen-like one-electron orbitals. The document includes the plots of the ns radial functions which students can change to view the np and nd functions. The calculus is used to demonstrate how to find the maximum and minimum for the functions. Other exercises include exploring the effect of nuclear charge on the function and extent (size) of the radial distribution function.
Real Gases: Defining the Standard State and Quantifying Deviations from Ideality
Theresa Julia Zielinski, Monmouth University
Joseph H. Noggle (1936 - 1998), Monmouth University
This document has four parts. In part 1 is an exploration of the PV behavior of gases using the Redlich-Kwong equation. Part 2 determines the energy required to raise the temperature of a sample of a gas and also explores the mathematical definition of the standard state the imperfection concept to account for real properties of a gas. In part 3 is the calculation of the temperature drop and the Joule-Thomson inversion temperature for the expansion of a SO2 gas.
Rejection of Data
Kevin Lehmann, Princeton University
The document provides a detailed presentation to the theory of rejection of data using a Gaussian distribution. The document discusses the conditions under which the Q-test is used. The exercises in the document give students opportunities to practice the concepts. The document provides a numerical example of how statistical methods can reduce the errors in information extracted from measurements with real, as oppose to Gaussian, noise characteristics.
Relating Qualitative Analysis to Equilibrium Principles JCE
Glenn V. Lo, Nicholls State University
Mathcad document intended for a prelab activity for qualitative analysis experiments. Focus is on calculation of solubilities etc.
Rotational States of Carbon Monoxide JCE
Theresa Julia Zielinski, Monmouth University
David M. Hanson, State University of New York at Stony Brook
This Mathcad document develops the components that contribute to the observed rotational absorption spectrum of CO.
sp3dn Orbital Hybrids and Molecular Geometry JCE
Mark David Ellison, Wittenberg University
A Mathcad document that gives students the opportunity to explore hybrid orbitals that give rise to octahedral, trigonal bipyramidal, square planar, and square pyramidal shapes. LCAO is use to generate the hybrid shapes.
Steepest Descents: Finding the Minimum in a Function
Theresa Julia Zielinski, Monmouth University
Jeffry D. Madura, Duquesne University
An introduction to the method of Steepest Descents for finding a minimum in a function. Countour maps are developed and search variables explored. A set of exercises using the 2-D Rosenbrock function, whichcontains a minimum that lies at the bottom of a banana shapedvalley completes the set of interactive exercises.
Studies in FT-IR
Scott Van Bramer, Widener University
This is a collection of six documents that can be used to teach and learn the basic concepts of FT-IR. ft_notes.mcd, provides an introduction to FT as a lecture handout or interactive student activity. In ft_01_6.mcd students study the parameters that affect resolution and spectral width. ft_02_06.mcd shows how two frequency components add to give a beat pattern. ft_03_6.mcd shows how cosine and sine waves describe the frequency and phase of a signal. In ft_04_6.mcd students examine the effect of an exponential decay in the time domain and frequency domain. The entire series of documents is summarized in ft_05_6.mcd which contains large fonts and no explanatory text and is designed for use in class lectures.
Studying Nonlinear Chemical Dynamics with Numerical Experiments JCE
John A Pojman, University of Southern Mississippi
Oscillating reactions are the main focus of this document. Also an Introduction of numerical integration for chemical rate expressions and examination of the limitations to standard methods and an illustration of the erroneous results that can be obtained for simple coupled differential equations using two algorithms in Mathcad. Oscillating reactions are the main focus of this document.
Symbolic Math Approach To Solve Particle-in-the-Box and H-Atom Problems JCE
Todd M. Hamilton, Georgetown College
After practicing with the particle-in-a-box problem, students solve the H atom symbolically using the derivatives feature in MathCad. The result is an expression for the energy levels of the hydrogen atom. Students will also prove that the radial part of the wavefunction is normalized and explore the concept of an orbital.
Temperature as a Measure of the Distribution of Particles Over Energy States: Would a Negative Absolute Temperature be Very Cold, or Very Hot? JCE
Arthur Ferguson, Worcester State College
This exercise explores the implications of the Boltzmann Equation for the population of energy states as a function of temperature. Plots present the relative population of the first three vibrational states of carbon monoxide from 0 K to very high temperatures and focuses attention on what happens to the relative populations of these states over that range, especially at the extremes of infinite and zero absolute temperature. It then seeks explors the implications of hypothetical negative absolute temperatures.
The Autocatalytic Reaction
Theresa Julia Zielinski, Monmouth University
An introduction autocatalytic reactions and the use of the Runge-Kutta method for finding numerical solutions to sets of coupled chemical reactions. This document can serve as one component in a series of studies on the standard chemical kinetics systems found in most standard physical chemistry texts.
The Carnot Cycle JCE
Harold H. Harris, University of Missouri-St. Louis
Mathcad document that allows students to investigate the Carnot cycle with numerical calculations on an ideal, monatomic gas. An advanced assignment is provided for a Van der Waals gas.
The Crystal Structure of Bromobenzoic Acid
George L. Hardgrove, St. Olaf College
The purpose of this Mathcad exercise is to resolve a two-dimensional image of the structure of bromobenzoic acid in order to find out which isomer (ortho, meta, or para) of bromobenzoic acid ispresent and to speculate on the pattern of possible hydrogen bonding.
The Diffusion Game - Using Symbolic Mathematics Software to Play the Game on a Large Scale JCE
W. Tandy Grubbs, Stetson University
A Mathcad instructional document that allows students to explore the diffusion. The goal of the activites in this document is to observe how the statistical 'microscopic' behavior of individual particles ultimately determines a certain 'macroscopic' behavior in a large system. The document provides simulations using the random number generator of Mathcad The outcome of the simulations are analyzed within the context of equilibrium and non-equilibrium states, entropy maximization, and Fick's law of diffusion, reinforcing one's understanding of the microscopic origin of these phenomena.
The Entropy of Lead
George L. Hardgrove, St. Olaf College
This Mathcad files present the determination of the third law entropy of lead using numerical methods, i.e. the trapezoid rule.
The Gibbs Free Energy of a Chemical Reaction System as a Function of the Extent of Reaction and the Prediction of Spontaneity JCE
Arthur Ferguson, Worcester State College
This Mathcad template is designed to help students develop a more nuanced understanding of the relationship between ΔrxG and spontaneity through the creation of a graph of G for a reaction mixture vs. the extent of reaction, x , and an exploration of how the shape of that graph is affected by the value of ΔrxG.
The Iodine Spectrum JCE
George Long, Indiana University of Pennsylvania
Theresa Julia Zielinski, Monmouth University
The goal of this document is to present a systematic development of the relationship between spectroscopic experiments and the determination of molecular bond lengths in the excited state of diatomic molecules for which a high resolution vibronic spectrum is available. This document can serve as a template for students to use for data analysis of the UV-vis spectrum of I2 or other diatomic gases.
The Joule-Thomson Effect and Enthalpy for SO2*
Theresa Julia Zielinski, Monmouth University
An introduction to thermodynamics of real gases, computing imperfection functions, Joule-Thomson coefficients, and Joule-Thonson inversion temperatures.
The Morse Oscillator
Kevin Lehmann, Princeton University
In this worksheet, we find a presentation of the vibrational motion of a diatomic molecule held together with a potential function of a special form known as the Morse Potential. Both the classical and quantum motion of the oscillator will be studied, and explicit expressions for eigenenergies and wavefunctions are given The effect of rotation is also discusses. The document contains embedded 20 exercises and 4 advanced problems for users to test their mastery of the topic.
The Steady-State and Equilibrium Approximations: Background Readings
Flick Coleman, Wellesley College
In this Mathcad document students obtain background information about the steady state approximation.
The Vibrations of Molecules - I : The Simple Harmonic Oscillator
George L. Hardgrove, St. Olaf College
This Mathcad file presents a basic introduction to simple harmonic motion and permit students to explore the effect of mass and force constant on this motion. It also gives a foundation to harmonic motion interms of Legrange's equation of motion.
The Vibrations of Molecules - II : The Carbon Dioxide Molecule
George L. Hardgrove, St. Olaf College
This Mathcad file providez a basic introduction to harmonic motion in a triatomicmolecule. Students explore the effect of initial velocity andposition on the mode of vibration obtained by solving classical equations of motion. The document can serve as an introduction to Molecular Dynamics.
Using a Computer to Help Understand How Symmetry Principles Reduce Calculations JCE
Louis Kijewski, Monmouth University
This Mathcad document uses symmetry to simplify the evaluation of an eigenvalue problem by reducing a matrix to sparse form. The sparse matrix is then transformed to block diagonal form. Group theory is used to set up the calculations without going through the proofs of the methods. The ability to use symmetry principles to get the vibrational frequencies and other information about molecules with symmetry is an important asset to any chemist.
Using Mathcad in Physical Chemistry - Calculus
Theresa Julia Zielinski, Monmouth University
R. D. Poshusta, Washington State University
Provides a set of documents to help students learn the fundamental operations of Mathcad. Basic usage, algebra, solving equations, graphing functions, arrays, and matrices are the topics covered. The templates are easy for students to use.
Using Mathcad in Physical Chemistry - Introductory Exercises
Theresa Julia Zielinski, Monmouth University
R. D. Poshusta, Washington State University
Provides a set of documents to help students learn the fundamental operations of Mathcad. Basic usage, algebra, solving equations, graphing functions, arrays, and matrices are the topics covered. The templates are easy for students to use.
van der Waals and Redlich Kwong: Fitting Two Parameter Equations to Gas Data
Theresa Julia Zielinski, Monmouth University
The Mathcad document given here is a highly annotated application of non-linear curve fitting for determining the a and b parameters for the van der Waals and Redlich-Kwong equations. The document goes through the development of the sum of squares of deviations SSD, applies the Levenberg-Marquradt to minimize the SSD and then uses the F-test to determine the equation that best represents the data. The determination of the standard deviation of the fitting parameters is done through the explicit construction and inversion of the Hessian matrix.
Variational Method Applied to the Harmonic Oscillator JCE
S. Keith Dunn, Centre College
Mathcad document that introduces students to the variational method using a simple, one-dimensional harmonic oscillator and sinusoidal particle-in-a-box wavefunctions as a basis set.
Variational Methods Applied to the Particle in the Box JCE
W. Tandy Grubbs, Stetson University
A Mathcad instructional document that allows students to explore the variational method. The variational method is used here to estimate the energy levels of the particle in a one-dimensional box, where exact expressions are available for the energies and wave functions. By practicing the variational method on a known system, students explore the factors that govern the accuracy of the estimated energy and thereby gain an appreciation and confidence in the variational method that is difficult to obtain in any other fashion. Exercises are included that allow students to try the variational method using a wide range of trial functions.
Vibronic Spectra of Diatomic Molecules and the Birge-Sponer Extrapolation JCE
Theresa Julia Zielinski, Monmouth University
George M. Shalhoub, La Salle University
The goal of this document is to examine the potential energy curve for both the ground and excited state of an electronic transition in order to set the stage for a Birge-Sponer determination of the Do and De dissociation energies of the excited state of a diatomic molecule from experimental UV-vis spectral data. After using this document students will be able to: explain the relationship between the potential energy curves for excited and ground states in a molecule; explain the significance of each parameter in a Morse potential energy function; and use a Birge-Sponer plot to determine dissociation energy of an electronic excited state of a diatomic molecule. IodineSpectrum.mcd, is the companion template for the analysis of the UV-vis spectrum of a diatomic molecule.
Visualization of Wavefunctions of the Ionized Hydrogen Molecule JCE
John L. Johnson, The Pennsylvania State University
Mathcad and the Shooting Method, are used to solve the Schrodinger equation for the H2+ molecule. The angular portions of the wavefunctions are the spheroidal harmonics, which when plotted show the shapes of molecular orbitals. These exact solutions are compared with those obtained from the LCAO method, using hydrogenic orbitals as basis functions. Plot of the electronic charge densities show the concentration of charge that distinguishes bonding and anti-bonding states. The energies of the wavefunctions as a function of internuclear distance for the eight lowest molecular energy states are calculated.
Visualizing Particle-in-a-Box Wavefunctions using Mathcad JCE
Edmund L. Tisko, University of Nebraska at Omaha
Using the built-in differential equation solvers and the graphical capabilities of Mathcad, students can visualize the wavefunctions of the particle-in-a-box potential. The document examines bound states and tunneling using the particle in a box, step potential and double well potential.
Work Done During Reversible and Irreversible Isothermal Expansion of an Ideal Gas JCE
Arthur Ferguson, Worcester State College
This Mathcad template explores the implications of the Boltzmann Equation for the population of energy states as a function of temperature.