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However, study of any of the underlying mathematics out of context tends to reduce mathematical activity to a series of clean, dry routines and procedures. Many students then struggle with applying the quantitative knowledge in the complicated chemical contexts they encounter. For example, we have Mathematics Chemistry context Ratios Mixing solutions with certain molarities, making dilutions Proportional reasoning Analysis of molecular structure; moles Algebra and graphs Analysis of experimental plots of reaction rates; gas laws Calculus Predicting and measuring rates of reaction in measurable experiments Units of measurements Making sense of real, complicated measurements Vectors Understanding crystal structure Logarithms Understanding pH Probability Drawing general conclusions from trials Suppose that a chemist achieved a good grade in GCSE mathematics or AS mathematics.

There are several possible reasons: Procedural thinking Mathematics exams can often be passed by learning the content procedurally.

## Mathematics and Chemistry: Partners in Understanding Our World

This means that students can answer certain types of question by following a recipe. The problems in chemistry arise because even minor deviations from the precise recipe cause the student to fail to know what to do. Inability to translate mathematical meaning to chemical meaning Students who are very skilled at mathematics might have trouble seeing how to relate the mathematical process to a real-world context; this hampers the use of common sense, so valuable in quantitative science.

Inability to make estimates or approximations Mathematical contexts in chemistry are rarely simple. In order to apply mathematics predictively, approximations will need to be made.

To make approximations requires the student to really understand the meaning and structure of the mathematics. Organizing data: Univariate and bivariate tables and graphs; histograms. Describing data: Measures of location, spread, and correlation. Theory: Probability; random variables; binomial and normal distributions. Drawing conclusions from data: Confidence intervals; hypothesis testing; z-tests, t-tests, and chi-square tests; one-way analysis of variance.

## Precalculus: the Mathematics of Numbers, Functions and Equations

Non-parametric methods. Acceptable for credit: CSU, UC Description: Introduction to analytic geometry, differential and integral calculus of algebraic function, particular attention paid to simple applications.

Acceptable for credit: CSU, UC Description: Continuation of differential and integral calculus: Transcendental functions, methods of integration, partial differentiation, and multiple integration with particular attention to applications. Class: 5 hours lecture GR Prerequisite s MATH or MATH or MATH or appropriate placement through multiple-measures assessment process Not open for credit to students who have completed or are currently enrolled in MATH ABCD Description: Basic algebraic operations: Linear equations and inequalities, relations and functions, factoring quadratic polynomials, solving quadratic equations, fractions, radicals and exponents, word problems, graphing, and number systems.

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Class: 3 hours lecture GR Prerequisite s MATH or MATH D or appropriate placement through multiple-measures assessment process Description: Introduction to plane geometry emphasizing mathematical logic and proofs: Geometric constructions, congruent triangles, parallel lines and parallelograms, proportions, similar triangles, circles, polygons, and area. Recommended preparation: MATH Not open for credit to students who have completed or are currently enrolled in MATH ABCD Description: Intermediate algebraic operations: Real number properties and operations; solutions and graphs of linear equations in one and two variables; absolute value equations; advanced factoring; complex numbers, quadratic equations and systems of quadratic equations; conics; determinants; solutions and graphs of first-degree, quadratic and rational inequalities; exponential and logarithmic functions; and sequences and series.

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Work these out. In a gas simulation you would use the derivative to calculate the forces on the atoms and would integrate Newton's equations to find out where the atoms will be next. Set the derivative of this to zero and find out where the van der Waals minimum is.

### Examples for

Differentiate again and show that the derivative is positive, therefore the well is a minimum, not a turning point. In a diatomic molecule the energy is expanded as the bond stretches in a polynomial. The 2nd and 3rd derivatives will then need to be evaluated to give the shape of the potential and hence the infra-red spectrum.

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A one-dimensional metal is modelled by an infinite chain of atoms picometres apart. If the metal is lithium each nucleus has charge 3 and its electrons are modelled by the function.

## Maths for chemists : Numbers, Functions and Calculus. Volume I

What constant must this function be multiplied by to ensure there are 3 electrons on each atom? This integral is a dimensionless number equal to the number of electrons, so we will have to multiply by a normalisation constant. Here we have modelled the density of electrons. Later in the second year you will see electronic structure more accurately described by functions for each independent electron called orbitals.