Maths in Chemistry

Numerical Methods for Physical and Analytical Chemistry

Prerna Bansal

• Publisher: De Gruyter
• Year: 2020
• eBook ISBN: 9783110695489
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Applied Mathematics

Numerical methods are the mathematical procedures that approximate the solution of complex mathematical problems into much simpler form and which find a wide variety of use while solving complex Physical Chemistry problems. This book aims to aide in understanding of such numerical methods including solving complex differential equations and numerical differentiation & integration. Moreover it also explains various statistical tests used in Analytical Chemistry for data analysis. The author has tried to include as many example from Chemistry problems for a better understanding of the methods.

Chapter 1 Units and measurements

1.1 Introduction

In science, especially physical chemistry, quite a lot of measurements are involved. A physical quantity holds a great importance. A physical quantity is the product of two quantities, that is, a unit and a number. While measuring any parameter in physical chemistry, number itself is not relevant unless it is hooked to a unit. For instance, 6 metre has the number “6” and unit “metre”. Also, while correlating two physical quantities, constant and variables are involved. A constant is a physical quantity that has a fixed value, for example, Planck’s constant (h = 6.626$\times$10⁻³⁴ J s) and Avogadro’s constant (N_A = 6.023$\times$10²³ mol⁻¹). A variable, on the other hand, is a quantity that can have any permissible values based on some other quantities for which the given variable is a function, for example, p is a function of V, T, n, that is,

(1.1)$p=nRT/V$

where p can be represented as $p=f(V,T,n).$

Further, the variables are classified as dependent and independent variables. In the earlier example, since p is a function of V, T and n, p is a dependent variable while the rest are independent variables.

1.2 Basic arithmetic operations

A physical quantity is often the result of correlation of other physical quantities that are related by mathematical operators. To do so, basic arithmetic operations are performed on these quantities. We are all familiar with basic mathematical operations: addition, subtraction, multiplication and division. In this section, we will simply sum up the basics learned so far (Table 1.1).

Table 1.1:Mathematical operations.

Operation	Expression	Ordering
Addition	$A+B$	$A+B=B+A$
Subtraction	$A-B$	$A-B\ne B-A$
Multiplication	$A\times B\mathrm{o}\mathrm{r}A.B\mathrm{o}\mathrm{r}AB$	$A\times B=B\times A$
Division	$A÷B\mathrm{o}\mathrm{r}{\displaystyle \frac{A}{B}}$	${\displaystyle \frac{A}{B}\ne \frac{B}{A}}$

Some of the equations used in physical chemistry involving these operations are listed below:

(1.2)$\mathrm{I}\mathrm{d}\mathrm{e}\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{a}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:pV=nRT$

(1.3)$\mathrm{H}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{o}\mathrm{n}-\mathrm{H}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{b}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{h}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:\mathrm{p}\mathrm{H}=\mathrm{p}{K}_{\mathrm{a}}+\mathrm{l}\mathrm{o}\mathrm{g}(\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{d}/\mathrm{s}\mathrm{a}\mathrm{l}\mathrm{t})$

(1.4)$\mathrm{E}\mathrm{i}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:E=m{c}^2$

(1.5)$\mathrm{N}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:E=E^0-(RT/nF)\mathrm{l}\mathrm{o}\mathrm{g}Q$

(1.6)$\mathrm{O}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:\wedge ={\wedge }^0-b\sqrt{c}$

(1.7)$\mathrm{A}\mathrm{r}\mathrm{r}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}:k=A\mathrm{e}\mathrm{x}\mathrm{p}(-E_{\mathrm{a}}/RT)$

1.3 Hierarchy of operations

Before performing arithm...

Table of contents

• Title Page
• Copyright
• Contents
• Acknowledgement
• Chapter 1 Units and measurements
• Chapter 2 Uncertainties and errors
• Chapter 3 Some mathematical functions
• Chapter 4 Descriptive statistics
• Chapter 5 Numerical curve fitting
• Chapter 6 Numerical integration
• Chapter 7 Differential equations
• Chapter 8 Numerical differentiation
• Chapter 9 Numerical root-finding methods
• Index