- Title Pages
- Foreword
- Epigraph
- Preface
- Index of symbols
- Index of units-of-measure
- List of Tables
- 1 Systems and states
- 2 Work and energy
- 3 Temperature and heat
- 4 Thermodynamics and mathematics
- 5 The First Law of Thermodynamics
- 6 Enthalpy and thermochemistry
- 7 Ideal gas processes – and two ideal gas case studies too
- 8 Spontaneous changes
- 9 The Second Law of Thermodynamics
- 10 Clausius, Kelvin, Planck, Carathéodory and Carnot
- 11 Order, information and time
- 12 The Third Law of Thermodynamics
- 13 Free energy
- 14 Chemical equilibrium and chemical kinetics
- 15 Phase equilibria
- 16 Reactions in solution
- 17 Acids, bases and buffer solutions
- 18 Boiling points and melting points
- 19 Mixing and osmosis
- 20 Electrochemistry
- 21 Mathematical round up
- 22 From ideal to real
- 23 The biochemical standard state
- 24 The bioenergetics of living cells
- 25 Macromolecular conformations and interactions
- 26 Thermodynamics today – and tomorrow
- Glossary
- Bibliography
- Index

# Thermodynamics and mathematics

# Thermodynamics and mathematics

- Chapter:
- (p.76) 4 Thermodynamics and mathematics
- Source:
- Modern Thermodynamics for Chemists and Biochemists
- Author(s):
### Dennis Sherwood

### Paul Dalby

- Publisher:
- Oxford University Press

Most school mathematics is about how one variable, *y*, varies with respect to one other variable, *x*, according to an equation such as *y* = 3*x*^{2}. Equations like this underpin the student’s knowledge of algebra, and differential and integral calculus. Thermodynamics, however, is necessarily about how a variable, such as the pressure *P*, varies with respect not to one but to three variables simultaneously – for example, the mole number *n*, the volume *V*, and the temperature *T*. This makes the algebra of thermodynamics more complex, and also implies that mutual changes between pairs of variables is described not in terms of total derivatives of the form d*y*/d*x*, but rather by partial derivatives of the form (∂*P*/∂*T*)_{V}. Many students find the leap from d*y*/d*x* to (∂*P*/∂*T*)_{V} very difficult - the purpose of this chapter is therefore to build the reader’s confidence in understanding, and manipulating, functions of two and three variables.

*Keywords:*
functions of two and three variables, partial derivatives, systems of constant mass, pure substances, mixtures

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- Title Pages
- Foreword
- Epigraph
- Preface
- Index of symbols
- Index of units-of-measure
- List of Tables
- 1 Systems and states
- 2 Work and energy
- 3 Temperature and heat
- 4 Thermodynamics and mathematics
- 5 The First Law of Thermodynamics
- 6 Enthalpy and thermochemistry
- 7 Ideal gas processes – and two ideal gas case studies too
- 8 Spontaneous changes
- 9 The Second Law of Thermodynamics
- 10 Clausius, Kelvin, Planck, Carathéodory and Carnot
- 11 Order, information and time
- 12 The Third Law of Thermodynamics
- 13 Free energy
- 14 Chemical equilibrium and chemical kinetics
- 15 Phase equilibria
- 16 Reactions in solution
- 17 Acids, bases and buffer solutions
- 18 Boiling points and melting points
- 19 Mixing and osmosis
- 20 Electrochemistry
- 21 Mathematical round up
- 22 From ideal to real
- 23 The biochemical standard state
- 24 The bioenergetics of living cells
- 25 Macromolecular conformations and interactions
- 26 Thermodynamics today – and tomorrow
- Glossary
- Bibliography
- Index