Chapter 11

# Chapter 11 - Chapter 11 Mathematical foundations Read BS...

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Unformatted text preview: Chapter 11 Mathematical foundations Read BS, Chapter 14.2-14.6, 17.5, 17.6 This chapter will serve as an introduction to some of the mathematical underpinnings of thermodynamics. Though the practicality is not immediately obvious to all, this analysis is a necessary precursor for building standard theories to describe chemical reactions, which have widespread application in a variety of engineering scenarios, including combustion, materials processing, and pollution control. 11.1 Maxwell relations We begin with a discussion of the so-called Maxwell relations, 1 named after the great nineteenth century physicist, James Clerk Maxwell, shown in Fig. 11.1. Figure 11.1: James Clerk Maxwell (1831-1879) Scottish physicist; image from http://www-history.mcs.st-and.ac.uk/ ∼ history/Biographies/Maxwell.html . 1 J. C. Maxwell, 1871, Theory of Heat , reprinted 2001, Dover, Mineola, New York, p. 169. 309 310 CHAPTER 11. MATHEMATICAL FOUNDATIONS Recall that if z = z ( x, y ), we have Eq. (4.3): dz = ∂z ∂x vextendsingle vextendsingle vextendsingle vextendsingle y dx + ∂z ∂y vextendsingle vextendsingle vextendsingle vextendsingle x dy. (11.1) Recall if dz = M ( x, y ) dx + N ( x, y ) dy , the requirement for an exact differential is ∂z ∂x vextendsingle vextendsingle vextendsingle vextendsingle y = M, ∂z ∂y vextendsingle vextendsingle vextendsingle vextendsingle x = N, (11.2) ∂ 2 z ∂y∂x = ∂M ∂y vextendsingle vextendsingle vextendsingle vextendsingle x , ∂ 2 z ∂x∂y = ∂N ∂x vextendsingle vextendsingle vextendsingle vextendsingle y . (11.3) These equations are the same as Eqs. (4.6,4.7). Because order of differentiation does not mat- ter for functions which are continuous and differentiable, we must have for exact differentials, Eq. (4.8): ∂N ∂x vextendsingle vextendsingle vextendsingle vextendsingle y = ∂M ∂y vextendsingle vextendsingle vextendsingle vextendsingle x . (11.4) Compare the Gibbs equation, Eq. (8.58), to our equation for dz : du = − Pdv + Tds, (11.5) dz = Mdx + Ndy. (11.6) We see the equivalences z → u, x → v, y → s, M → − P N → T. (11.7) and just as one expects z = z ( x, y ), one then expects the natural, or canonical form of u = u ( v, s ) . (11.8) Application of Eq. (11.4) to the Gibbs equation, Eq. (8.58), gives then ∂T ∂v vextendsingle vextendsingle vextendsingle vextendsingle s = − ∂P ∂s vextendsingle vextendsingle vextendsingle vextendsingle v . (11.9) Equation (11.9) is known as a Maxwell relation . Moreover, specialization of Eq. (11.2) to the Gibbs equation gives ∂u ∂v vextendsingle vextendsingle vextendsingle vextendsingle s = − P, ∂u ∂s vextendsingle vextendsingle vextendsingle vextendsingle v = T. (11.10) CC BY-NC-ND. 2011, J. M. Powers. 11.2. TWO INDEPENDENT VARIABLES 311 11.2 Two independent variables Consider a general implicit function linking three variables, x , y , z : f ( x, y, z ) = 0 . (11.11) In x − y − z space, this will represent a surface. If the function can be inverted, it will bespace, this will represent a surface....
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Chapter 11 - Chapter 11 Mathematical foundations Read BS...

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