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Unformatted text preview: Chapter 11 Mathematical foundations Read BS, Chapter 14.214.6, 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 socalled Maxwell relations, 1 named after the great nineteenth century physicist, James Clerk Maxwell, shown in Fig. 11.1. Figure 11.1: James Clerk Maxwell (18311879) Scottish physicist; image from http://wwwhistory.mcs.stand.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 BYNCND. 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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 Spring '10
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