Chapter 4 - Chapter 4 Work and heat Read BS, Chapter 4 In...

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Unformatted text preview: Chapter 4 Work and heat Read BS, Chapter 4 In this chapter, we consider work and heat. We will see that these often accompany a thermodynamic process. We will first consider some general mathematical notions, which will be pertinent. 4.1 Mathematical preliminaries: exact differentials Here we review some notions from calculus of many variables. Recall in thermodynamics, we are often concerned with functions of two independent variables, e.g. P = P ( v, T ), as is found in an equation of state. Here, let us consider z = z ( x, y ) for a general analysis. 4.1.1 Partial derivatives Recall if z = z ( x, y ), then the partial derivative of z can be taken if one of the variables is held constant. Example 4.1 If z = radicalbig x 2 + y 2 , find the partial of z with respect to x and then with respect to y . First let us get the derivative with respect to x . We take z x vextendsingle vextendsingle vextendsingle vextendsingle y = x radicalbig x 2 + y 2 . (4.1) Next for the derivative with respect to y , we have z y vextendsingle vextendsingle vextendsingle vextendsingle x = y radicalbig x 2 + y 2 . (4.2) 77 78 CHAPTER 4. WORK AND HEAT 4.1.2 Total derivative If z = z ( x, y ), for every x and y , we have a z . We also have the total differential dz = z x vextendsingle vextendsingle vextendsingle vextendsingle y dx + z y vextendsingle vextendsingle vextendsingle vextendsingle x dy. (4.3) Now we can integrate dz along a variety of paths C in the x y plane. Two paths from z 1 to z 2 are shown in Fig. 4.1. Integrating, we get x y z one.inferior ztwo.inferior path A path B Figure 4.1: Sketch of two paths from z 1 to z 2 in the x y plane. integraldisplay 2 1 dz = integraldisplay C parenleftBigg z x vextendsingle vextendsingle vextendsingle vextendsingle y dx + z y vextendsingle vextendsingle vextendsingle vextendsingle x dy parenrightBigg . (4.4) Now because z = z ( x, y ), it will not matter which path we choose. Conversely, if we were given dz = M ( x, y ) dx + N ( x, y ) dy, (4.5) the associated integrals are path independent iff z ( x, y ) can be found by solving. M = z x vextendsingle vextendsingle vextendsingle vextendsingle y , N = z y vextendsingle vextendsingle vextendsingle vextendsingle x . (4.6) One easy way to check this is to form the following two partial derivatives of Eqs. (4.6): M y vextendsingle vextendsingle vextendsingle vextendsingle x = 2 z yx , N x vextendsingle vextendsingle vextendsingle vextendsingle y = 2 z xy . (4.7) CC BY-NC-ND. 2011, J. M. Powers. 4.1. MATHEMATICAL PRELIMINARIES: EXACT DIFFERENTIALS 79 Now if z ( x, y ) and all its partial derivatives are continuous and differentiable, it is easy to prove the order of differentiation does not matter: 2 z/xy = 2 z/yx . Thus if z = z ( x, y ), we must insist that M y vextendsingle vextendsingle vextendsingle vextendsingle x = N x vextendsingle vextendsingle vextendsingle...
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This note was uploaded on 03/02/2012 for the course THERMO 20231 taught by Professor Powers during the Spring '10 term at Notre Dame.

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Chapter 4 - Chapter 4 Work and heat Read BS, Chapter 4 In...

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