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Chapter 4

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

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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

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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 ∂y∂x , ∂N ∂x vextendsingle vextendsingle vextendsingle vextendsingle y = 2 z ∂x∂y . (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/∂x∂y = 2 z/∂y∂x . Thus if z = z ( x,y ), we must insist that ∂M ∂y vextendsingle vextendsingle vextendsingle vextendsingle x = ∂N ∂x vextendsingle vextendsingle vextendsingle vextendsingle y . (4.8) We define the following: exact differential : a differential which yields a path-independent integral.

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

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