Example 44 show that work is path dependent we have

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Example 4.4 Show that work is path-dependent. We have δW = PdV. In terms of intensive variables, assuming path-independence, we would have dw = Pdv. (4.35) If w were a path-independent property, we could have w = w ( P, v ), which would admit the exact dw = ∂w ∂v vextendsingle vextendsingle vextendsingle vextendsingle P bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright = P dv + ∂w ∂P vextendsingle vextendsingle vextendsingle vextendsingle v bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright =0 dP. (4.36) Our physics of dw = Pdv + 0 dP tells us by comparison that we would need ∂w ∂v vextendsingle vextendsingle vextendsingle vextendsingle P = P, and ∂w ∂P vextendsingle vextendsingle vextendsingle vextendsingle v = 0 . (4.37) Integrating the first gives w = Pv + f ( P ) . (4.38) Differentiating with respect to P gives ∂w ∂P vextendsingle vextendsingle vextendsingle vextendsingle v = v + df ( P ) dP = 0 . (4.39) CC BY-NC-ND. 2011, J. M. Powers.
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84 CHAPTER 4. WORK AND HEAT Thus df ( P ) dP = v. (4.40) Functions of P cannot be functions of v if P and v are independent. Therefore dw is not exact, w negationslash = w ( P, v ), and integraltext 2 1 Pdv is path-dependent. We can also see the path-dependence of 1 W 2 by realizing that 1 W 2 = integraltext 2 1 PdV represents the area under a curve in a P V diagram. Consider two paths, A and B from the same points 1 to 2 as depicted in the P V space of Fig. 4.4. The area under the curve defined V P V P W = P dV 1 2 1 2 1 2 1 2 W =∫ P dV 1 2 1 2 P ath A P a t h B Area A Area B Figure 4.4: P V diagram for work for two different processes connecting the same states. by Path A is clearly different from that under the curve defined by Path B. Clearly, the work 1 W 2 depends on the path selected, and not simply the end points. Obviously then, to calculate the work, we will need full information on P ( V ) for the process under consideration. Many processes in thermodynamics are well modeled as a Polytropic process : a process which is described well by an equation of the form PV n = constant = C. Here n is known as the polytropic exponent. Example 4.5 Find the work for a gas undergoing a polytropic process with n negationslash = 1. A polytropic process has P ( V ) = C V n . (4.41) CC BY-NC-ND. 2011, J. M. Powers.
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4.2. WORK 85 So the work is 1 W 2 = integraldisplay 2 1 C V n dV = C integraldisplay 2 1 dV V n , (4.42) = C 1 n V 1 n vextendsingle vextendsingle 2 1 , (4.43) = C 1 n ( V 1 n 2 V 1 n 1 ) . (4.44) Now C = P 1 V n 1 = P 2 V n 2 , so 1 W 2 = P 2 V 2 P 1 V 1 1 n . (4.45) Note this formula is singular if n = 1. Now if n = 1, we have PV = C , which corresponds to an isothermal process. We need to consider such processes as well. Example 4.6 Find the work for a gas undergoing a polytropic process with n = 1. For this process, we have P ( V ) = C V . (4.46) Therefore the work is 1 W 2 = integraldisplay 2 1 C dV V = C ln V 2 V 1 . (4.47) Since P 1 V 1 = C , we can say 1 W 2 = P 1 V 1 ln V 2 V 1 . (4.48) Example 4.7 Find the work for an isobaric process.
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