Example 44 show that work is path dependent we have

• Notes
• 23

This preview shows page 7 - 10 out of 23 pages.

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.

Subscribe to view the full document.

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.
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.

Subscribe to view the full document.

You've reached the end of this preview.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern