Lectures_12-13 - Part IV Work Lecture 12 PV work Recall the...

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Part IV — Work Lecture 12 — PV work Recall the notion of a thermodynamic process: change of state via some external intervention. Common processes include: Adding or removing heat Expanding or compressing the system Often, the process involves requires or generates work. We say, work is done on the system, or done by the system. Sign convention: work done on the system is positive, work done by the system is negative. (We are keeping track of the energy of the system.) Recall that being able to calculate the required amount of work for a process, and designing a process so that the minimum amount of work is required, or the maximum amount of work is generated, are central goals of thermodynamics. An external force acting through a small displacement does a small amount of work dW = F ext dx = - P ext dV We integrate to find the total work done during the process W = - V 2 V 1 P ext dV Sign convention: when we compress the system, dx > 0 (displacement in direction of force applied) and dV < 0. In compressing, we do work on the system, increasing its energy (like compressing a spring). Work is not a state variable — it depends on the path (i.e., the details of the process). For example: it is harder to compress a hot gas than a cold one. So consider a process in which we increase T by some amount and decrease V by some amount. If we warm first, the work will be greater than if we warm after compressing. (sketch) We will see more specific examples of this next lecture. To compute W , we need to know how P ext varies as we do the volume change. If we do the change slowly enough, P ext = P ( V, T ) (the equilibrium system pressure) to a close approximation — nearly balanced forces, very slow motion of the piston. This we call a reversible change — at every point along the way, we are very close to equilibrium. Then, we can compute the work, if we have an EOS to tell us P ( V, T ). In general, if the process is done at a finite rate, it won’t be “reversible”. In that case, W = W rev + W lost 30
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We always have W lost > 0 (This is a postulate , that is consistent with our common experience — and detailed experiment besides. We shall meet other equivalent forms of this assumption in later lectures.) If we are doing work on the system ( W > 0), it takes more work than the minimum W rev to e ff ect the change. If the system is doing work ( W < 0), we get less work than the maximum - W rev . Example: we slowly heat an ideal gas confined by a fixed external pressure P ext . “Slowly” means P P ext during the process. Because the external pressure is fixed, the work is simple to com- pute. Pv = RT implies Δ v = R Δ T/P , hence W = - R Δ T . The gas does positive work on the surroundings (pushes in the direction of motion), so the surroundings do negative work on the gas ( W < 0).
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