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One of the most important examples is to have the system of interest be a gas and to look at the

Section 2.10: Quasi-Static Work Done by Pressure. One of the most important examples is to have the system of interest be a gas and to look at the Quasi-Static Work done by Pressure on the gas.

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One of the most important examples is to have the system of interest be a gas and to look at the

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  1. Section 2.10: Quasi-Static Work Done by Pressure

  2. One of the most important examples is to have thesystem of interest be a gasand to look at the Quasi-Static Work done by Pressure on the gas. • Consider the situation in the figure, which is a gas confined to a container of volume V, with a piston at the top. There is a weight on the top of the piston, which is changed by adding small lead shot to it, as shown. • Initially, the piston & the gas are in equilibrium. If the weight is increased, the piston will push down on the gas, increasing the pressure p & doing work ON the gas. If the weight is decreased, the gas will push up on it, decreasing the pressure p & doing work ON the piston.

  3. From elementary physics, the differential work đW done by the gas when the piston undergoes a vertical displacement ds is: đW = F ds F = Total vertical force on the piston. Definition of (mean) Pressurep: F = pA A =piston cross sectional area. V = As= gas volume. So,đW = p Ads = pdV • So, the work done by the gas as the volume changes from Vi to Vf is given by the integral of the pressure p as a function of V: Obviously, this is the areaunderthep(V)vsVcurve!

  4. Note:There are many possible ways to take the gas from an initial state i to final state f. the work done is, in general, different for each. This is consistent with the fact that đW is an inexact differential! Figures (a) & (b) are only 2 of the many possible processes!

  5. Figures (c), (d), (e), (f) 4 more of the many possible processes!

  6. Section 2.11: Brief Math Discussion of Exact & Inexact Differentials

  7. We’ve seen that,for infinitesimal, quasi-static processes,the First Law of Thermodynamicsis đQ = dĒ + đW dĒ is an Exact DifferentialđQ, đW are Inexact Differentials • Lets first briefly review what is meant by an Exact Differential

  8. Exact Differentials Let F(x,y) =an arbitrary function of x &y. • F(x,y) is a well behaved function satisfying all the math criteria for being an analytic function of x & y. • It’s Exact Differential is: dF(x,y) ≡ A(x,y) dx + B(x,y) dy whereA(x,y) ≡ (∂F/∂x)y& B(x,y) ≡ (∂F/∂y)x. • If F(x,y) is an analytic function, then its 2nd cross partial derivatives must be equal: (∂2F/∂x∂y)≡ (∂2F/∂y∂x) • Also, if F(x,y) is an analytic function, the integral of dF between any 2 arbitrary points 1 & 2 in the xy plane is independent of the path between 1 & 2.

  9. For an Arbitrary Analytic Function F(x,y) The integral of dF around an arbitrary closed path vanishes dF = 0

  10. 3 Tests for an Exact Differential F(x,y) =an arbitrary analytic function of x &y.

  11. đQ + đW does not depend on the path For a Gas: The Quasi-Static Work Done by Pressure đW is clearly path dependent ∆E

  12. Summary The Differential dF = Adx + Bdy is Exact if:

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