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. 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.

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) Pressure p: 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 V i to V f is given by the integral of the pressure p as a function of V: Obviously, this is the area under the p(V) vs V curve!

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!

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

Section 2.11: Brief Math Discussion of Exact & Inexact Differentials

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

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 where A(x,y) ≡ (∂F/∂x) y & B(x,y) ≡ (∂F/∂y) x. If F(x,y) is an analytic function, then its 2 nd cross partial derivatives must be equal: (∂ 2 F/∂x∂y) ≡ (∂ 2 F/∂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. Exact Differentials

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

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

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

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