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 Vi to Vf 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 + đWdĒ 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 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.

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

U đQ + đW does not depend on the path∆E

Summary The Differential dF = Adx + Bdy

is Exact if:

1.

2. 0

3. is independent of pathb

a

A B

y x

dF

dF