Section 2.9: Quasi-Static Processes

The discussion so far has been general concerning 2 systems A & A' interacting with each other. Now, we consider a Simpler (Ideal) Special Case Definition: Quasi-Static Process This is defined to be a general process by which system A interacts with system A', but the interaction is carried out So Slowly that system A remains arbitrarily close to equilibrium at all stages of the process. How slowly does this interaction have to take place? This depends on the system, but the process must be much slower than the time it takes system A to return to equilibrium if it is suddenly disturbed.

Quasi-Static Processes or Quasi-Equilibrium Processes Sufficiently slow processes that any intermediate state can be considered an equilibrium state. (Macroscopic parameters are well defined for all intermediate states). Advantages of Quasi-Static Processes The macrostate of a system that participates in such a process can be described with the same (small) number of macroscopic parameters as for a system in equilibrium (for a gas, this could be T & P). By contrast, for non-equilibrium processes (e.g. turbulent flow of a gas), a huge number of macroscopic parameters is needed.

See figure for an Example A gas confined to a container of volume V. At the top is a moveable piston. Sand is placed on the top of that to weigh it down. Quasi-Static Processes Several different types of Quasi-Static Processes can be carried out on this system. These processes & the terminology used to describe them are: Isochoric Process:  V = constant Isobaric Process:  P = constant Isothermal Process:  T = constant Adiabatic Processes:  Q = 0

Quasi-Static Processes Let the external parameters of system A be: x 1,x 2,x 3,…x n. For this case, the energy of the r th quantum state of the system may abstractly be written: E r = E r (x 1,x 2,…x n ) When the values of one or more external parameters are changed, the energy E r also obviously changes. Let each external parameter change by an infinitesimal amount: x α x α + dx α How does E r change? From calculus, we know: dE r = ∑ α (∂E r /∂x α )dx α

The exact differential of E r is: dE r = ∑ α (∂E r /∂x α )dx α As we’ve already noted, if the external parameters change, some mechanical work must be done. For the system in it’s r th quantum state, that work may be written: đW r = - dE r = - ∑ α (∂E r /∂x α )dx α ≡ ∑ α X α,r dx α Here, X α,r ≡ - (∂E r /∂x α ) X α,r ≡ The Generalized Force associated with the external parameter x α

The Macroscopic Work done when the system’s external parameters change is related to the change in it’s mean internal energy Ē by: đW = - dĒ ≡ ∑ α dx α Here, ≡ - (∂Ē/∂x α ) ≡ Mean Generalized Force associated with external parameter x α

External Parameter x α Mean Generalized Force Macroscopic Work đW = dx α Position Coordinate x Mean Mechanical Force đW = dx Volume V Mean Pressure đW = dV Electric Potential U Mean Electric Charge đW = dU Magnetic Field B Mean Magnetic Moment đW = dB Electric Field E Mean Electric Dipole Moment đW = dE Examples: External parameters & mean generalized forces:

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. A gas is confined to a container, volume V, with a piston at the top. A weight is 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.

Elementary Physics: Differential work đW done by gas when piston undergoes differential vertical displacement ds is: đW = F ds F = Total vertical force on the piston. Definition of (mean) pressure p: F = pA A = Cross sectional area. V = As = gas volume. So, đW = pAds = pdV So, the work done by the gas as the volume changes from V i to V f is the integral of the pressure p as a function of V: Obviously, this is the area under the p(V) vs. V curve!

There are Many Possible Paths in the P-V Plane to take the gas from 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. Figs. (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: 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. To understand what an Inexact Differential is, it helps to 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. Math Theorem If F(x,y) is an analytic function, then It’s 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): The integral of dF over an arbitrary closed path in the x-y plane is zero. = 0

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

Gases: Quasi-Static Work Done by Pressure đW is clearly path dependent ∆Ē = đQ + đW does not depend on the path.

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