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Section 2.9: Quasi-Static Processes

Section 2.9: Quasi-Static Processes. The discussion so far has been general concerning 2 systems A & A ' interacting with each other. Now, consider a Simpler (Ideal) Special Case Definition : Quasi-Static Process.

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Section 2.9: Quasi-Static Processes

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  1. Section 2.9:Quasi-Static Processes

  2. The discussion so far has been general concerning 2 systems A & A'interacting with each other. Now, consider a • Simpler (Ideal) Special Case • Definition: • Quasi-Static Process

  3. The discussion so far has been general concerning 2 systems A & A'interacting with each other. Now, consider a • Simpler (Ideal) Special Case • Definition: • Quasi-Static Process • This is definedto be a general process by which system Ainteracts with system A', but the interaction is carried out so slowlythat systemA remains arbitrarily close to equilibrium at all stages of the process.

  4. The discussion so far has been general concerning 2 systems A & A'interacting with each other. Now, consider a • Simpler (Ideal) Special Case • Definition: • Quasi-Static Process • This is definedto be a general process by which system Ainteracts with system A', but the interaction is carried out so slowlythat systemA 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 slowerthan the time it takes system A to return to equilibrium if it is suddenly disturbed.

  5. Quasi-Static Processes or Quasi-Equilibrium Processes • These are defined to be sufficiently • slow processes that any • intermediate state can be • considered an equilibrium state. • The macroscopic parameters must • be well defined for all intermediate • states.

  6. 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 • parametersis needed.

  7. Quasi-Static Processes • See figure for an Example • A gas is 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. • Several different types of Quasi-Static Processescan 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

  8. Quasi-Static Processes • Let the external parameters of system A be: • x1,x2,x3,…xn. • For this case, the energy of the rth quantum state of the system may abstractly be written: • Er =Er(x1,x2,…xn) • When the values of one or more external parameters are changed, the energy Eralso obviously changes. Let each external parameter change by an infinitesimal amount: • xα xα + dxα • How does Erchange? From calculus, we know: • dEr = ∑α(∂Er/∂xα)dxα

  9. The exact differential of Er is: • dEr = ∑α(∂Er/∂xα)dxα • As we’ve already noted, if the external parameters change, some mechanical work must be done. For the system in it’s rth quantum state, that work may be written: • đWr = - dEr = - ∑α(∂Er/∂xα)dxα≡ ∑αXα,rdxαHere, Xα,r ≡ - (∂Er/∂xα) • Xα,r ≡ The Generalized Force • associated with the external parameterxα

  10. The Macroscopic Workdone when the system’s external parameters change is related to the change in it’s mean internal energy Ē by: • đW = - dĒ≡ ∑α<Xα>dxα • Here, • <Xα> ≡ - (∂Ē/∂xα) • <Xα> ≡Mean Generalized Force • associated with external parameterxα

  11. Examples:External parameters & mean generalized forces:

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

  13. One of the most important examples is to have the system of interest be a gas & to look at the Quasi-Static Work Done by Pressure on the gas. • See the figure. A gas is confined to a container of volumeV, 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.

  14. Quasi-Static Work Done by Pressure on the gas. • 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.

  15. Elementary Physics: Differential work đW done by gas when piston undergoes differential vertical displacement ds is: • đW = F ds • F = Total vertical force on piston. • Definition of (mean) pressure p: F = pA A = Cross sectional area. V = As = gas volume. So, đW = pAds = pdV

  16. đW = pAds = pdV • So, the work done by the gas as the volume changes from Vi to Vf is the integral of the pressure p as a function of V: Obviously, this is the area underthe p(V) vs. Vcurve!

  17. There are Many Possible Paths in the P-V Planeto 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!

  18. Figures (c), (d), (e), (f): 4 more of the Many Possible Processes!

  19. Section 2.11: Brief Math Discussion: Exact & Inexact Differentials

  20. We’ve seen that, for infinitesimal, quasi-static processes, the First Law of Thermodynamicsis đQ = dĒ + đW dĒis an Exact Differential. đQ, đWare Inexact Differentials. • To understand what an Inexact Differential is, it helps to first briefly review what is meant by an Exact Differential

  21. 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 • 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 2nd cross partial derivatives MUST be equal: • (∂2F/∂x∂y)≡ (∂2F/∂y∂x)

  22. The Exact Differential of the function F(x,y) 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. • As just discussed, if F(x,y) is an analytic function, then it must be true that: • (∂A/∂y)x ≡ (∂B/∂x)y • or • (∂2F/∂x∂y)≡ (∂2F/∂y∂x)

  23. It can further be shown that, if F(x,y) is an analytic function, the integral of dF between any 2 arbitrary points 1 & 2 in thexyplane is • Independent of the path • between 1 & 2.

  24. Also, it can be shown that, for an • Arbitrary Analytic Function F(x,y): • The integral of dFover an arbitrary closed path in the x-y plane is zero. = 0

  25. 3 Tests for an Exact Differential F(x,y) =arbitrary analytic function of x&y.

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

  27. Summary: The Differential dF = Adx + Bdy is Exact if: or (∂2F/∂x∂y)≡ (∂2F/∂y∂x)

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