1. Section 2.9: Quasi-Static Processes

2. Simpler (Ideal) Special Case
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. Simpler (Ideal) Special Case
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 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.

4. Simpler (Ideal) Special Case
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 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.

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
parameters is 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 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

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 Er also obviously changes. Let each external parameter change by an infinitesimal amount:
xα xα + dxα
How does Er change? From calculus, we know:
dEr = ∑α(∂Er/∂xα)dxα

9. Xα,r ≡ The Generalized Force
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,rdxα Here, Xα,r ≡ - (∂Er/∂xα)
Xα,r ≡ The Generalized Force
associated with the external parameter xα

10. <Xα> ≡ - (∂Ē/∂xα)
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<Xα>dxα
Here,
<Xα> ≡ - (∂Ē/∂xα)
<Xα> ≡ Mean Generalized Force
associated with external parameter xα

11. đW = <p>dV đW = <q>dU đW = <μ>dB đW = <P>dE
Examples: External parameters & mean generalized forces:
External Parameter xα
Mean Generalized Force <Xα>
Macroscopic Work đW = <Xα>dxα
Position Coordinate x
Mean Mechanical Force <Fx>
đW = <Fx>dx
Volume V
Mean Pressure <p>
đW = <p>dV
Electric Potential U
Mean Electric Charge <q>
đW = <q>dU
Magnetic Field B
Mean Magnetic Moment <μ>
đW = <μ>dB
Electric Field E
Mean Electric Dipole Moment <P>
đW = <P>dE

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

13. Quasi-Static Work Done
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
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.

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. F = pA đW = F ds A = Cross sectional area. V = As = gas volume.
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 under the
p(V) vs. V curve!

17. Figs. (a) & (b) are only 2 of the Many Possible Processes!
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!

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

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

20. đQ = dĒ + đW dĒ is an Exact Differential.
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

21. dF(x,y) ≡ A(x,y) dx + B(x,y) dy
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. (∂2F/∂x∂y) ≡ (∂2F/∂y∂x) (∂A/∂y)x ≡ (∂B/∂x)y or
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. Independent of the path
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 the xy plane is
Independent of the path
between 1 & 2.

24. Arbitrary Analytic Function F(x,y):
Also, it can be shown that, 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
All of this stuff is derived in Appendix A of the text book. I will provide you with a copy of the text book (or a photocopy).
It is the property of exactness that makes, e.g. P, V. T, state variables, i.e. how you reach a particular state (the path) is unimportant. Therefore, a state defined by a set of state variables is unique.
Work and heat are not state variables, i.e. a system may do work or exchange heat with its surroundings in many different ways. In other words, if a system starts of in one state, and it performs some work. The final state is not uniquely defined. For instance, if a gas does work adiabatically, it must have cooled (remember the adiabatic expansion demo I did in the first class – see Mocko web page). On the other hand, if the gas did work under isothermal conditions, its temperature does not change. Furthermore, heat must have flowed into the gas to compensate the cooling that would have occurred under adiabatic conditions. We will discuss heat next week.
Oddly enough, there exists an infinite number of ways a system can exchange heat with its surroundings, whilst also doing work, that will take it from one unique state (P1,V1,T1) to another unique state (P2,V2,T2). Therefore, there exists a combination of dW and dQ that is exact, even though they themselves are not!!! This is the first law of thermodynamics which we will discuss in the next chapter.

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
∆Ē =
đQ + đW
does not depend on the path.

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