1. Interaction Between Macroscopic Systems

2. (f = # of degrees of freedom of the system ~ 1024).
We’ve been focusing on isolated Macroscopic Systems. So far, we’ve been interested in the statistical treatment of the dependence of the number of accessible states Ω(E) on the system energy E. We’ve found that
Ω(E)  Ef δE
(f = # of degrees of freedom of the system ~ 1024).
Now, we want to focus on how to characterize the Macroscopic properties of the system & to do this especially when it isn’t isolated, but is allowed to interact with another macroscopic system
 “The Outside World”.

3. The System & It’s Surroundings
or The System & The Universe!
Abstract Sketch An Example

4. 3 General Kinds of Systems
1. Open Systems
Systems that can exchange both matter & energy with their surroundings.
2. Closed Systems
Systems that can exchange energy with
their surroundings, but not matter.
3. Isolated Systems:
Systems that do not exchange matter or
energy with their surroundings.

5. External Parameters: x1,x2,x3,…xn
We now want to characterize the macroscopic properties of a system, when it isn’t isolated, but it is interacting with another macroscopic system
 “The Outside World”.
To describe the system’s properties, we specify it’s
Average Energy Ē
Plus some (usually a small number n) of measurable
External Parameters: x1,x2,x3,…xn
Of course, the quantum mechanical energy levels of the system depend on these external parameters, through the equations of motion.

6. Examples of External Parameters x1,x2,x3,…xn
Example 1: System = A mass m (or more than one mass) interacting with its environment:
The position coordinates x1,x2,x3
are external parameters.
Example 2: System = A gas confined to a container:
The container volume V
is an external parameter.

7. More Examples of External
Parameters x1,x2,x3,…xn
Example 3: System = An electric charge q (or more than one charge) interacting with its environment:
An Applied Electric Field E
is an external parameter.
Example 4: System = A magnetic dipole  (or more than one dipole):
An Applied Magnetic Field B

8. Er(x1,x2,…xn) Average Energy Ē. Ω(E) = AEf
The energy of a system’s many body, quantum mechanical Microstate, labeled r, is specified by it’s quantized energies:
Er(x1,x2,…xn)
The Macrostate of the same system can be Defined by specifying the system’s
Average Energy Ē.
For an ensemble of similar systems, all in the same Macrostate, we can find any one of these systems in a HUGE NUMBER of different Microstates.
From our previous discussion, these are characterized by
Ω(E) = AEf

9. Interaction  Exchange of Energy.
Consider 2 macroscopic systems A & A', interacting with each other & in thermal equilibrium (note that we still haven’t yet rigorously defined thermal equilibrium!). It is reasonable that
Interaction  Exchange of Energy.
We assume that the total system
Ao = A + A', is isolated & at equilibrium. A & A' are interacting.

10. Interaction  Exchange of Energy.
Assume that the total system Ao = A + A', is isolated & at equilibrium. A & A' are interacting. An ensemble of similar systems is shown schematically in the figure. We focus attention on system A. A A' A A' A A'

11. 2. Mechanical Interactions
There are 2 General Kinds of Interactions
between systems. These are:
1. Thermal Interactions
External parameters x1,x2,x3,…xn remain fixed.
The quantum energy levels Er(x1,x2,…xn)
are unchanged.
But, the POPULATIONS of these levels change, so the occupation probability of these levels also changes.
2. Mechanical Interactions
External parameters x1,x2,x3,…xn DO change
 The quantum energy levels Er(x1,x2,…xn)
are shifted.

12. Section 2.6: Thermal Interactions
Consider 2 macroscopic systems A & A', interacting with each other & in thermal equilibrium. Consider the case where there are Thermal Interactions Only, no mechanical interactions. See the Figure A A'
We’ll now focus on system A with mean internal energy Ē(x1,x2,x3,…xn)

13. Thermal Interactions Only
No mechanical interactions F A A'
A’s mean internal energy = Ē(x1,x2,x3,…xn)
No mechanical interaction so all external parameters x1,x2,x3,…xn remain fixed (no mechanical work is done!)
 xi = constant, i = 1,…n
The total system Ao = A + A', is isolated & at equilibrium. So
 The energy of system Ao is conserved, so
Ēo = Ē + Ē' = constant

14. A Very Important Definition!
Heat ≡ The mean energy transferred from one system to another as a result of a purely thermal interaction.
More precisely, due to it’s interaction with A', the mean energy of A is changed by Ē
Ē ≡ Q ≡ heat absorbed (or emitted) by A
(Q can be positive or negative)
Similarly, for , A', the mean energy change is
Ē' ≡ Q' ≡ heat absorbed (or emitted) by A'
Ēo = Ē + Ē' = const  Ēo = 0 = Ē + Ē'
Or Q + Q' = 0; Q = - Q'
 The heat absorbed (given off) by A
= - heat given off (absorbed) by A'.
Conservation of Total System Energy!!

15. Section 2.7: Mechanical Interactions
Consider again 2 macroscopic systems A & A', interacting with each other & in thermal equilibrium.
Consider the case where there are Mechanical Interactions only, & no Thermal Interactions. This requires that they are thermally isolated (insulated) from each other. This is achieved by surrounding the systems with an
“Adiabatic Envelope”

16. Macroscopic systems A & A', interacting & in equilibrium
Macroscopic systems A & A', interacting & in equilibrium. Mechanical Interactions only, no Thermal Interactions. This requires them to be completely thermally isolated (insulated) from each other. This is achieved by surrounding the systems with an
“Adiabatic Envelope” .
≡ An ideal partition which separates the 2 systems A & A', in which external parameters are fixed & each of which is in internal equilibrium, such that each subsystem remains in its Macrostate indefinitely. Obviously, this is an idealization!

17. Physically, this Adiabatic Envelope is such that no energy (heat) transfer can occur between the two systems A & A'. This is clearly, an idealization!!! But many materials behave approximately as is shown in the Figure: A' A

18. Changes in their External Parameters ≡ Mechanical Interaction
Adiabatic Envelope  No energy (heat) transfer can occur between the two systems A & A'. Idealization!!! Many materials behave approximately as in the Figure: A A'
When 2 systems, A, A', are thermally insulated from each other, they are STILL capable of interacting. How? Through
Changes in their External Parameters
≡ Mechanical Interaction
In this case, the 2 systems do MECHANICAL WORK on each other & This Work Can Be Measured.

19. Freshman Physics Example!
A gas is enclosed in a vertical cylinder closed by a piston of weight W. The piston is thermally insulated from the gas.
System A  gas + cylinder
System A'  piston + weight
Consider system A which has a mechanical interaction with A' w
gravity s
gas
A’s external parameters change:  So does it’s mean
energy. Call this change xĒ. The macroscopic
work done ON the system is then defined as W  xĒ.

20. W = -W  - xĒ Freshman Example w s gas
continued
Initially, the piston is clamped in position at height si. It is released & the height changes to some final height sf (higher or lower than si).
System A  gas + cylinder
System A'  piston + weight
Their interaction involves changes
in the system’s external parameters
(gas volume, piston height s). w
gravity s
gas
A’s external parameters change
due to its interaction with A'.
So, it’s mean energy change is
xĒ. The macroscopic work done
ON the system is: W  xĒ
The macroscopic work done BY the system is defined to be the negative of this:
W = -W  - xĒ

21. Freshman Example, Continued
Macroscopic work done BY the system: W = -W  - xĒ
Energy Conservation for combined system: W = -W ',
or W + W ' = 0. Mechanical interaction between the systems due to changes in their external parameters causes changes in their energy levels & also changes in their occupancy. See figure!

22. Section 2.8: General Interaction

23. Thermal Interaction  Interaction with no mechanical work.
Consider 2 interacting macroscopic systems A, A', in thermal equilibrium.
Thermal Interaction  Interaction with no mechanical work.
 The energy exchange between A, A' is
Heat Exchange:
Conservation of energy for the combined system gives:
Ēo = 0 = Ē + Ē'
Ē ≡ Q = heat absorbed (emitted) by A
Ē' ≡ Q' = heat absorbed (emitted) by A'
So, Q + Q' = 0 or Q = - Q'

24. Mechanical Interaction  One in which A,
Consider 2 interacting macroscopic systems A, A', in thermal equilibrium.
Mechanical Interaction  One in which A,
A', are thermally insulated from each other.
 No heat exchange is possible.
They interact by doing MECHANICAL WORK on each other through changes in their external parameters.
The work done ON A is W  xĒ.
The work done BY A: W = -W  - xĒ.
Conservation of energy for the combined system gives
xĒo = 0 = xĒ + xĒ' So, W + W' = 0

25. Mechanical Interactions at the same time.
The most general case is one with Thermal &
Mechanical Interactions at the same time.
The external parameters are NOT fixed. A, A' are NOT thermally insulated from each other.
As a result of this General Interaction, the mean energy of A is changed BOTH by a change in external parameters AND by a transfer of thermal energy. This mean energy change may be written:
Ē = W + Q or Ē = - W + Q
Q = Ē + W Q

26. Summary
In the most general case with Thermal & Mechanical Interactions at the same time, the mean energy of A can be changed BOTH by a change in external parameters AND by a transfer of thermal energy. The resulting change in the mean energy of A can be written:
Q = Ē + W Q

27. The First Law of Thermodynamics The physics of this law is simply
Q = Ē + W Q
This result is a statement of
The First Law of Thermodynamics
The physics of this law is simply
Conservation of Total Energy
Total Energy ≡
Heat + Mechanical Energy
This result could also be viewed as a definition of the heat absorbed (emitted) by a system.

28. 2nd Law of Thermodynamics!
Q = Ē + W Q
The First Law of Thermodynamics
The physics of this law is
Conservation of Total Energy
Total Energy ≡ Heat + Mechanical Energy
Note that, for two interacting systems at equilibrium,
The 1st Law of Thermodynamics says that
total energy is conserved,
Note that it says nothing about the DIRECTION of energy transfer between systems. For that, we will need the
2nd Law of Thermodynamics!

29. Classical Thermodynamics
Q = Ē + W Q
In this course we are studying this law in order to obtain a fundamental understanding of the relation between thermal & mechanical interactions.
This type of study is called
Classical Thermodynamics

30. Comments on Units Q = Ē + W
Work, Heat, & Internal Energy obviously all have the same units.
The SI Energy units are Joules (J).
But, the old units for heat are calories (C) & sometimes we’ll use them.
Using calories for heat units is widespread in Biology, the Life Sciences, Medicine, Chemistry & some Engineering disciplines.

31. As a simple Example, consider two gases, A & A' confined to a container & separated by a moveable piston, as shown in the figure. s
Moveable
Piston A' A
Q = Ē + W Q
Now, we’ll analyze this system using the 1st Law of thermodynamics in 4 different situations.

32. That is, nothing happens!
Example
Q = Ē + W Q A A'
At least 4 possible cases to consider:
1. The piston is clamped & thermally insulating.  A, A' don’t interact.
That is, nothing happens!
Moveable
Piston
2. The piston is clamped & NOT thermally insulating.
 A, A' interact thermally. So, the pressures change.
In this case, A, A' exchange heat Q,
but no mechanical work W is done.
Q = Ē Q

33. Example Q = Ē + W 0 = Ē + W At least 4 possible cases to consider:
3. The piston is thermally insulating & free to move.  A, A' interact mechanically. So, the pressures &
Moveable
Piston
the volumes both change. In this case, no heat Q is exchanged between A, A', but one of them does mechanical work W on the other.
0 = Ē + W

34. Example Q = Ē + W Q = Ē + W At least 4 possible cases to consider:
4. The piston is NOT thermally insulating & is free to move.  A, A' interact both thermally &
Moveable
Piston
mechanically. So, the pressures & volumes both change. A, A' exchange heat Q, and one of them does mechanical work W on the other.
Q = Ē + W Q

35. The First Law of Thermodynamics
Q = Ē + W
Ē = Change in the internal energy of the system.
Q = NET heat transferred to the system.
W = Work done BY the system.
The 1st Law is deceptively simple looking. It’s
obviously a form of the general
Law of Conservation of Total Energy.
But
Be careful about the sign conventions!
Positive Q is heat transferred to the system.
Positive W is work done by the system.

36. The First Law of Thermodynamics
In words:
The change in the internal energy OF a system depends only on the NET heat transferred to the system & the net work done BY the system, & is independent of the processes involved.

37. The 1st Law of Thermodynamics
Q = Ē + W
This form is valid for macroscopic processes in which A, A' are interacting with each other.
Now, consider infinitesimal changes in system A’s mean energy dĒ, resulting from interaction with A'.

38. The Differential Form of The 1st Law of Thermodynamics
Let system A’s mean energy undergo an infinitesimal change dĒ as a result of its interaction with A'.
In this process, the infinitesimal amount of heat absorbed by A due to interaction with A' is written as đQ & the infinitesimal amount of work done is written as đW.
So, the Differential Form of the 1st Law of Thermodynamics has the form:
đQ = dĒ + đW

39. Differential Form of the 1st Law
đQ = dĒ + đW
đQ & đW are special symbols which signify that
The heat absorbed & the work done are NOT exact differentials.
A more detailed discussion follows!

40. Differential Form of the 1st Law: đQ = dĒ + đW
Meaning of the Symbols đQ & đW
đQ & đW are symbols indicating that the heat absorbed & the work done are NOT exact differentials. That is, they are NOT differentials in the rigorous math sense. This is a contrast to dĒ, which is an exact differential.
Note: For any process for which system A starts out in state 1 & ends up in state 2,
It makes no sense to write:
dQ = Q2 – Q1 (1)
(1) would (incorrectly!) imply the existence of a “heat function” Q, which depends on system A properties & that this “heat function” is changed when A moves from macrostate 1 to macrostate 2.
It makes no sense to
talk about “the heat
of or in a system”!

41. Differential Form of the 1st Law: đQ = dĒ + đW
Meaning of the Symbols đQ & đW
Similar to the heat exchange discussion, when mechanical work is done & system A starts out in state 1 & ends up in state 2,
It makes no sense to write:
dW = W2 – W1 (2)
(2) would (incorrectly!) imply the existence of a “work function” W, which depends on the system A properties & that this “work function” is changed when A moves from macrostate 1 to macrostate 2.
It makes no sense to
talk about “the work
of or in a system”!

42. The 1st Law of Thermodynamics: Summary Q = Ē + W
For Macroscopic Processes:
Q = Ē + W Q
Some Terminology: Processes & the System Path
Process: System change from an initial state to a final state.
Path: The total of all intermediate steps between the initial state and the final state in a change of state.
Types of Processes
Isobaric: Carried out at constant pressure, p1 = p2 = psur.
Isochoric: Carried out at constant volume, V1 = V2.
Isothermal: Carried out at constant temperature,
T1 = T2 = Tsur.
Adiabatic: Carried out with no heat exchange, Q = 0.
Cyclic: Carried out with the Initial State = the Final State.

43. The Laws of Thermodynamics & Spontaneous & Non-Spontaneous Processes
A brief, somewhat philosophical preview of some of Ch. 3 topics:
Spontaneous Processes
Spontaneous Processes are those that will naturally occur in the absence of external driving forces. Such processes must obey
The 1st Law of Thermodynamics
(Total Energy Conservation)
Example: A ball rolls off a table & falls to the floor.

44. The 1st Law of Thermodynamics The 2nd Law of Thermodynamics
Non-Spontaneous Processes
Non-Spontaneous Processes are those that are the reverse of spontaneous processes.
This does not mean that non-spontaneous processes don’t happen! They just don’t happen by themselves, but they need an outside influence (force) to take place.
Such processes must obey
The 1st Law of Thermodynamics
(Total Energy Conservation)
The 1st Law is a necessary condition, but its not a sufficient condition for them to take place.
As we’ll see in Ch. 3, to take place, they must also obey
The 2nd Law of Thermodynamics
(Increasing Entropy)