Interaction Between Macroscopic Systems. We’ve been focusing on isolated Macroscopic Systems. So far, we’ve been interested in the statistical treatment.

Interaction Between Macroscopic Systems

• 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”.

The System & It’s Surroundingsor The System & The Universe!

Abstract Sketch An Example

3 General Kinds of Systems1. 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 orenergy with their surroundings.

• 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.

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.

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 is an external parameter.

• 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

• 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.

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'

• There are 2 General Kinds of Interactionsbetween 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.

• 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

Section 2.6: Thermal Interactions

A'

A'

• We’ll now focus on system A with mean internal energy Ē(x1,x2,x3,…xn)

Thermal Interactions OnlyNo mechanical interactions F

A

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,…nThe total system Ao = A + A', is isolated & at equilibrium. So The energy of system Ao is conserved, so

Ēo = Ē + Ē' = constant

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 ĒĒ ≡ 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!!

• 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”

Section 2.7: Mechanical Interactions

• 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!

• 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'

• 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.

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'

• 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.

s gas

w gravity

Freshman Examplecontinued

• 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 changesin the system’s external parameters

(gas volume, piston height s).

s gas

w gravity

• 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

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!

Section 2.8: General Interaction

• 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'

• 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

Q

Q = Ē + W

• 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

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

Q = Ē + W

• This result is a statement of

The First Law of Thermodynamics The physics of this law is simplyConservation 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.

Q

Q = Ē + W

The First Law of ThermodynamicsThe physics of this law is

Conservation of Total EnergyTotal Energy ≡ Heat + Mechanical Energy

• Note that, for two interacting systems at equilibrium,

The 1st Law of Thermodynamics says thattotal 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!

Q

Q = Ē + W

• 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

Q

Q = Ē + W

Comments on Units• 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.

Q

Q = Ē + W

• 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

Q = Ē + W

• Now, we’ll analyze this system using the 1st Law of thermodynamics in 4 different situations.

Example

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

Q

Q = Ē + W

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

Q = Ē

A A'

• At least 4 possible cases to consider:3. The piston is thermally insulating &

free to move. A, A' interact mechanically. So, the pressures &

Q

Q = Ē + W

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

0 = Ē + W

Example

Moveable Piston

A A'

• At least 4 possible cases to consider:4. The piston is NOT thermally

insulating & is free to move. A, A' interact both thermally &mechanically. So, the pressures & volumes both change. A, A' exchange heat Q, and one of them does mechanical work W on the other.

Q

Q = Ē + W

Moveable Piston

Q

Q = Ē + W Example

The First Law of ThermodynamicsQ = 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’sobviously 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.

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.

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'.

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

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!

Differential Form of the 1st Law: đQ = dĒ + đWMeaning 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 totalk about “the heatof or in a system”!

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 totalk about “the workof or in a system”!

The 1st Law of Thermodynamics: Summary

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.

Q

Q = Ē + W For Macroscopic Processes:

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.

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 obeyThe 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)