Interaction Between Macroscopic Systems Weve been focusing on

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) f E δE (f = # of degrees of freedom of 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 Surroundings or The System & The Universe! Abstract Sketch An Example

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.

• We now want to characterize the macroscopic properties of a system, when it isn’t isolated, but 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: x 1, x 2, x 3, …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 x 1, x 2, x 3, …xn • Example 1: System = A mass m (or more than one mass) interacting with its environment: The position coordinates x 1, x 2, x 3 are external parameters.

Examples of External Parameters x 1, x 2, x 3, …xn • Example 1: System = A mass m (or more than one mass) interacting with its environment: The position coordinates x 1, x 2, x 3 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 x 1, x 2, x 3, …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.

More Examples of External Parameters x 1, x 2, x 3, …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(x 1, x 2, …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.

• The situation is shown schematically in the figure, assuming the external parameter is the volume V. A (N, V, E) A' (N', V', E') • Because it is isolated, the combined system A 0 has a constant energy E 0: E 0 = E + E' = constant

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'

• There are 2 General Kinds of Interactions between systems. These are: 1. Thermal Interactions • External parameters x 1, x 2, x 3, …xn remain fixed. The quantum energy levels Er(x 1, x 2, …xn) are unchanged. But, the POPULATIONS of these levels change, so the occupation probability of these levels also changes. 2. Mechanical Interactions • External parameters x 1, x 2, x 3, …xn DO change The quantum energy levels Er(x 1, x 2, …xn) are shifted.

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 A' • We’ll now focus on system A with mean internal energy Ē(x 1, x 2, x 3, …xn)

Thermal Interactions Only: No Mechanical Interactions A' A’s mean internal energy = Ē(x 1, x 2, x 3, …xn) • No mechanical interaction so all external parameters x 1, x 2, x 3, …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: 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 Ē Ē ≡ Q ≡ heat absorbed (emitted) by A (Q can be positive or negative)

• Similarly, for , A', due to its interaction with A, its mean energy change is Ē' ≡ Q' ≡ heat absorbed (or emitted) by A' • Now, consider the total isolated system Ao. • Its subsystems A' & A' are interacting & exchanging energy as described. • Because Ao is isolated, its total energy is conserved. That is: Ēo = Ē + Ē' = const

• Because Ao is isolated, its total energy is conserved. That is: o Ē = Ē + Ē' = const o Ē = 0 = Ē + Ē' Or, from the definition of heat: Q + Q' = 0; Q = - Q' The heat absorbed (given off) by A = - heat given off (absorbed) by A'. Conservation of Total System Energy!!

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”

• 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.

A 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 w s gravity gas • 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.

Freshman Example w continued gravity • 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 s gas • Their interaction involves changes in system’s external parameters (gas volume, piston height s). • A’s external parameters change • The macroscopic work done due to its interaction with A'. BY the system is defined to be So, it’s mean energy change is xĒ. The macroscopic work done the negative of this: ON the system is: W xĒ 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 with no mechanical work. The energy exchange between A, A' is Heat Exchange: • Conservation of energy for system Ao 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

• 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 Q = Ē + W

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

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

Q Q = Ē + W ≡ The st 1 Law of Thermodynamics Physics: Conservation of Total Energy • So, for two interacting systems at equilibrium, The st 1 Law of Thermo says that total energy is conserved, • Note that it says nothing about the DIRECTION of energy transfer!! For that, we will need the nd 2 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.

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

Example Q Q = Ē + W 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

Example Q Q = Ē + W A A' • At least 4 possible cases to consider: 2. The piston is clamped & NOT Moveable Piston 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

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

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

The 1 st Law of Thermodynamics Q = Ē + W Ē = Change in internal energy of the system. Q = NET heat transferred to the system. W = Work done BY the system. • The 1 st 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.

The 1 st 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”.