Interaction Between Macroscopic Systems Were focusing on isolated

Interaction Between Macroscopic Systems

• We’re 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 ~ 1024) where f is the number of degrees of freedom of the system. • Now, we want to focus on how to characterize the macroscopic properties of the system & to do this especially when it is not isolated, but is allowed to interact with another macroscopic system “the outside world”.

• Characterizing 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 macroscopic properties, we specify it’s Average Energy Ē Along with some (usually a small number n) of measurable External Parameters: x 1, x 2, x 3, …xn Some examples of a system’s external parameters are Volume, Applied Electric field, Applied Magnetic field, etc. – Of course, the quantum mechanical energy levels of this system depend on these external parameters, through the equations of motion.

• The energy of a system’s many body, quantum mechanical Microstate, labeled r, is specified by it’s quantized energy levels 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 the previous discussion, these are characterized by Ω(E) = AEf

• Now, consider 2 macroscopic systems A & A', interacting with each other & in thermal equilibrium (we still haven’t yet rigorously defined 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. An ensemble of similar systems is shown schematically in the figure. We focus attention on system A. A A'

• There are 2 Kinds of Interactions between systems. These are: 1. Thermal Interactions – The external parameters x 1, x 2, x 3, …xn remain fixed The quantum energy levels Er(x 1, x 2, …xn) are unchanged. – However, the POPULATIONS of these levels change, so the occupation probability of these levels changes. 2. Mechanical Interactions – The 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 Interaction • Consider 2 macroscopic systems A & A', interacting with each other & in thermal equilibrium. Consider the case where there is Thermal interaction only, & no mechanical interactions. See the Figure A A' • Now, focus on system A with mean internal energy Ē(x 1, x 2, x 3, …xn) No mechanical interaction means that all external parameters x 1, x 2, x 3, …xn remain fixed (so that no 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' = const.

• 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 ≡ the heat absorbed (or emitted) by A (Q can be positive or negative) – Similarly, for , A', the mean energy 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'. This is just Conservation of Total Energy for the combined system!

Section 2. 7: Mechanical Interaction • Consider again 2 macroscopic systems A & A', interacting with each other & in thermal equilibrium. Consider the case where there is Mechanical Interaction only, & no thermal interactions. For this to occur, they need to be 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'. Clearly, an idealization!!! But many materials behave approximately as is shown 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 work can ≡ Mechanical Interaction The be measured In this case, the 2 systems do MECHANICAL WORK on each other.

A Freshman Physics Example!! • A gas is enclosed in vertical cylinder closed by a piston of weight W. The piston is thermally insulated from the gas. Initially, the piston is clamped in position at height si. It is released & the height changes to some final height sf s (higher or lower than si). System A gas + cylinder System A' piston + weight • The interaction between them involves changes in the system’s external parameters (i. e. gas volume & piston height s). Consider system A. Due to mechanical interaction with system A', it’s external parameters change: so does it’s mean energy. Denote this as xĒ. The macroscopic work done ON the system is defined as W xĒ.

Freshman Example, Continued A gas + cylinder, A' piston + weight • Interaction involves changes in the system external parameters (gas volume & piston height s). • A in mechanical interaction with A', it’s external parameters change: it’s mean energy change is xĒ. The macroscopic work done ON the system is: s W xĒ • The macroscopic work done BY the system is defined to be the negative of this: W = -W - xĒ

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

Section 2. 8: General Interaction

Consider 2 macroscopic systems A, A', interacting & in thermal equilibrium. Thermal Interaction: One in which no mechanical work is done. The energy exchange between A, A' is HEAT EXCHANGE: Conservation of energy for the combined system: Ēo = 0 = Ē + Ē' Ē ≡ Q = heat absorbed (emitted) by A, Ē' ≡ Q' = heat absorbed (emitted) by A' So, Q + Q' = 0; Q = - Q' Mechanical Interaction: One in which A, A', are thermally No heat exchange is possible. They interact by doing MECHANICAL WORK on each other insulated from each other. through changes in their external parameters The macroscopic work done ON A: W xĒ The macroscopic work done BY A: W = -W - xĒ. Conservation of energy for the combined system: 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 Ē = - W + Q or so oo Q = Ē + W • The boxed equation is a statement of THE FIRST LAW OF THERMODYNAMICS The physics of this law is simply Conservation of Total Energy ≡ Heat + Mechanical Energy Could also be viewed as a definition of the heat absorbed (emitted) by a system.

o o. Q = Ē + W THE FIRST LAW OF THERMODYNAMICS The physics of this law is Conservation of Total Energy ≡ Heat + Mechanical Energy • Notice that, for two interacting systems at equilibrium, the 1 st Law says that total energy is conserved, but it says nothing about the DIRECTION of energy transfer between the two systems. To understand that, we will need the 2 nd Law!

Q = Ē + W • A goal of this course is to study this law and to obtain a fundamental understanding of the relation between thermal & mechanical interactions. This type of study is called Classical Thermodynamics Comment • Work, Heat, & Internal Energy obviously all have the same units. The SI units are Joules (J). But, 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 = Ē + W Example: A A' There are several possibilities: Moveable Piston 1. The piston is clamped & thermally insulating. A, A' don’t interact. 2. The piston is clamped & NOT thermally insulating. A, A' interact thermally. Pressures change. Heat Q is exchanged, but no mechanical work W is done. 3. The piston is thermally insulating & free to move. A, A' interact mechanically. Pressures, volumes change. No heat Q is exchanged. Mechanical work W is done. 4. The piston is NOT thermally insulating & is free to move. A, A' interact both thermally & mechanically. Pressures & volumes both change. Heat Q is exchanged & mechanical work W is done.

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 This First Law relation is deceptively simple looking. It is obviously one of the forms of the general Law of Conservation of Energy. BUT! Be careful about the sign conventions! Positive Q is heat transferred to the system. Positive W is work done by the system. In words: The change in the internal energy OF a system depends only on the NET heat transferred to the system and the net work done BY the system, and is independent of the particular processes involved.

1 st Law of Thermodynamics: Q = Ē + W • For infinitesimal changes in system A’s energy, the small change of mean energy resulting from interaction with A' is written dĒ. • Similarly, the infinitesimal amount of heat absorbed by A due to interaction with A' is written đQ & the infinitesimal amount of work done is denoted by đW. • đQ & đW are special symbols which signify that The heat absorbed & the work done are NOT exact differentials. • That is, they are not differentials in the rigorous math sense. • This is in contrast to dĒ, which is an exact differential. • A more detailed discussion of this follows.

First Law of Thermodynamics: Q = Ē + W • For infinitesimal processes, this becomes: đQ = dĒ + đW The meaning of the symbols đQ & đW: • First, note that for any process for which system A starts out in state 1 & ends up in state 2, it makes no sense to write: d. Q ≠ Q 2 – Q 1 (1) • Neither does it make sense to write: d. W ≠ W 2 – W 1 (2) It makes no sense to talk about the work or heat of or in a system. • (1) implies the existence of a “heat function” Q, which depends on the system A properties & that this “heat function” is changed when A moves from macrostate 1 to macrostate 2. • (2) implies the existence of a “work function” W, which depends on the system A properties & that this “workfunction” is changed when A moves from macrostate 1 to macrostate 2.

Section 2. 9: Quasi-Static Processes

• The discussion so far has been general concerning two systems A & A' interacting with each other. Now, lets consider a SIMPLER 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 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.

Quasi-Static Processes Or quasi-equilibrium processes These are sufficiently slow processes, that any intermediate state can be considered as an equilibrium state (the macroscopic parameters are well-defined for all intermediate states). Advantages: The state of a system that participates in such a process can be described with the same (small) number of macro parameters as for a system in equilibrium (e. g. , for a gas, this could be T and P). By contrast, for non-equilibrium processes (e. g. turbulent flow of gas), we need a huge number of macro parameters. Examples of quasiequilibrium processes: isochoric: isobaric: isothermal: adiabatic: V = const P = const T = const Q=0 For quasi-equilibrium processes, P, V, T are well-defined – the “path” between two states is a continuous lines in the P, P V, T space. 2 T 1 V

Quasi-Static Processes • Let the external parameters of system A have values x 1, x 2, x 3, …xn. • For this case, the energy of the rth quantum state of the system may abstractly be written: Er = Er(x 1, x 2, …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 simple calculus, we can write: d. Er = ∑α(∂Er/∂xα)dxα

d. Er = ∑α(∂Er/∂xα)dxα • 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 = - d. Er = - ∑α(∂Er/∂xα)dxα ≡ ∑αXα, rdxα Here, Xα, r ≡ - (∂Er/∂xα) Xα, r ≡ the Generalized Force associated with the external parameter 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> ≡ the Mean Generalized Force associated with the external parameter xα

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