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The Laws of Thermodynamics - Revisited

Arthur Shavit, Professor Emeritus

Department of Mechanical Engineering

Technion – Israel Institue of Technology

Haifa, ISRAEL

Usual statement

A PMM2 is impossible

A PMM2 is cyclic device that produces positive work while interacting with a single reservoir

Questions:

Is that a law of thermodynamics?

What is a law of thermodynamics?

To answer these question some introduction is needed

The Structure of Thermodynamics

- • Definitions

- • Experimental facts

- • Laws (axioms)

- • Theorems and Corollaries

• Applications

Definitions

- Body

- Environment

- Primitive property

- State

- Allowed states

- Identical states

- System

- Change of state

- Path

- Interaction

- Process

- Cycle

BodyA body is a part of space enclosed by a well defined boundary.

The boundary may be physical or mathematical, fixed or changing

in time, closed or open to passage of matter.

Environment Everything outside the boundary of the body

Primitive Property

Primitive property of a body is specified by subjecting the body to an operation or a test, that requires no previous knowledge of the body, the result of which at a specific time is the value of the primitive property at that time.

A primitive property may be determined without the need to change the conditions of the body.

State The condition of the body identified by all its primitive properties

Identical statesStates that have the same values of thecorresponding primitive properties.

Change of stateOccurs when the value of at least one primitive property is changed.

Allowed states Allowed states of a body are all the states which the

body may inherently attain consistent with the definition of the body.

An idealization of a body that may includes only part of the allowed states of the body. It is also required that the system could be isolated from its environment

The allowed states may be given as an explicit list. or implicitly, by

describing one state and all possible variations of state.

These variations must be consistent with: 1. the laws of matter,

2. the constraints.

3. the passive resistances.

Closed System A system where matter may not cross its boundary.

Open System A system where matter may cross its boundary.

An observable characteristic of the system, whose change between two end states is independent of the path.

Derived property: A property that is not primitive

Examples: – Ampere-hour on a battery.

– Life time of an incandescent lamp.

- Primitive – Derived

- Extensive – Intensive

- Independent – Dependent

- Conservative – Non conservative

Equilibrium

An equilibrium state is one that can not be changed without a corresponding change in the environment.

4 types according to the changes required in the environment

Unstable

Stable

Neutral

Metastable

Mutual equilibrium

Types of Equilibria

The type of equilibrium is characterized by the required change in the environment for a finite change in the system

Neutral vs Themodynamic property

Neutral Property A property of neutral equilibrium that can change in both directions by only temporary changes in the environment.

Example: The horizontal position of the system in a gravity field.

Substate A state different from others only by neutral properties.

Thermodynamic Property Any property that is not neutral.

Note: A thermodynamic property may have several substates.

Thermodynamic State A state that includes only thermodynamic properties

Work is an interaction between two systems such that whatever happened in each system and its boundary could be repeated exactly while the sole external effect is a change of level of a weight.

Measure of work The work of a system equals the number of weights, in the test, that underwent a unit change of level.

Adiabatic processA process having no interactions other than work.

Dislacement of a wire pulled by a force

Modes of quasistatic work

Change of volume under pressure.

Change of magnetization in magnetic field.

Change of surface area with surface tension .

Etc.

First law

The work of a system undergoing an adiabatic process depends only on the end states.

Energy

A property whose change between two end states is determined by the adiabatic work.

The Laws of Thermodynamics

A law is a generalization of all known experimental facts.

Zeroth Law

Maxwell, 1891

First Law

Clausius, 1850 (Joule, 1848)

Clausius, 1850 (Carnot, 1824)

Second Law

State principle

Kline & Koenig, 1957

Third Law

Nernst, 1906

Work of a system in a stable state

Theorem:

A system, in a stable equilibrium state, cannot change its state while the only external effect is the rise of the level of weight.

Proof

Assume that the theorem is incorrect then there should be at least one case where a the stable state changes while the only external effect is a rise in a level of a weight. It possible to lower the weight and impart a velocity to the system.

In this case the net external effect is zero while the state of the system changed from a stable state to another.

That violates the definition of a stable state.

Quasi-stable State

Some non equilibrium states can be made stable by eliminating some of the allowed stated, while retaining others. This can be achieved by altering passive resistances and constraints.

A state made stable by altering passive resistances and/or constraints is called a quasi-stable state.

The stable state so produced is called corresponding stable state.

Heat Interaction

Heat is an interaction between two systems each in a stable state

with no change in the constraints and the passive resistances.

Heat Interaction between systems not in stable states.

Interaction during which the system vary only through the corresponding stable states.

Zeroth Law

If two systems, A and B, are each in mutual equilibrium with a system C

then they are in mutual equilibrium with each other.

Is that trivial???

Temperature

Thermometer

Temperature is a property that is common to all systems in

mutual equilibrium.

State Principle

The stable state of a system bounded by a fixed boundary and subjected to prescribed force fields is fully determined by its energy.

The state principle fixes the number of independent properties of a system in a stable state. These are the parameters of the boundary and the force fields and the energy.

A closed system that undergoes a cycle while having interactions

Heat Machine

Heat Engine

Heat Pump - Refrigerator

The Second Law

Reservoir A system in a stable state whose temperaturestays constant under finite interactions.

PMM2 A heat engine that communicates with a single reservoir

The Second Law

(Two statements)

• A PMM2 is not possible

• It is not possible to transfer heat from a reservoir at a low

temperature to one at a higher with no other effects

Are these really Laws (axioms)???

(a corollary of the second law)

For a reversible process

Leads to define a property

The Law of Stable Equilibrium

A system having specified allowed states can reach, from any given state, one and only one stable state and leave no effect on the environment.

Gibbs Principle of General Inertia

A finite rate of change (or a finite rate of a rate of change) cannot be stopped by means of infinitesimal alteration in the circumstances.

(J.W. Gibbs, Collected Works, Yale University. Press, Vol. 1 p.56,1948)

Second law

First law

State principle

Law of stable

equilibrium

Zeroth Law

Gibbs Principle

Prague 14.04.2003

Structure of Thermodynamics

Pressure is a thermodynamic property.

weight

A weight is an idealized body whose only independent

property is its level in a gravitational field.

Theorem

A process involving no effects except the lowering

of weights is impossible.

Work Interaction

Work is an interaction between two systems such that whatever happened in each system and its boundary could be repeated exactly while the sole external effect is a change of level of a weight.

Proof of the State Principle

According to the Law of Stable States one and only one stable state is possible for a system of fixed constraints (and passive resistances) that undergoes no interactions (constant energy)

It follows that the stable state is determined by the constraints

Thus

Where is any property of the system in stable equilibrium

are the constraints

are the passive resistances

Entropy

Define entropy

Criterion of Equilibrium

It is necessary and sufficient for equilibrium of an isolated system,

not subdivided by adiabatic walls, that all possible variation in

state satisfy: (dS)E ≤ 0

Stable Equilibrium

For every possible variation for which

Unstable Equilibrium

For at least one possible variation for which

Metastable Equilibrium

For all possible variation for which

And for p.v. smaller than a certain value

And for some possible variation, larger than that value, at least one p.v. is

Proof of the Zeroth Law

Any property

Thus

Consider two systems A and B in mutual equilibrium

39

Krakow 12.09.2011

The Laws of Thermodynamics Revisited

Proof of the Zeroth Law (cont.)

For equilibrium

Temperature

Kelvin scale

For a system in equilibrium with the reservoir

For the triple point of water

41

Krakow 12.09.2011

The Laws of Thermodynamics Revisited

Alternative Criteria of Equilibrium

For all possible variations

to states of

equal E

equal S

}

uniform T

uniform and equal T

}

uniform T and equal T

Alternative Criteria of Equilibrium

For all possible variations

From states of

to states of

}

uniform T

equal S

uniform p

equal p and S

}

uniform p and T

equal p

uniform p and T

equal p and T

Alternative Criteria of Equilibrium

Kelvin scale

For a system in equilibrium with the reservoir

Select

For the triple point of water

Discussions ???

A,aHelmholz free energy xdegree of reaction

E,e energy xpassive resistance

G,gGibbs free energy bconstraint

H,henthalpy mchemical potential

mmass ffugacity

nnumber of moles Wwork

ppressure Qheat

S,sentropy

Ttemperature

U,uinternal energy

V,vvolume

Simple system

A system that has only one boundary quasistatic work parameter.

If the parameter is the volume the system is called

a simple compressible system.

Such a system has exactly 2 independent properties:

the volume and the energy. (V and E)

The volume, v, and energy, e, of a simple system in a stable state are two independent properties.

Namely, any property:

etc.

Solving for e and v yields:

For example if and then

Equation of state

or

U = Internal energy

Internal energy is the functional relation

of two independent properties.

for a stable state.

In general

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