Physical significance of entropy or of the second law

by Joseph Frederic Klein

J. F. Klein.

Lehigh University, October, 1910.

TABLE OF CONTENTS

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		PAGE
	INTRODUCTION
	Purpose, acknowledgments, the two methods of approach and outline of treatment	[ 1]
	PART I
	THE DEFINITIONS, GENERAL PRELIMINARIES, DEVELOPMENT, CURRENT AND PRECISE STATEMENTS OF THE MATTERS CONSIDERED
	SECTION A
	(1) The "state" of a body and its "change of state"	[ 5]
	The two points of view; the microscopic and the macroscopic observer; the micro-state and macro-state or aggregate	[ 5]
	The selected and the rejected micro-states; the use of the hypothesis of "elementary chaos"	[ 7]
	PLANCK'S fuller description of what constitutes the state of a physical system	[10]
	(2) Further elucidation of the essential prerequisite, "elementary chaos." Sundry aspects of haphazard	[11]
	BOLTZMANN'S service to science in this field and his view of what constitute the necessary features of haphazard	[12]
	BURBURY'S simplification of haphazard necessary and his example of "elementary chaos"	[15]
	Haphazard as expressed by a system possessing an extraordinary number of degrees of freedom	[17]
	(3) Settled and unsettled states; distinction between final stage of "elementary chaos" and its preceding stages	[18]
	Each stage has sufficient haphazard; examples and characteristics of the settled and unsettled stages of "elementary chaos"; all micro-states not equally likely; the assumed state of "chaos" does not eliminate adequate haphazard; two anticipatory remarks	[19]
	SECTION B
	CONCERNING THE APPLICATION OF THE CALCULUS OF PROBABILITIES
	(1) The probability concept, its usefulness in the past, its present necessity, and its universality	[22]
	Popular objection to its use; Boltzmann's justification of this concept; its usefulness in the past and in other fields; some of its good points; the haphazard features necessary for its use	[23]
	(2) What is meant by probability of a state? Example	[27]
	SECTION C
	(1) The existence, definition, measure, properties, relations and scope of irreversibility and reversibility	[29]
	Inference from experience; inference from the H-theorem or calculus of probabilities; definitions of irreversible and reversible processes; examples of each	[30]
	(2) Character of process decided by limiting states	[32]
	Nature's preference for a state; measure of this preference	[33]
	Entropy both the criterion and the measure of irreversibility	[33]
	(3) All the irreversible processes stand or fall together	[34]
	(4) Convenience of the fiction, the reversible processes	[35]
	Entropy the only universal measure of irreversibility. Outcome of the whole study of irreversibility	[36]
	SECTION D
	(1) The gradual development of the idea that entropy depends on probability or number of complexions	[37]
	Why it is difficult to conceive of entropy. Origin and first definition due to CLAUSIUS; some formulas for it available from the start. Its statistical character early appreciated; lack of precise physical meaning; its dependence on probability; number of complexions a synonym for probability	[37]
	(2) PLANCK'S formula for the relation between entropy and the number of complexions	[40]
	Certain features of entropy	[41]
	SECTION E
	Equivalents of change of entropy in more or less general physical terms or aspects	[41]
	Not surprising that its many forms should have been a reproach to the second law	[41]
	General principles for comparing these aspects. Various aspects of growth of entropy from the experiential and from the atomic point of view	[42]
	SECTION F
	More precise and specific statements of the second law	[44]
	General arrangement and the principles for comparison	[44]
	Ten different statements of the law and comments thereon	[44]
	PART II
	ANALYTICAL EXPRESSIONS FOR A FEW PRIMARY RELATIONS
	Procedure followed	[48]
	SECTION A
	Maxwell's law of distribution of molecular velocities	[48]
	Outline of proof, illustration, and consequences of this law	[48]
	SECTION B
	Simple analytical expression for dependence of entropy on probability	[53]
	PLANCK'S derivation; illustration, limitations, consequences, features and comments	[53]
	SECTION C
	Determination of a precise, numerical expression for the entropy of any physical configuration	[56]
	BOLTZMANN'S pioneer work, PLANCK'S exposition, and the six main steps	[56]
	Step a
	Determination of the general expression for the of a given configuration of a known aggregate state	[57]
	Step b
	Determination of the general expression for the entropy of a given configuration of a known aggregate state	[63]
	Step c
	Special case of (b), namely, expression for the entropy of the state of thermal equilibrium of a monatomic gas	[63]
	Step d
	Confirmation, by equating this value of with that found thermodynamically and then deriving known results	[64]
	Step e
	PLANCK'S conversion of the expressions of (b) and (c) into more precise ones by finding numerical value of	[66]
	Step f
	Determination of the dimensions of the universal constant and therefore also of entropy in general	[67]
	PART III
	THE PHYSICAL INTERPRETATIONS
	SECTION A
	Of the simple reversible operations in thermodynamics
	Isometric, isobaric, isothermal, and isentropic change	[69]
	SECTION B
	Of the fundamentally irreversible processes
	Heat conduction, work into heat of friction, expansion without work, and diffusion of gases	[72]
	SECTION C
	Of negative change of entropy
	Some of its physical features and necessary accompaniments	[78]
	SECTION D
	Physical significance of the equivalents for growth of entropy given on pp. 42-43	[80]
	SECTION E
	Physical significance of the more specific statements of second law given on pp. 44-47	[81]
	PART IV
	SUMMARY OF THE CONNECTION BETWEEN PROBABILITY, IRREVERSIBILITY, ENTROPY, AND THE SECOND LAW
	SECTION A
	(1) Prerequisites and conditions necessary for the application of the theory of probabilities
	(a) Atomic theory; (b) like particles; (c) very numerous particles; (d) "elementary chaos"	[83]
	(2) Differences in the states of "elementary chaos"	[85]
	(3) Number of complexions, or probability of a chaotic state	[86]
	SECTION B
	Irreversibility	[86]
	SECTION C
	Entropy	[87]
	SECTION D
	The Second Law
	Its basis and best statements; it has no independent significance	[88]
	PART V
	REACH OR SCOPE OF THE SECOND LAW
	SECTION A
	Its extension to all bodies
	PLANCK'S presentation; fifteen steps in the proof	[91]
	SECTION B
	General conclusion as to entropy changes	[98]

THE PHYSICAL SIGNIFICANCE OF ENTROPY
AND OF THE SECOND LAW

[There is no difference between change of Entropy and Second Law, when each is fully defined.]

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[INTRODUCTION]
PURPOSE, ACKNOWLEDGMENTS, THE TWO METHODS OF APPROACH
AND OUTLINE OF TREATMENT

THIS article is intended for those students of engineering who already have some elementary knowledge of thermodynamics. It is intended to clear up a difficulty that has beset every earnest beginner of this subject. The difficulty is not one of application to engineering problems, although here too there have been widespread misconceptions,[1] for the expressions developed by CLAUSIUS are simple, have long been known and much used by engineers and physicists. The difficulty is rather as to the ultimate physical meaning of entropy. This term has long been known as a sort of property of the state of the body, has long been surmised to be of essentially a statistical nature, but with it all there was a sense that it was a sort of mathematical fiction, that it was somehow unreal and elusive, so it is no wonder that in certain engineering quarters it was dubbed the "ghostly quantity."