Various statements of the law Second law of thermodynamics

1 various statements of law

1.1 carnot s principle
1.2 clausius statement
1.3 kelvin statement
1.4 equivalence of clausius , kelvin statements
1.5 planck s proposition
1.6 relation between kelvin s statement , planck s proposition
1.7 planck s statement
1.8 principle of carathéodory
1.9 planck s principle
1.10 statement system has known expression of internal energy function of extensive state variables

various statements of law

the second law of thermodynamics may expressed in many specific ways, prominent classical statements being statement rudolf clausius (1854), statement lord kelvin (1851), , statement in axiomatic thermodynamics constantin carathéodory (1909). these statements cast law in general physical terms citing impossibility of processes. clausius , kelvin statements have been shown equivalent.

carnot s principle

the historical origin of second law of thermodynamics in carnot s principle. refers cycle of carnot heat engine, fictively operated in limiting mode of extreme slowness known quasi-static, heat , work transfers between subsystems in own internal states of thermodynamic equilibrium. carnot engine idealized device of special interest engineers concerned efficiency of heat engines. carnot s principle recognized carnot @ time when caloric theory of heat considered, before recognition of first law of thermodynamics, , before mathematical expression of concept of entropy. interpreted in light of first law, physically equivalent second law of thermodynamics, , remains valid today. states

the efficiency of quasi-static or reversible carnot cycle depends on temperatures of 2 heat reservoirs, , same, whatever working substance. carnot engine operated in way efficient possible heat engine using 2 temperatures.

clausius statement

the german scientist rudolf clausius laid foundation second law of thermodynamics in 1850 examining relation between heat transfer , work. formulation of second law, published in german in 1854, known clausius statement:

heat can never pass colder warmer body without other change, connected therewith, occurring @ same time.

the statement clausius uses concept of passage of heat . usual in thermodynamic discussions, means net transfer of energy heat , , not refer contributory transfers 1 way , other.

heat cannot spontaneously flow cold regions hot regions without external work being performed on system, evident ordinary experience of refrigeration, example. in refrigerator, heat flows cold hot, when forced external agent, refrigeration system.

kelvin statement

lord kelvin expressed second law as

it impossible, means of inanimate material agency, derive mechanical effect portion of matter cooling below temperature of coldest of surrounding objects.



equivalence of clausius , kelvin statements

derive kelvin statement clausius statement

suppose there engine violating kelvin statement: i.e., 1 drains heat , converts work in cyclic fashion without other result. pair reversed carnot engine shown figure. net , sole effect of newly created engine consisting of 2 engines mentioned transferring heat



Δ
q
=
q

(


1
η


−
1
)



{\displaystyle \delta q=q\left({\frac {1}{\eta }}-1\right)}

cooler reservoir hotter one, violates clausius statement. violation of kelvin statement implies violation of clausius statement, i.e. clausius statement implies kelvin statement. can prove in similar manner kelvin statement implies clausius statement, , hence 2 equivalent.

planck s proposition

planck offered following proposition derived directly experience. regarded statement of second law, regarded starting point derivation of second law.

it impossible construct engine work in complete cycle, , produce no effect except raising of weight , cooling of heat reservoir.



relation between kelvin s statement , planck s proposition

it customary in textbooks speak of kelvin-planck statement of law, example in text ter haar , wergeland. 1 text gives statement planck s proposition, attributes kelvin without mention of planck. 1 monograph quotes planck s proposition kelvin-planck formulation, text naming kelvin author, though correctly cites planck in references. reader may compare 2 statements quoted above here.

the kelvin–planck statement (or heat engine statement) of second law of thermodynamics states that

it impossible devise cyclically operating device, sole effect of absorb energy in form of heat single thermal reservoir , deliver equivalent amount of work.



planck s statement

planck stated second law follows.

every process occurring in nature proceeds in sense in sum of entropies of bodies taking part in process increased. in limit, i.e. reversible processes, sum of entropies remains unchanged.

rather planck s statement of uhlenbeck , ford irreversible phenomena.

... in irreversible or spontaneous change 1 equilibrium state (as example equalization of temperature of 2 bodies , b, when brought in contact) entropy increases.



principle of carathéodory

constantin carathéodory formulated thermodynamics on purely mathematical axiomatic foundation. statement of second law known principle of carathéodory, may formulated follows:

in every neighborhood of state s of adiabatically enclosed system there states inaccessible s.

with formulation, described concept of adiabatic accessibility first time , provided foundation new subfield of classical thermodynamics, called geometrical thermodynamics. follows carathéodory s principle quantity of energy quasi-statically transferred heat holonomic process function, in other words,



δ
q
=
t
d
s


{\displaystyle \delta q=tds}

.

though customary in textbooks carathéodory s principle expresses second law , treat equivalent clausius or kelvin-planck statements, such not case. content of second law, carathéodory s principle needs supplemented planck s principle, isochoric work increases internal energy of closed system in own internal thermodynamic equilibrium.

planck s principle

in 1926, max planck wrote important paper on basics of thermodynamics. indicated principle

the internal energy of closed system increased adiabatic process, throughout duration of which, volume of system remains constant.

this formulation not mention heat , not mention temperature, nor entropy, , not implicitly rely on concepts, implies content of second law. closely related statement frictional pressure never positive work. using now-obsolete form of words, planck himself wrote: production of heat friction irreversible.

not mentioning entropy, principle of planck stated in physical terms. closely related kelvin statement given above. relevant system @ constant volume , mole numbers, entropy monotonic function of internal energy. nevertheless, principle of planck not planck s preferred statement of second law, quoted above, in previous sub-section of present section of present article, , relies on concept of entropy.

a statement in sense complementary planck s principle made borgnakke , sonntag. not offer full statement of second law:

... there 1 way in entropy of [closed] system can decreased, , transfer heat system.

differing planck s foregoing principle, 1 explicitly in terms of entropy change. of course, removal of matter system can decrease entropy.

statement system has known expression of internal energy function of extensive state variables

the second law has been shown equivalent internal energy u being weakly convex function, when written function of extensive properties (mass, volume, entropy, ...).