Basic Theoretical Physics: A Concise Overview P35 doc

46.2 On the Impossibility of Perpetual Motion of the Second Kind 3551 It is not possible to construct an ideal heat pump in which heat can flow from a colder body to a warmer body without

By transforming from an extensive expression for the work done, for

ex-ample,

δA = + HdmH + ,

to the corresponding intensive quantity, e.g.,

δA  := δA − d(HmH ) = − mH dH + , one obtains, instead of the Helmholtz free energy F (T , V, m H , N ), the Gibbs free energy

Fg (T , V, H, N ) := F (T , V, m H (T , V, H, N ), N ) − mHH ,1

with

dF g=−pdV + μdN − mH dH − SdT

and the corresponding relations

∂F g

∂H =−mH , ∂F g

∂T =−S ,

etc The second law is now

dF g ≤ δA  − SdT The Gibbs free energy F g (T , V, H, N ), and not Helmholtz free energy

F (T , V, mH, N ), is appropriate for Hamiltonians with a Zeeman term, e.g.,

the Ising Hamiltonian (42.5), because the magnetic field is already taken into account in the energy values Whereas for non-magnetic systems one can show that

F (T , V, N ) ≡ −kBT · ln Z(T, V, N) ,

for magnetic systems

Fg (T , V, H, N ) ≡ −kBT · ln Z  (T , V, H, N ) , with

Z  (T , V, H, N ) ≡

l

e−βE l (V,H,N ) 2

46.2 On the Impossibility of Perpetual Motion

of the Second Kind

The following six statements can be regarded as equivalent formulations of the second law of thermodynamics:

1 Similarly, for magnetic systems one may also distinguish, e.g.,

be-tween a Helmholtz enthalpy IHelmholtz(T, p, m H , N ) and a Gibbs enthalpy

IGibbs(T, p, H, N ).

2

Here the prime is usually omitted

46.2 On the Impossibility of Perpetual Motion of the Second Kind 355

1) It is not possible to construct an ideal heat pump in which heat can flow

from a colder body to a warmer body without any work being done to accomplish this flow This formulation is due to Robert Clausius (1857)

In other words, energy will not flow spontaneously from a low temperature object to a higher temperature object

2) An alternative statement due to William Thomson, later Lord Kelvin,

is essentially that you cannot create an ideal heat engine which extracts

heat and converts it all to useful work

Thus a perpetual motion machine of the second kind, which is a hypotheti-cal device undergoing a cyclic process that does nothing more than convert heat into mechanical (or other) work, clearly does not exist, since it would contradict the second law of thermodynamics On the other hand, as we shall

see later, a cyclically operating (or reciprocating) real heat pump is perfectly feasible, where an amount of heat Q2 is absorbed at a low temperature T2

and a greater amount of heat Q1 = Q2+ ΔQ is given off at a higher tem-perature T1 (> T2), but – to agree with the first law of thermodynamics –

the difference ΔQ must be provided by mechanical work done on the system (hence the term “heat pump”).

3a) All Carnot heat engines3 (irrespective of their operating substance) have

the same maximum efficiency

η := ΔA

Q1

=Q1− Q2

Q1

, where η ≤ T1− T2

T1

is also valid The equal sign holds for a reversible process

The following (apparently reciprocal) statement is also valid:

3b)All Carnot heat pumps have the same maximum efficiency

η  := Q1

ΔA =

Q1

Q1− Q2

, with η  ≤ T1

T1− T2

.

The equal sign again applies for a reversible process

The efficiencies η and η  are defined in a reciprocal way, and in the

op-timum case also give reciprocal values However the inequalities are not reciprocal; the irreversibility of statistical physics expresses itself here, since in both cases we have the inequality sign≤, due to, for example,

frictional losses

Figure 46.1 shows the (T , V )-diagram for a Carnot process Depending on

whether we are considering a Carnot heat engine or a Carnot heat pump

the cycle (which comprises two isotherms and two adiabatics) runs either clockwise or anticlockwise.

3

Usually more practical than Carnot machines are Stirling machines, for which the adiabatics of the Carnot cycle (see below) are replaced by isochores, i.e.,

V = constant.

Fig 46.1 Carnot process in a (V, T )

rep-resentation Carnot cycle (running clock-wise:→ heat engine; running anticlockwise:

→ heat pump) in a (V, T )-diagram with

isotherms at T1 = 2 and T2 = 1.5 (arbi-trary units) and adiabatics T · V 5/3 = 1.5 and 0.5, as for an ideal gas Further

expla-nation is given in the text

We shall now calculate the value of the optimal efficiency for a Carnot cycle

η opt. = 1/η 

opt.

by assuming that the working substance is an ideal gas Firstly, there is an

isothermal expansion from the upper-left point 1 to the upper-right point

1 in Fig 46.1 at constant temperature T1 The heat absorbed during this expansion is Q1 Then we have a further expansion from 1 down to 2, i.e., from the upper-right to the lower-right (i.e., from T1= 2 to T2= 1.5), which occurs adiabatically according to

pV κ = T V κ −1 = const , where κ = C p

Cv =

5

3 , down to point 2 at the lower temperature T2 Subsequently there is an isothermal compression to the left, i.e., from 2  → 2 at T2, with heat

released Q2 Then the closed cycle W is completed with an adiabatic

compression leading from 2 up to the initial point 1 It follows that

W

dU ≡ 0 ;

on the other hand we have

W

dU = Q1− Q2− ΔA , where the last term is the work done by the system.

The efficiency

η := ΔA

Q ≡ Q1− Q2

Q

46.2 On the Impossibility of Perpetual Motion of the Second Kind 357 can now be obtained by calculating the work done during the isothermal expansion

A1→1  =

1



1

pdV ≡ Q1,

since for an ideal gas

ΔU1→1  = 0

Furthermore

1



1

pdV = N kBT1

1



1

dV

V , i.e. Q1= N k B T1ln

V1

V1

.

Similarly one can show that

Q2= N k B T2lnV2

V2

.

For the adiabatic sections of the process

(1 , T

1)→ (2  , T

2) and (2, T2)→ (1, T1)

we have

T1V1κ   = T2V2κ   and T1V1κ  = T2V2κ  ,

where

κ  :=C P

CV − 1 = 2

3 .

We then obtain

V1

V1

=V2

V2

and η = 1 − T2ln

V 2

V2

T1lnV 1

V1

,

finally giving

η = 1 − T2

T1

,

as stated

We shall now return to the concept of entropy by discussing a fourth

equivalent version of the second law:

4) The quantity

S :=

2



1

δQrev

T

is a variable of state (where “rev” stands for “reversible”), i.e., the integral does not depend on the path Using this definition of the state variable

entropy we may write:

dS ≥ δQ

T ,

with the equality symbol for reversible heat flow

We now wish to show the equivalence between 4) and 3) by considering

a Carnot heat engine According to 4) we have

δQ

Q1

T1 − Q2

T2 ,

but from 3) for a reversible process

η = 1 − Q1

Q2 ≡ 1 − T2

T1

, i.e.,

δQrev

T ≡ 0

For an irreversible process the amount of heat given out at the lower

tem-perature T2is greater than in the reversible case Thus

dS > δQ

rev

One may approximate general reciprocating (or cyclic) processes by in-troducing Carnot coordinates as indicated in Fig 46.2 below:

Fig 46.2 Carnot coordinates This (V, T ) diagram shows a single curved line

segment in the upper-right hand part of the diagram and two sets of four horizontal

isotherms and four non-vertically curved adiabatics, respectively, forming a grid

N of so-called Carnot coordinates For an asymptotic improvement of the grid

the single curve is extended to the bounding line ∂G of a two-dimensional region

G, and since the internal contributions cancel each other in a pairwise manner,

one may show analogously to Stokes’s integral theorem that the following is valid:H

∂G

δQrev

Carnot processes∈N

δQrev

T Finally, instead of P

∈N

one may also write the integralR

47 Shannon’s Information Entropy

What actually is entropy ? Many answers seem to be rather vague: Entropy

is a measure of disorder, for example An increase in entropy indicates loss of information, perhaps However the fact that entropy is a quantitative

mea-sure of information, which is very important in both physics and chemistry, becomes clear when addressing the following questions:

– How large is the number N of microstates or configurations of a fluid consisting of N molecules (ν = 1, , N , e.g., N ∼ 1023), where each

molecule is found with a probability p j in one of f different orthogonal states ψ j (j = 1, , f )?

– How large is the number N of texts of length N (= number of digits

per text) which can be transmitted through a cable, where each digit originates from an alphabet{Aj}j=1, ,f of length f and the various A j occur with the probabilities (= relative frequencies) p j?

– How many times N must a language student guess the next letter of

a foreign-language text of length N , if he has no previous knowledge of the language and only knows that it has f letters A j with probabilities

pj , j = 1, , f ?

These three questions are of course identical in principle, and the answer

is (as shown below):

N = 2 N ·I= eN·S kB , where I = 1

NldN

is the so-called Shannon information entropy Apart from a normalization

factor, which arose historically, this expression is identical to the entropy used

by physicists Whereas Shannon and other information theorists used binary logarithms ld (logarithms to the base 2) physicists adopt natural logarithms (to the base e = exp(1) = 2.781 , ln x = (0.693 ) · ldx) Then we have

S

kB =

1

N lnN , which apart from unessential factors is the same as I Proof of the above

relation between N and I (or S

k B) is obtained by using some “permutation

gymnastics” together with the so-called Stirling approximation, i.e., as

fol-lows

The number of configurations is

N1!N2!· · Nf!

because, for a length of text N there are N ! permutations where exchange

gives in general a new text, except when the digits are exchanged amongst each other We then use the Stirling approximation: for

2πN



N e

N

·



1 +O

 1

N



;

i.e., by neglecting terms which do not increase exponentially with N , we may

write

N ! ≈



N e

N

Thus

N ∼=

N

e

(N1+N2+ +N f)

N

1

e

N1

· ·N f

e

N f , or

lnN ∼=−

f



j=1

N j · ln



Nj N



≡ −N

f



j=1

p j · ln pj ,

which gives

N ∼= eN · S

kB , with S

kB ≡ −

f



j=1

pj · ln pj (47.1)

Using the same basic formula for S one can also calculate the thermody-namic entropy S(T ), e.g., with the Boltzmann-Gibbs probabilities

pj= e

−βE j

Z(T ) ,

in agreement with the expressions

Z(T ) =

j

e−βE j , U (T ) = − d ln Z

dβ , F (T ) = −kBT · ln Z(T ) ,

with F (T ) = U (T ) − T · S(T ) , i.e S(T ) ≡ − ∂F

∂T .

The relative error made in this calculation is

O

ln√ 2πN

which for N

47 Shannon’s Information Entropy 361

To end this section we shall mention two further, particularly neat for-mulations of the second law:

5) A spontaneously running process in a closed system can only be reversed

by doing work on the system

This is equivalent to stating:

6) Heat only flows spontaneously from a higher to a lower temperature This last formulation due to Max Planck goes back to Robert Clausius (see above).1 At the same time he recognized that it is not easy to prove the second law in a statistical-physical way This is possible, but only using stochastic methods.2

Work done on a closed system, δA > 0, always leads to the release of heat from the system, δQ < 0, since

dU ( ≡ δA + δQ) = 0

The entropy must therefore have increased along the “ first leg of the cycle ”, since5

dS = 0 The requirement that we are dealing with a closed system, i.e., dU = 0, is thus unnecessarily special, since one can always modify an open system, e.g., with heat input, δQ > 0, to be a closed one by including

the heat source In this sense, closed modified systems are the “ most general ” type of system, and we shall see in the following sections in what ways they may play an important part

1 We should like to thank Rainer H¨ollinger for pointing this out

2

see, e.g., the script Quantenstatistik by U.K.

in Phenomenological Thermodynamics

48.1 Closed Systems and Microcanonical Ensembles

The starting point for the following considerations is a closed system,

corre-sponding to the so-called microcanonical ensemble The appropriate function

of state is S(U, V, N ) The relevant extremum principle is S = max., so that!

ΔS > 0 until equilibrium is reached Furthermore

dS = δQ

rev

dU − δA

dU + pdV − μdN

1

T =



∂S

∂U



V,N

T =



∂S

∂V



U,N

and μ

T =−



∂S

∂N



U,V

The probabilities p j are given by

pj= 1

N , if U − δU < Ej ≤ U otherwise p j = 0 Here, δU is, e.g., an instrumental uncertainty, and

N (= N (U − δU, U))

is the total number of states with

U − δU < Ej ≤ U

We shall see below that the value ofδU U is not significant unless it is extremely small in magnitude

48.2 The Entropy of an Ideal Gas

from the Microcanonical Ensemble

Inserting the relation

pj ≡ N1 for U − δU < Ej ≤ U ,

364 48 Canonical Ensembles in Phenomenological Thermodynamics

otherwise≡ 0, into the relation for S gives

S = kB · ln N , with

N = N (U − δU, U) = V N

h 3N N ! ·

⎧

⎪

⎪



U −δU< p21+ +p2 N

d3N p

⎫

⎪

⎪ .

The braced multidimensional integral in p-space is the difference in volume

between two spherical shells inR 3N (p) with radii

R1(U ) := √

2mU and R2(U − δU) :=2m · (U − δU)

This gives

N = V N

h 3N N ! Ω(3N )(2m)

3N/2 ·"U 3N/2 − (U − δU) 3N/2#

,

where Ω(d) is the volume of a unit sphere in d-dimensional space.

Here it is important to note that the whole second term, (U − δU) 3N/2,

can be neglected compared to the first one, U 3N/2, except for extremely small

δU , since normally we have



U − δU U

3N/2

 1 ,

because N is so large1, i.e., one is dealing with an exponentially small ratio

We therefore have

N ≡ N (U) = V N Ω(3N )(2mU ) 3N/2

S = k BlnN = kB N ·



ln V

V0 +

3

2ln

U

U0+ ln



s0V0U03



.

Here, V0 and U0 are volume and energy units (the exact value is not signifi-cant; so they can be arbitrarily chosen) The additional factor appearing in the formula, the “atomic entropy constant”

s 

0:= ln



s0V0U03



is obtained from the requirement that (for N

N · ln[s0V0U03]= ln!

,

(2m) 3N/2 Ω(3N )

h 3N N ! V

N

0 U 3N2

0

-.

1 E.g., one should replace U by 1 and (U − δU) by 0.9 und study the sequence

(U − δU) N

for N = 1, 2, 3, 4,

6) Heat only flows spontaneously from a higher to a lower temperature This last formulation due to Max Planck goes back to Robert Clausius (see above).1 At the same time he...

47 Shannon’s Information Entropy 361

To end this section we shall mention two further, particularly