Basic Theoretical Physics: A Concise Overview P32 ppt

322 41 The First and Second Laws of Thermodynamicspass through the cross-sections S i at given positions x i left and right of the throttle valve in a given time interval dt.. In order t

The second law can also be transformed by expressing the entropy (wholly

or partially) as a function of intensive variables: e.g., S(T , p, H, N, ) The

Maxwell relations can be extended to cover this case, e.g.,

∂I

∂p = V − T ∂V

∂T ,

∂I

∂H = T

∂H

∂T − H ,

or, in general

∂I

∂f j

= T ∂X j

We now come to the Joule-Thomson effect, which deals with the station-ary flow of a fluid through a pipe of uniform cross-sectional area (S)1 on the input side of a so-called throttle valve,V thr On the output side there is

also a pipe of uniform cross-section (S)2, where (S)2 = (S)1 We shall see

that, as a result, p1 = p2 We shall consider stationary conditions where the

temperatures T on each side of the throttle valve V thr are everywhere the

same Furthermore the pipe is thermally insulated (δQ = 0) In the region

ofV thr itself irreversible processes involving turbulence may occur However

we are not interested in these processes themselves, only in the regions of stationary flow well away from the throttle valve A schematic diagram is shown in Fig 41.1

We shall now show that it is not the internal energy U (= N · u) of the

fluid which remains constant in this stationary flow process, but the specific enthalpy,

i(x) := I(T , p(x), N )

in contrast to the Gay-Lussac experiment Here x represents the length

co-ordinate in the pipe, where the region of the throttle valve is not included

In order to prove the statement for i(x) we must remember that at any time t the same number of fluid particles

dN1= dN2

Fig 41.1 Joule-Thomson process A fluid (liquid or gas) flowing in a stationary

manner in a pipe passes through a throttle valve The pressure is p1on the left and

p2(< p1) on the right of the valve There are equal numbers of molecules on average

in the shaded volumes left and right of the valve In contrast to p, the temperatures

are everywhere the same on both sides of the valve

322 41 The First and Second Laws of Thermodynamics

pass through the cross-sections S i at given positions x i left and right of the

throttle valve in a given time interval dt Under adiabatic conditions the

difference in internal energy

δU := U2− U1

for the two shaded sets S i · dx i in Fig 41.1, containing the same number of

particles, is identical to δA, i.e.,

δA = −p1· S1· dx1+ p2 · S2· dx2,

since on the input side (left) the fluid works against the pressure while on the output side (right) work is done on the fluid Because of the thermal

isolation, δQ = 0, we thus have:

dU = −p1dV1+ p2dV2≡ (−p1v1+ p2v2)dN ;

i.e., we have shown that

u2+ p2 v2= u1 + p1 v1, or i2= i1 , viz the specific enthalpy of the gas is unchanged.2

The Joule-Thomson process has very important technical consequences, since it forms the basis for present-day cryotechnology and the Linde gas liquefaction process

In order to illustrate this, let us make a quantitative calculation of the Joule-Thomson cooling effect (or heating effect (!), as we will see below for

an important exceptional case):



dT dp



i

=−

∂i

∂p

∂i

∂T

.

(This can be described by − ∂i

∂p

c(0)p

, with the molecular heat capacity c(0)p :=C p

N ;

an analogous relation applies to the Gay-Lussac process.)

Using the van der Waals equation

p = − a

v2 + k B T

v − b ,

see next section, for a real gas together with the Maxwell relation

∂i

∂p = v − T ∂v

∂T and



∂v

∂T



p

=−

∂p

∂T

∂p

∂v

,

2

If the temperature is different on each side of the throttle valve, we have a situ-ation corresponding to the hotly discussed topic of non-equilibrium transport

phenomena (the so-called Keldysh theory ).

one obtains



dT dp



i

c(0)p

2a

k B T

(v −b)2

v2 − b

1− 2a

vk B T

(v −b)2

v2

The first term of the van der Waals equation, − a

v2, represents a negative

internal pressure which corresponds to the long-range attractive r −6 term of

the Lennard-Jones potential (see below) The parameter b in the term k B T

v −b,

the so-called co-volume, corresponds to a short-range hard-core repulsion

∝ r −12:

b ≈ 4πσ3

3 .

σ is the characteristic radius of the Lennard-Jones potentials, which are

usu-ally written as3

VL.-J.(r) = 4ε·σ

r

12

−σ r

6

In the gaseous state, v



dT dp



i

≈ v

c(0)p

T Inv.

T

1− T Inv.

T

,

where the so-called inversion temperature

T Inv.:= 2a

k B v .

As long as T < T Inv., which is normally the case because the inversion tem-perature usually (but not always!) lies above room temtem-perature, we obtain

a cooling effect for a pressure drop across the throttle valve, whereas for

T > T Inv.a heating effect would occur

In order to improve the efficiency of Joule-Thomson cooling, a counter-current principle is used The cooled fluid is fed back over the fluid to be cooled in a heat exchanger and successively cooled until it eventually

lique-fies Air contains 78.08 % nitrogen (N2), 20.95 % oxygen (O2), 0.93 % argon (Ar) and less than 0.01 % of other noble gases (N e, He, Kr, Xe) and

hydro-gen; these are the proportions of permanent gases A further 0.03 % consists

of non-permanent components (H2 O, CO and CO2, SO2, CH4, O3, etc.). The liquefaction of air sets in at 90 K at normal pressure For nitrogen the

liquefaction temperature is 77 K; for hydrogen 20 K and helium 4.2 K The

inversion temperature for helium is only 20 K; thus in order to liquefy helium using this method, one must first pre-cool it to a temperature

T < T Inv. = 20 K (He above 20 K is an example of the “exceptional case” mentioned above.)

3

see (4), Chap 3, in [37]

324 41 The First and Second Laws of Thermodynamics

41.7 Adiabatic Expansion of an Ideal Gas

The Gay-Lussac and Joule-Thomson effects take place at constant U (T , V, N )

and constant

i(T , P ) := U + pV

respectively These subsidiary conditions do not have a special name, but calorific effects which take place under conditions of thermal insulation (with

no heat loss, δQ ≡ 0), i.e., where the entropy S(T, V, N) is constant, are called adiabatic.

In the following we shall consider the adiabatic expansion of an ideal gas.

We remind ourselves firstly that for an isothermal change of state of an ideal

gas Boyle-Mariotte’s law is valid:

pV = N k B T = constant

On the other hand, for an adiabatic change, we shall show that

pV κ = constant , where κ := C p

C V

.

dT dV



S

=−

∂S

∂V

∂S

∂T

=

∂U

∂V + p

C V

,

since

dS = dU − δA

C v dT

∂U

∂V + p dV

Using the Maxwell relation

∂U

∂V + p = T

∂p

∂T ,

dT dV



S

=T

∂p

∂T

C V

.

In addition, we have for an ideal gas:

(C p − C V) =



∂U

∂V + p

 

dV dT



p

≡ p



dV dT



p

= p V

T = N k B ,

so that finally we obtain



dT dV



S

=− C p − C V

C V · T

V .

Using the abbreviation

˜

κ := C p − C V

C V

,

we may therefore write

dT

T + ˜κ

dV

V = 0 , i.e.,

d

ln T

T0 + ln



V

V0

˜κ

= 0 , also

ln T

T0 + ln



V

V0

˜κ

= constant , and

T · V˜κ = constant

With

κ := C p

C V = 1 + ˜κ and p =

N k B T V

we finally obtain:

pV κ = constant

In a V, T diagram one describes the lines

$

dT dV



S

≡ constant

%

as adiabatics For an ideal gas, since ˜ κ > 0, they are steeper than isotherms

T = constant In a V, p diagram the isotherms and adiabatics also form

a non-trivial coordinate network with negative slope, where, since κ > 1, the

adiabatics show a more strongly negative slope than the isotherms This will become important below and gives rise to Fig 46.2

One can also treat the adiabatic expansion of a photon gas in a similar way

42 Phase Changes, van der Waals Theory

and Related Topics

As we have seen through the above example of the Joule-Thomson effect, the van der Waals equation provides a useful means of introducing the topic of liquefaction This equation will now be discussed systematically in its own right

42.1 Van der Waals Theory

In textbooks on chemistry the van der Waals equation is usually written as



p + A

V2



· (V − B) = nMolR0T

In physics however one favors the equivalent form:

p = − a

v2+ k B T

with

a := A

N , v :=

V

N , N k B = nMolR0, b :=

B

N .

The meaning of the terms have already been explained in connection with Lennard-Jones potentials (see (41.8))

Consider the following p, v-diagram with three typical isotherms (i.e., p versus v at constant T , see Fig 42.1):

a) At high temperatures, well above T c, the behavior approximates that of

an ideal gas,

p ≈ k B T

v ,

i.e with negative first derivative

dp

dv ;

but in the second derivative,

d2p

dv2 ,

there is already a slight depression in the region of v ≈ v c, which corre-sponds later to the critical density This becomes more pronounced as the temperature is lowered

b) At exactly T c, for the so-called critical isotherm, which is defined by both derivatives being simultaneously zero:

dp

dv |Tc,pc ≡d2p

dv 2 |Tc,pc≡ 0

c = v −1

c is also determined by T c

and p c

c) An isotherm in the so-called coexistence region,

T ≡ T3< T c

This region is characterized such that in an interval

v1(3)< v < v(3)2

the slope

dp dv

of the formal solution p(T3 , v) of (42.1) is no longer negative, but becomes

positive In this region the solution is thermodynamically unstable, since

an increase in pressure would increase the volume, not cause a decrease

The bounding points v(3)i of the region of instability for i = 1, 2 (e.g., the

lowest point on the third curve in Fig 42.1) define the so-called “spin-odal line”, a fictitious curve along which the above-mentioned isothermal compressibility diverges

On the other hand, more important are the coexistence lines, which are

given by p i (v) for the liquid and vapor states, respectively (i = 1, 2).

These can be found in every relevant textbook

Fig 42.1 Three typical solutions to van

der Waals’ equation are shown, from top

to bottom: p = −0.05/v2

+ T /(v − 0.05) for, (i), T = 4; (ii) T ≡ T c= 3; and, (iii),

T = 2

42.1 Van der Waals Theory 329

The solution for p(T3 , v) is stable outside the spinodal At sufficiently small v, or sufficiently large v, respectively, i.e., for

v < v1liquid or v > v2vapor,

the system is in the liquid or vapor state, respectively One obtains the bound-ing points of the transition,

v1liquid and v2vapor,

by means of a so-called Maxwell construction, defining the coexistence region.

Here one joins both bounding points of the curve (42.1) by a straight line

which defines the saturation vapor pressure p s (T3) for the temperature T =

T3:

p(T3, vliquid1 ) = p(T3 , vvapor2 ) = p s (T3)

This straight line section is divided approximately in the middle by the solu-tion curve in such a way that (this is the definisolu-tion of the Maxwell

construc-tion) both parts above the right part and below the left part of the straight line up to the curve are exactly equal in area The construction is described

in Fig 42.2

This means that the integral of the work done,

vdp or −

pdv ,

for the closed path from

(v1liquid, p s ) ,

firstly along the straight line towards

(vvapor2 , p s)

and then back along the solution curve p(T3 , v) to the van der Waals’ equation

(42.1), is exactly zero, which implies, as we shall see later, that our fluid

Fig 42.2 Maxwell construction The

fig-ure shows the fictitious function p(V ) =

−(V + 1) · V · (V − 1), which corresponds

qualitatively to the Maxwell theory; i.e., the l.h.s and r.h.s represent the marginal values

of the liquid and vapor phases, respectively The saturation-pressure line is represented by the “Maxwell segment”, i.e., the straight line

p ≡ 0 between V = −1 and V = +1, where according to Maxwell’s construction the

ar-eas above and below the straight line on the r.h.s and the l.h.s are identical

system must possess the same value of the chemical potential μ in both the

liquid and vapor states:

μ1(ps , T ) = μ2(ps , T )

If v lies between both bounding values, the value of v gives the ratio of the

fluid system that is either in the liquid or the vapor state:

v = x · vliquid

1 + (1− x) · vvapor

2 , where 0≤ x ≤ 1

Only this straight line (= genuine “Maxwell part”) of the coexistence curve

and the condition μ1 = μ2 then still have a meaning, even though the van der Waals curve itself loses its validity This is due to the increasing influence

of thermal density fluctuations which occur in particular on approaching the critical point

T = T c , p = p c , v = v c Near this point according to the van der Waals theory the following Taylor expansion would be valid (with terms that can be neglected written as + ):

−(p − p c ) = A · (T − T c)· (v − v c ) + B · (v − v c)3+ , (42.2)

since for T ≡ T c there is a saddle-point with negative slope, and for T > T c

the slope is always negative, whereas for p = p s < p c the equation p ≡ p s

leads to three real solutions The coefficients A and B of the above Taylor

expansion are therefore positive

42.2 Magnetic Phase Changes; The Arrott Equation

In magnetism the so-called Arrott equation,

H = A · (T − T0)· m h + B · (m h)3 (42.3)

has analogous properties to the van der Waals equation in the neighborhood

of the critical point, (42.2)

T0is the Curie temperature As in the van der Waals equation, the

coef-ficients A and B are positive For T ≡ T0, i.e., on the critical isotherm,

H = B · (m h)δ , with a critical exponent δ = 3 For T 0 we have

H = A · (T − T0)· m H , i.e., so-called Curie-Weiss behavior Finally, for T < T0 and positive ΔH

we have a linear increase in H, if m H increases linearly from the so-called

spontaneous boundary value m(0) that arises for H = 0+, together with

42.2 Magnetic Phase Changes; The Arrott Equation 331

a discontinuous transition for negative ΔH This corresponds to a coexis-tence region of positively and negatively magnetized domains, respectively, along a straight line section on the x-axis, which corresponds to the Maxwell

straight line, i.e., for

−m(0)

H < m H < m(0)H

Therefore, one can also perform a Maxwell construction here, but the result,

the axis H ≡ 0, is trivial, due to reasons of symmetry However, the fact that the Maxwell line corresponds to a domain structure with H ≡ 0, and

thus a corresponding droplet structure in the case of a fluid, is by no means trivial

The spontaneous magnetization m(0)H follows from (42.3) with H ≡ 0 for

T < T0, and behaves as

m(0)H ∝ (T0− T ) β , with a critical exponent

β = 1

2 .

Finally, one can also show that the magnetic susceptibility

χ = dm H dH

=



dH

dm H

−1

,

is divergent, viz for T > T0 as

A · (T − T0).

On the other hand, for T < T0it behaves as

2A · (T0− T ) ,

i.e., generally as

χ ∝ |T − T0| γ , with γ = 1

Fig 42.3 The Arrott equation The figure

shows three typical solutions to the Arrott equation, which are discussed in the text The

equation is H = A · (T − T0)· M + B · M3

and

is plotted for three cases: (i), T = 3, T0 = 1;

(ii), T = T0= 1, and (iii), T = −0.5, T0 = 1

In all cases we have assumed A ≡ 1

One can also treat the adiabatic expansion of a photon gas in a similar way