41 The First and Second Lawsof Thermodynamics 41.1 Introduction: Work An increment of heat absorbed by a system is written here as δQ and not dQ, in order to emphasize the fact that, in
Trang 1310 40 Phenomenological Thermodynamics: Temperature and Heat
and postpone the evaluation of the real part to the very end (i.e., in effect
we omit it) Below the surface z < 0 we assume that
T (z, t) = T ∞ + b1ei(k1z −ω1t) + b2ei(k2z −ω2t) , with complex (!) wavenumbers kj, j = 1, 2 The heat diffusion equation then leads for a given (real) frequency ω1or ω2= 365ω1to the following formula for calculating the wavenumbers:
iω j = D w k2j , giving
D w k j=√ ω
j
√
i
With
√
i = 1 + i√
2
one obtains for the real part k(1)j and imaginary part k(2)j of the
wavenum-ber k j in each case the equation
k j(1)= k j(2)=
√ ω j
√ 2Dw . The real part k1(1)gives a phase shift of the temperature rise in the ground
relative to the surface, as follows: whereas for z = 0 the maximum daytime
temperature average over the year occurs on the 21st June, below the
ground (for z < 0) the temperature maximum may occur much later The imaginary part k1(2) determines the temperature variation below ground;
it is much smaller than at the surface, e.g., for the seasonal variation of
the daytime average we get instead of T ∞ ± b1:
T ± (z) = T ∞ ± e −k(2)
1 |z| · b1;
however, the average value over the year, T ∞ , does not depend on z The
ground therefore only melts at the surface, whereas below a certain depth
|z| c it remains frozen throughout the year, provided that
T ∞+ e−k
(2)
1 |z| c · b1
lies below 0◦ C It turns out that the seasonal rhythm ω1 influences the penetration depth, not the daily time period ω2 From the measured pene-tration depth (typically a few decimetres) one can determine the diffusion
constant D w
2) With regard to Green’s functions, it can be shown by direct differentiation
that the function
G(x, t) := e
− x2
4Dw t
√ 4πDw t
is a solution to the heat diffusion equation (40.3) It represents a special solution of general importance for this equation, since on the one hand
Trang 240.4 Solutions of the Diffusion Equation 311
for t → ∞, G(x, t) propagates itself more and more, becoming flatter and broader On the other hand, for t → 0 + ε, with positive infinitesimal
ε6, G(x, t) becomes increasingly larger and narrower In fact, for t → 0+,
G(x, t) tends towards the Dirac delta function,
G(x, t → 0+)→ δ(t) , since e−
x2
2σ2
√ 2πσ2 for t > 0 gives a Gaussian curve gσ(x) of width σ2 = 2Dw t; i.e., σ → 0 for t → 0,
but always with unit area (i.e.,3∞
−∞
dxg σ (x) ≡ 1 as well as 3∞
−∞
dxx2g σ (x) ≡
σ2 , ∀σ) Both of these expressions can be obtained by using a couple
of integration tricks: the “squaring” trick and the “exponent derivative” trick, which we cannot go into here due to lack of space
Since the diffusion equation is a linear differential equation, the principle
of superposition holds, i.e., superposition of the functions f (x )· G(x −
x , t), with any weighting f (x ) and at any positions x , is also a solution
of the diffusion equation, and thus one can satisfy the Cauchy problem
for the real axis, for given initial condition T (x, 0) = f (x), as follows:
T (x, t) =
∞
−∞
dx f (x )· G(x − x , t) (40.4)
For one-dimensional problems on the real axis, with the boundary condi-tion that for|x| → ∞ there remains only a time dependence, the Green’s
function is identical to the fundamental solution given above If special
boundary conditions at finite x have to be satisfied, one can modify the
fundamental solution with a suitable (more or less harmless) perturbation and thus obtain the Green’s function for the problem, as in electrostatics Similar results apply in three dimensions, with the analogous fundamental solution
G 3d(r, t) =: e−
r2
4Dw t
(4πDw t) 3/2
These conclusions to this chapter have been largely “mathematical” How-ever, one should not forget that diffusion is a very general “physical” process;
we shall return to the results from this topic later, when treating the kinetic theory of gases.
6
One also writes t → 0+
Trang 3
41 The First and Second Laws
of Thermodynamics
41.1 Introduction: Work
An increment of heat absorbed by a system is written here as δQ and not
dQ, in order to emphasize the fact that, in contrast to the variables of state
U and S (see below), it is not a total differential The same applies to work
A, where we write its increment as δA, and not as dA.
Some formulae:
α) Compressional work :
The incremental work done during compression of a fluid (gas or liquid),
is given by δA = −pdV [Work is given by the scalar product of the applied force and the distance over which it acts, i.e., δA = F · dz =
pΔS(2) · −dV/ΔS(2) =−pdV ]
β) Magnetic work :
In order to increase the magnetic dipole moment m Hof a magnetic sample
(e.g., a fluid system of paramagnetic molecules) in a magnetic field H we
must do an amount of work
δA = H · dm H (For a proof : see below.)
Explanation: Magnetic moment is actually a vector quantity like magnetic
fieldH However, we shall not concern ourselves with directional aspects
here Nevertheless the following remarks are in order A magnetic dipole momentm H atr produces a magnetic fieldH:
H (m)(r) = −grad m H · (r − r )
4πμ0|r − r |3 , where μ0 is the vacuum permeability On the other hand, in a field H
a magnetic dipole experiences forces and torques given by
F (m)= (m H · ∇)H and D (m)=m H × H
This is already a somewhat complicated situation which becomes even more complicated, when in a magnetically polarizable material we have
Trang 4314 41 The First and Second Laws of Thermodynamics
to take the difference between the magnetic field H and the magnetic
induction B into account As a reminder, B, not H, is “divergence free”,
i.e.,
B = μ0H + J ,
where J is the magnetic polarization, which is related directly to the
dipole momentm H by the expression
m H=
V
d3r J = V · J ,
where V is the volume of our homogeneously magnetized system.
In the literature we often find the term “magnetization” instead of “magnetic polarization” used for J or even for m H, even though, in the mksA system,
the term magnetization is reserved for the vector
M := J/μ0.
One should not allow this multiplicity of terminology or different conventions
for one and the same quantity dmH to confuse the issue Ultimately it “boils
down” to the pseudo-problem of μ0 and the normalization of volume In any case we shall intentionally write here the precise form
δA = H · dm H (and not = H · dM)
In order to verify the first relation, we proceed as follows The inside of
a coil carrying a current is filled with material of interest, and the work done
on changing the current is calculated (This is well known exercise.) The work
is divided into two parts, the first of which changes the vacuum field energy density μ0H2
2 , while the second causes a change in magnetic moment This is based on the law of conservation of energy applied to Maxwell’s theory For
changes in w, the volume density of the electromagnetic field energy, we have
δw ≡ E · δD + H · δB − j · E · δt ,
where E is the electric field, j the current density and D the dielectric
polarization; i.e., the relevant term is the last-but-one expression,
H · δB , with δB = μ0δ H + δJ ,
i.e., here the term∝ δJ is essential for the material, whereas the term ∝ δH,
as mentioned, only enhances the field-energy
If we introduce the particle number N as a variable, then its conjugated quantity is μ, the chemical potential, and for the work done on our material
system we have:
δA = −pdV + Hdm H + μdN =:
i
f i dX I , (41.1)
where dXi is the differential of the work variables The quantities dV , dmH and dN (i.e., dXi) are extensive variables, viz they double when the system size is doubled, while p, H and μ (i.e., fi) are intensive variables.
Trang 541.2 First and Second Laws: Equivalent Formulations 315
41.2 First and Second Laws: Equivalent Formulations
a) The First Law of Thermodynamics states that there exists a certain vari-able of state U (T , V, m H , N ), the so-called internal energy U (T , X i), such that (neither the infinitesimal increment δQ of the heat gained nor the in-finitesimal increment δA of the work done alone, but ) the sum, δQ + δA, forms the total differential of the function U , i.e.,
δQ + δA ≡ dU = ∂U
∂T dT +
∂U
∂V dV +
∂U
∂m H
dmH+ ∂U
∂N dN ,
or more generally
= ∂U
∂T dT +
i
∂U
∂X i dXi
(This means that the integrals3
W
δQ and 3
W
δA may indeed depend on the integration path W , but not their sum3
W
(δQ + δA) = 3
W
dU For a closed
integration path both5
δQ and5
δA may thus be non-zero, but their sum
is always zero:5
(δQ + δA) =5
dU ≡ 0.) b) As preparation for the Second Law we shall introduce the term irre-versibility: Heat can either flow reversibly (i.e., without frictional heat
or any other losses occurring) or irreversibly (i.e., with frictional heat),
and as we shall see immediately this is a very important difference In
contrast, the formula δA = −pdV + is valid independently of the type
of process leading to a change of state variable
The Second Law states that a variable of state S(T , V, mH , N ) exists, the so-called entropy, generally S(T , Xi), such that
dS ≥ δQ
where the equality sign holds exactly when heat is transferred reversibly What is the significance of entropy? As a provisional answer we could say that
it is a quantitative measure for the complexity of a system Disordered sys-tems (such as gases, etc.) are generally more complex than regularly ordered systems (such as crystalline substances) and, therefore, they have a higher entropy
There is also the following important difference between energy and en-tropy: The energy of a system is a well defined quantity only apart from an additive constant, whereas the entropy is completely defined We shall see later that
S
k B ≡ −
j
p j · ln p j , where pj are the probabilities for the orthogonal system states, i.e.,
p j ≥ 0 and p j ≡ 1
Trang 6316 41 The First and Second Laws of Thermodynamics
We shall also mention here the so-called Third Law of Thermodynamics
or Nernst’s Heat Theorem: The limit of the entropy as T tends to zero, S(T → 0, X i), is also zero, except if the ground state is degenerate.1 As
a consequence, which will be explained in more detail later, the absolute
zero of temperature T = 0 (in degrees Kelvin) cannot be reached in a finite
number of steps
The Third Law follows essentially from the above statistical-physical for-mula for the entropy including basic quantum mechanics, i.e., an energy gap
between the (g0-fold) ground state and the lowest excited state (g1-fold) Therefore, it is superfluous in essence However, one should be aware of the above consequence At the time Nernst’s Heat Theorem was proposed (in 1905) neither the statistical formula nor the above-mentioned consequence was known
There are important consequences from the first two laws with regard to the coefficients of the associated, so-called Pfaff forms or first order differential forms We shall write, for example,
δQ + δA = dU =
i
a i(x1, , x f)dxi , with x i = T , V, mH , N
The differential forms for dU are “total”, i.e., they possess a stem function
U (x1, , x f), such that, e.g.,
a i= ∂U
∂x i . Therefore, similar to the so-called holonomous subsidiary conditions in
me-chanics, see Part I, the following integrability conditions are valid:
∂a i
∂x k =
∂a k
∂x i for all i, k = 1, , f Analogous relations are valid for S.
All this will be treated in more depth in later sections We shall begin as follows
41.3 Some Typical Applications: CV and ∂U ∂V;
The Maxwell Relation
We may write
dU (T , V ) = ∂U
∂T dT +
∂U
∂V dV = δQ + δA
1
If, for example, the ground state of the system is spin-degenerate, which
presup-poses that H ≡ 0 for all atoms, according to the previous formula we would then
have S(T → 0) = k B N ln 2.
Trang 741.3 Some Typical Applications: C V and ∂V; The Maxwell Relation 317
The heat capacity at constant volume (dV = 0) and constant N (dN = 0), since δA = 0, is thus given by
C V (T , V, N ) = ∂U (T , V, N )
Since
∂
∂V
∂U
∂T
= ∂
∂T
∂U
∂V
,
we may also write
∂C V (T , V, N )
∂
∂T
∂U
∂V
.
A well-known experiment by Gay-Lussac, where an ideal gas streams out
of a cylinder through a valve, produces no thermal effects, i.e.,
∂
∂T
∂U
∂V
= 0
This means that for an ideal gas the internal energy does not depend on the
volume, U (T , V ) ≡ U(T ) and C V = CV (T ), for fixed N
According to the Second Law
dS = δQ |reversible
dU − δA |rev
dU + pdV
∂U
∂T
dT
T +
∂U
∂V + p
dV
T ,
i.e., ∂S
∂T =
1
T
∂U
∂T and
∂S
∂V =
1
T
∂U
∂V + p
.
By equating mixed derivatives,
∂2S
∂V ∂T =
∂2S
∂T ∂V , after a short calculation this gives the so-called Maxwell relation
∂U
∂V ≡ T ∂p
As a consequence, the caloric equation of state, U (T , V, N ), is not
re-quired for calculating ∂U ∂V ; it is sufficient that the thermal equation of state, p(T , V, N ), is known.
If we consider the van der Waals equation (see below), which is perhaps
the most important equation of state for describing the behavior of real gases:
p = − a
v2+ k B T
v − b ,
it follows with
v := V
N and u :=
U
N that
∂U
∂V =
∂u
∂v = T
∂p
∂T − p = + a
v2 > 0
Trang 8318 41 The First and Second Laws of Thermodynamics
The consequence of this is that when a real gas streams out of a pressurized cylinder (Gay-Lussac experiment with a real gas) then U tends to increase, whereas in the case of thermal isolation, i.e., for constant U , the temperature
T must decrease:
dT
dV |U=− ∂U/∂V
∂U/∂T ∝ − a
v2 .
(In order to calculate the temperature change for a volume increase at
con-stant U we have used the following relation:
0= dU =! ∂U
∂T dT +
∂U
∂V dV ,
and therefore
dT
dV |U=− ∂V ∂U
∂U
∂T
The negative sign on the r.h.s of this equation should not be overlooked.)
41.4 General Maxwell Relations
A set of Maxwell relations is obtained in an analogous way for other extensive
variables Xi (reminder: δA =
i f idXi) In addition to
∂U
∂V = T
∂p
∂T −p we also have ∂U
∂m H
= H −T ∂H
∂T and
∂U
∂N = μ −T ∂μ
∂T ,
and in general
∂U
∂X i
= f i − T ∂f i
41.5 The Heat Capacity Differences
Cp − CV and CH − Cm
In order to calculate the difference Cp −C V, we begin with the three relations
C p=δQ |p
dT =
dU − δA
dT , dU =
∂U
∂T dT +
∂U
∂V dV and δA = −pdV ,
and obtain: C p= δQ |p
dT =
dU − δA
∂U
∂T +
∂U
∂V + p
dV dT
|p
Therefore,
C p ≡ C V +
∂U
∂V + p
dV dT
|p , or C p − C V =
∂U
∂V + p
dV dT
|p
Trang 941.6 Enthalpy and the Joule-Thomson Experiment; Liquefaction of Air 319 Using the above Maxwell relation we then obtain
C p − C v = T ∂p
∂T ·
dV dT
|p
, and finally C p − C V =−T ∂p
∂T ·
∂p
∂T
∂p
∂V
Analogously for CH − C mwe obtain:
C H − C m = T ∂H
∂T ·
∂H
∂T
∂H
∂m H
.
These are general results from which we can learn several things, for
example, that the difference Cp − C V is proportional to the isothermal com-pressibility
κ T :=−1
V
∂V
∂p . For incompressible systems the difference Cp − C V is therefore zero, and for solids it is generally very small
In the magnetic case, instead of compressibility κT , the magnetic suscep-tibility χ = ∂m H
∂H is the equivalent quantity
For an ideal gas, from the thermal equation of state,
p = N
V k B T ,
one obtains the compact result
C p − C V = N kB (Chemists write: Cp − C V = nMolR0) It is left to the reader to obtain the analogous expression for an ideal paramagnetic material
41.6 Enthalpy and the Joule-Thomson Experiment; Liquefaction of Air
In what follows, instead of the internal energy U (T , V, ) we shall introduce
a new variable of state, the enthalpy I(T , p, ) This is obtained from the
internal energy through a type of Legendre transformation, in a similar way
to the Hamilton function in mechanics, which is obtained by a Legendre transform from the Lagrange function Firstly, we shall write
I = U + p · V and then eliminate V using the thermal equation of state
p = p(T , V, N, ) ,
Trang 10320 41 The First and Second Laws of Thermodynamics
resulting in I(T , p, mH , N, ) We can also proceed with other extensive variables, e.g., mH, where one retains at least one extensive variable, usually
N , such that I ∝ N, and then obtains “enthalpies” in which the extensive variables Xi(= V , mH or N ) are replaced wholly (or partially, see above) by the intensive variables fi = p, H, μ etc., giving the enthalpy
I(T , p, H, N, ) = U + pV − m H H ,
or generally
I(T , f1, , f k , X k+1 , ) = U (T , X1, , X k , X k+1 , ) −
k
i=1
f i X i
We can proceed in a similar manner with the work done Instead of ex-tensive work
δA =
i
f idXi ,
we define a quantity the intensive work
δA := δA − d
i
f i X i ,
such that
δA =−
i
X idfi
is valid The intensive work δA is just as “good” or “bad” as the extensive work δA For example, we can visualize the expression for intensive work
δA = +V dp
by bringing an additional weight onto the movable piston of a cylinder con-taining a fluid, letting the pressure rise by moving heavy loads from below onto the piston As a second example consider the magnetic case, where the expression for intensive magnetic work
δA =−m H dH
is the change in energy of a magnetic dipole m H in a variable magnetic field,
H → H + dH ,
at constant magnetic moment
From the above we obtain the following equivalent formulation of the first law
A variable of state, called enthalpy I(T , p, H, N, ), exists whose total
differential is equal to the sum
δQ + δA .
... equivalent formulation of the first lawA variable of state, called enthalpy I(T , p, H, N, ), exists whose total
differential is equal to the sum
δQ + ? ?A ...
X idfi
is valid The intensive work ? ?A is just as “good” or “bad” as the extensive work ? ?A For example, we can visualize the expression...
is the change in energy of a magnetic dipole m H in a variable magnetic field,
H → H + dH ,
at constant magnetic moment
From the above we obtain the following