310 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 ∞ + b 1 e i(k 1 z−ω 1 t) + b 2 e i(k 2 z−ω 2 t) , with complex (!) wavenumbers k j , j =1, 2. The heat diffusion equation then leads for a given (real) frequency ω 1 or ω 2 = 365ω 1 to the following formula for calculating the wavenumbers: iω j = D w k 2 j , 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 (1) j = k (2) j = √ ω j √ 2D w . The real part k (1) 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 k (2) 1 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 ∞ ± b 1 : T ± (z)=T ∞ ± e −k (2) 1 |z| · b 1 ; 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 ·b 1 lies below 0 ◦ C. It turns out that the seasonal rhythm ω 1 influences the penetration depth, not the daily time period ω 2 . 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 − x 2 4D w t √ 4πD w 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 40.4 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 ε 6 , 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 − x 2 2σ 2 √ 2πσ 2 for t>0 gives a Gaussian curve g σ (x)ofwidthσ 2 =2D w t; i.e., σ → 0fort → 0, but always with unit area (i.e., ∞ −∞ dxg σ (x) ≡ 1aswellas ∞ −∞ dxx 2 g σ (x) ≡ σ 2 , ∀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 − r 2 4D w t (4πD w 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 + . 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,andnot 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 H of 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 :seebelow.) Explanation: Magnetic moment is actually a vector quantity like magnetic field H. However, we shall not concern ourselves with directional aspects here. Nevertheless the following remarks are in order. A magnetic dipole moment m H at r produces a magnetic field H: H (m) (r)=−grad m H ·(r − r ) 4πμ 0 |r −r | 3 , where μ 0 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 314 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,notH, is “divergence free”, i.e., B = μ 0 H + J , where J is the magnetic polarization, which is related directly to the dipole moment m H by the expression m H = V d 3 rJ = 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/μ 0 . One should not allow this multiplicity of terminology or different conventions for one and the same quantity dm H to confuse the issue. Ultimately it “boils down” to the pseudo-problem of μ 0 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 μ 0 H 2 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 = μ 0 δH + 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,thechemical 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 dX i is the differential of the work variables. The quantities dV ,dm H and dN (i.e., dX i )areextensive variables, viz they double when the system size is doubled, while p, H and μ (i.e., f i )areintensive variables. 41.2 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 )thesum, δQ + δA, forms the total differential of the function U , i.e., δQ + δA ≡ dU = ∂U ∂T dT + ∂U ∂V dV + ∂U ∂m H dm H + ∂U ∂N dN, or more generally = ∂U ∂T dT + i ∂U ∂X i dX i . (This means that the integrals W δQ and W δA may indeed depend on the integration path W , but not their sum W (δQ + δA)= W dU.Foraclosed integration path both δQ and δA may thus be non-zero, but their sum is always zero: (δQ + δA)= 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,m H ,N) exists, the so-called entropy, generally S(T,X i ), such that dS ≥ δQ T , (41.2) 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 p j are the probabilities for the orthogonal system states, i.e., p j ≥ 0and j p j ≡ 1 . 316 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 (g 0 -fold) ground state and the lowest excited state (g 1 -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 (x 1 , ,x f )dx i , with x i = T,V,m H ,N . The differential forms for dU are “total”, i.e., they possess a stem function U(x 1 , , 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: C V 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. 41.3 Some Typical Applications: C V and ∂U ∂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) ∂T . Since ∂ ∂V ∂U ∂T = ∂ ∂T ∂U ∂V , we may also write ∂C V (T,V,N) ∂V = ∂ ∂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 )andC V = C V (T ), for fixed N . According to the Second Law dS = δQ |reversible T = dU −δA |rev T = dU + pdV T = ∂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, ∂ 2 S ∂V ∂T = ∂ 2 S ∂T∂V , after a short calculation this gives the so-called Maxwell relation ∂U ∂V ≡ T ∂p ∂T − p. (41.3) 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 v 2 + 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 v 2 > 0 . 318 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 v 2 . (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 = − ∂U ∂V ∂U ∂T . (41.4) 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 X i (reminder: δA = i f i dX i ). 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 , andingeneral ∂U ∂X i = f i − T ∂f i ∂T . (41.5) 41.5 The Heat Capacity Differences C p − C V and C H − C m In order to calculate the difference C p −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 dT = ∂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 . 41.6 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 C H − C m we 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 C p −C V is proportional to the isothermal com- pressibility κ T := − 1 V ∂V ∂p . For incompressible systems the difference C p − C V is therefore zero, and for solids it is generally very small. In the magnetic case, instead of compressibility κ T ,themagnetic 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 = Nk B . (Chemists write: C p − C V = n Mol R 0 ). 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, ) , 320 41 The First and Second Laws of Thermodynamics resulting in I(T,p,m H ,N, ). We can also proceed with other extensive variables, e.g., m H , where one retains at least one extensive variable, usually N, such that I ∝ N, and then obtains “enthalpies” in which the extensive variables X i (= V , m H or N) are replaced wholly (or partially, see above) by the intensive variables f i = p, H, μ etc., giving the enthalpy I(T,p,H,N, )=U + pV − m H H, or generally I(T,f 1 , ,f k ,X k+1 , )=U(T,X 1 , ,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 i dX i , we define a quantity the intensive work δA := δA −d i f i X i , such that δA = − i X i df i 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 . . 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 A, andnot as dA. Some. temperature maximum may occur much later. The imaginary part k (2) 1 determines the temperature variation below ground; it is much smaller than at the surface, e.g., for the seasonal variation. 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