Introduction 1. Foundations of thermodynamics • Thermodynamics: phenomenological description of 1.1. Fundamental thermodynamical equilibrium bulk properties of matter in terms of only concepts a few “state variables” and thermodynamical laws. System : macroscopic entity under consideration. • Statistical physics: microscopic foundation of thermodynamics Environment : world outside of the system (infinite). • ∼ 1023 degrees of freedom → 2–3 state variables! Open system : can exchange matter and heat with the environment. • “Everything should be made as simple as possible, but no simpler” (A. Einstein) Closed system : can exchange heat with the environment while keeping the number of particles fixed. Summary of contents: Isolated system : can exchange neither matter nor heat with the environment. Can (possibly) still do work • Review of thermodynamics by e.g. expanding. • Thermodynamical potentials Thermodynamical equilibrium: • Phase space and probability • No macroscopic changes. • Quantum mechanical ensembles • Uniquely described by (a few) external variables of • Equilibrium ensembles state. • Ideal fluids • System forgets its past: no memory effects, no hysteresis. • Bosonic systems • Often the term global equilibrium is used, as opposed • Fermionic systems to local equilibrium, which is not full equilibrium at all (next page)! • Interacting systems Nonequilibrium: • Phase transitions and critical phenomena • Generally much more complicated than equilibrium state. • Simplest case: isolated systems each in an equilibrium state. • In a local thermodynamical equilibrium small regions are locally in equilibrium, but neighbour regions in different equilibria ⇒ particles, heat etc. will flow. Example: fluid (water) with non-homogeneous temperature. • Stronger nonequilibrium systems usually relax to a local equilibrium. Degrees of freedom (d.o.f.) is the number of quantities needed for the exact description of the microscopic state. Example: classical ideal gas with N particles: 3N coordinates (x,y,z), 3N momenta (px,py,pz). State variables are parameters characterizing the macroscopic thermodynamical state. These are all extensive or intensive: Extensive variable: change value when the size (spatial volume and the number of degrees of freedom) is changed: volume V , particle number N, internal energy U, entropy S, total magnetic moment d3r M. R 1 Intensive variable: independent of the size of the system, and can be determined for every 1 = 2 semimicroscopical volume element: e.g. temperature T , pressure p, chemical potential µ, magnetic field H, ratios of extensive variables like ρ = N/V , s = S/N,.... Conjugated variables: A and B appear in pairs in dp = dU = · · · =0. expressions for the differential of the energy (or I1→2 I1→2 more generally, some state variable), i.e. in (B) The total change of an exact differential is forms ±A dB or ±B dA; one is always extensive independent on the path of integration: and the other intensive. 1 b Example: pressure p and volume V ; change in internal energy U when V is changed (adiabatically, at constant S) is dU = −pdV . a 2 Process is a change in the state. Reversible process: advances via states dU − dU =0, a b infinitesimally close to equilibrium, Z Z so that we can write quasistatically (“slow process”). The direction of a reversible process can be reversed, obtaining 2 the initial state (for system + environment!) U(2) = U(1) + dU Z1 Isothermal process : T constant. Isobaric process : p constant. Exact differentials Isochoric process : V constant. Let us denote by dF¯ a differential which is not necessarily Isentropic or adiabatic process: S constant. exact (i.e. integrals can depend on the path). Assuming Irreversible process is a sudden or spontaneous it depends on 2 variables x, y, the differential change during which the system is far from equilibrium. In the intermediate steps global dF¯ = F1(x, y) dx + F2(x, y) dy state variables (p, T , . .) are usually not well is exact differential if defined. ∂F ∂F Cyclic process consists of cycles which take the 1 = 2 . system every time to its initial state. ∂y ∂x ∂F (x,y) Then ∃F (x, y) so that F1(x, y)= ∂x and ∂F (x,y) 1.2. State variables and exact F2(x, y)= and differentials ∂y Let us suppose that, for example, the state of the system 2 can be uniquely described by state variables T , V ja N. dF¯ = F (2) − F (1) 1 Other state variables are then their unique functions: Z is independent on the path, and integrable. In this case p = p(T,V,N) (x, F1) and (y, F2) are pairs of conjugated variables with U = U(T,V,N) respect to F . Examples: are the following differentials exact? S = S(T,V,N) . dF¯ = y dx + x dy By applying differential calculus, the differential of p, for example, is dF¯ = x dx + x dy ∂p ∂p ∂p All physical state variables are exact differentials! This dp = dT + dV + dN ∂T ∂V ∂N will enable us to derive various identities between state V,N T,N T,V . variables. Integrating factor The differentials of state variables, If dF¯ = F1dx + F2dy is not exact, there exists an dp, dT , dV , . ., are exact differentials. These have the integrating factor λ(x, y) so that in the neighbourhood of following properties the point (x, y) (A) Their total change evaluated over a closed path vanishes: λdF¯ = λF1dx + λF2dy = df 2 is an exact differential. λ and f are state variables. 1.3. Equations of state Example: find λ for the differential Encodes (some of the) physical properties of the equilibrium system. Usually these relate “mechanical” dF¯ = x dx + x dy . readily observable variables, like p, T , N, V ; not “internal” variables like S, internal energy U etc. A typical example: pressure of some gas as a function of T Legendre transformations and density ρ. Legendre transformations can be used to make changes in Some examples: the set of the indepependent state variables. For example, let us look at the function f(x, y) of two variables. We Classical ideal gas denote ∂f(x, y) z = fy = ∂y pV = NkB T and define the function where N = number of molecules T = absolute temperature g = f − yfy = f − yz. −23 kB =1.3807 · 10 J/K = Boltzmann constant. Chemists use often the form (Note: z,y is a congjugated pair with respect to f!) Now pV = nRT dg = df − y dz − z dy = fxdx + fydy − y dz − z dy n = N/N0 = number of moles = fxdx − y dz. R = kB N0 =8.315J/K mol Thus we can take x and z as independent variables of the = gas constant function g, i.e. g = g(x, z). Obviously 23 N0 = 6.0221 · 10 = Avogadro’s number. ∂g(x, z) y = − . If the gas is composed of m different species of molecules ∂z the equation of state is still Corresponding to the Legendre transformation f → g there is the inverse transformation g → f pV = NkBT, f = g − zgz = g + yz. where now m N = Ni i=1 Often needed identities X and Let F = F (x, y), x = x(y,z), y = y(x, z) and z = z(x, y). If we want to give F in terms of (x, z), we can write p = pi, pi = NikBT/V, i X F (x, y)= F (x, y(x, z)). where pi is the partial pressure of the i:th component Applying differential rules we obtain identities Virial expansion of real gases ∂F ∂F ∂F ∂y When the interactions between gas molecules are taken = + ∂x ∂x ∂y ∂x into account, the ideal gas law receives corrections which z y x z are suppressed by powers of density ρ = N/V : ∂F ∂F ∂y = 2 3 ∂z ∂y ∂z p = kB T ρ + ρ B2(T )+ ρ B3(T )+ · · · x x x One can show that Here Bn is the n:th virial coefficient. ∂x 1 = ∂y ∂y z Van der Waals equation ∂x z The molecules of real gases interact ∂x ∂y ∂z = −1 • repulsively at short distances; every particle needs at ∂y ∂z ∂x > z x y least the volume b ⇒ V ∼Nb. and 6 ∂F • attractively (potential ∼ (r/r0) ) at large distances ∂x ∂y due to the induced dipole momenta. The pressure = z . ∂y ∂F decreases when two particles are separated by the z ∂x z attraction distance. The probability of this is ∝ (N/V )2. 3 We improve the ideal gas state equation where ′ ′ p V = NkBT P = electric polarization = atomic total dipole momentum/volume so that D = electric flux density ′ −12 V = V − Nb ǫ0 = 8.8542 · 10 As/Vm p = p′ − aρ2 = true pressure. = vacuum permeability. then In homogenous dielectric material one has (p + aρ2)(V − Nb)= Nk T. B b P = a + E, T Solid substances where a and b are almost constant and a,b ≥ 0. The thermal expansion coefficient Curie’s law 1 ∂V When a piece of paramagnetic material is in magnetic α = p V ∂T field H we write p,N and the isothermal compressibility B = µ0(H + M), 1 ∂V where κ = − T V ∂p T,N M = magnetic polarization of solid materials are very small, so the Taylor series = atomic total magnetic moment/volume B = magnetic flux density V = V0(1 + αpT − κT p) −7 µ0 = 4π · 10 Vs/Am = vacuum permeability. is a good approximation. Polarization obeys roughly Curie’s law Typically ρC −10 M = H, κT ≈ 10 /Pa T −4 αp ≈ 10 /K. where ρ is the number density of paramagnetic atoms and C an experimental constant related to the individual atom. Stretched wire Note: Use as a thermometer: measure the quantity Tension [N/m2] M/H. σ = E(T )(L − L0)/L0, where L0 is the length of the wire when σ = 0 and E(T ) is the temperature dependent elasticity coefficient. Surface tension t n σ = σ 1 − 0 t′ t = temperature ◦C t′ and n experimental constants, < < 1∼n∼2 ◦ σ0 = surface tension when t =0 C.