Thermodynamics following Callen 1985 book
1—The problem and the postulates 2
Temporal nature of macroscopic measurements • Complete (microscopic) description of (macroscopic) matter: positions and momenta of all molecules … ~10 23 values (~ N A ) • vs. macroscopic coordinates aka thermodynamic coordinates . • Macroscopic measurements are extremely slow on the atomic scale of time (atomic vibrations ~10 − 15 s), and are extremely coarse on the atomic scale of distance (~10 − 9 m). • By definition ( ← macroscopic observations), thermodynamics describes only static states of macroscopic systems. • Regarding coarseness of time scale: ~ N A molecules and their coordinates (positions, momenta) → only few are time-independent. Obvious candidates for thermodynamic variables are total E (energy), total p (momentum), total L (angular momentum). 3
Spatial nature of macroscopic measurements • Macroscopic measurements are extremely slow on the atomic scale of time (atomic vibrations ~10 − 15 s), and are extremely coarse on the atomic scale of distance (~10 − 9 m). • Macroscopic observations, using “blunt” instruments, only sense coarse spatial averages of atomic coordinates. • For example, of all the vibrational modes at various wavelengths (and frequencies), only the longest wavelength mode “survives” the spatial and the time averaging → volume in macroscopic description. • Thermodynamics is concerned with the macroscopic consequences of all those coordinates, that do not appear explicitly in a macroscopic description of a system. • These “hidden” modes may act as repositories of energy. Energy transfer via hidden modes → heat. (Compare to energy transfer associated with a macroscopic coordinate, e.g., changing volume → mechanical work − P dV .) 4
Composition of thermodynamic systems • We (temporarily) restrict our attention to simple systems, defined as systems that are: • macroscopically homogeneous, • isotropic, • uncharged, • large enough to neglect surface e ff ects, • not acted on by electric, magnetic, gravity fields. • Relevant quantities (parameters): • V …volume (m 3 ) … mechanical parameter • N k (k = 1,…r) … number of molecules (in units of moles) in each of the chemically pure components of which the system is a mixture … chemical composition • Both V and N k are extensive parameters = increase in system’s size proportionally increases V and N k Avogadro constant V V N k N k 2 V 6.022 140 76 x 10 23 mol -1 Numerical value 2 N k (exact) Standard uncertainty (exact) Relative standard uncertainty 5 6.022 140 76 x 10 23 mol -1 https://physics.nist.gov/cgi-bin/cuu/Value?na Concise form
Internal energy • Conservation of energy … a principle developed over ~2.5 centuries (1693 Leibnitz through 1930 Pauli) • 1798 Count Rumford … thermal e ff ects as he bore cannons → Humphry Davy, Sadi Carnot, Robert Mayer, James Joule … heat as a form of energy transfer • Macroscopic systems have definite and precise energies, subject to a definite conservation principle. Postulate (0) For a macroscopic system there exists an energy function U , the internal energy . U is an extensive parameter and is measured relative to the energy in a fiducial state, the energy of which is arbitrarily taken as zero. 6
Thermodynamic equilibrium • Macroscopic systems exhibit some memory. This memory eventually fades (fast or very slowly) toward a simple state, independent of memory. • In all systems there is a tendency to evolve toward states in which the properties are determined by intrinsic factors and not by previously applied external influences. Such simple states are, by definition, time independent. They are called equilibrium states. Thermodynamics describes these equilibrium states . • Experimental observations as well as formal simplicity suggest (and ultimately verified by the derived theory’s success): Postulate 1 There exist particular states ( equilibrium states ) of simple systems that, macro- scopically, are characterized completely by the internal energy U , the volume V , and the mole numbers N 1 ,…,N r of the chemical components. • N.B.: • There will be more parameters for more involved systems. • Is system in equilibrium? Quiescence not su ffi cient condition. Inconsistencies with thermodynamics formalism are a sign of non-equilibrium. • equilibrium—ergodicity… • Few real systems (…none?) are in equilibrium. 7
Walls and constraints • Manipulations of a wall (separation from surrounding, boundary condition) of a thermodynamics system → redistribution of a quantity. • Walls can be restrictive or non-restrictive to a change in a particular extensive parameter (i.e., a wall may constrain the parameter or allow it to change). For example, a rigidly fixed piston in a cylinder constitutes a wall restrictive with respect to volume, whereas a movable piston is non- restrictive w.r.t volume. • Walls can be restrictive / non-restrictive w.r.t.: • flow of heat … adiabatic vs. diathermal • flux of matter • doing work Callen calls this “closed” 8
Measurability of the energy • Ok, ∃ U (internal energy). But can we control it a measure it? • Stirring an ice+water system (in a container) causes it to melt faster (transfer of mechanical energy). Shining sun also causes it to melt faster (inflow of heat); changing the walls (e.g., di ff erent material) can decrease the rate of ice melting. • Wall impermeable (restrictive) to heat flow = adiabatic Wall permeable to heat flow = diathermal Wall allowing the flux of neither work nor heat = restrictive w.r.t. energy Wall restrictive w.r.t. U and V and N k = closed • Simple system enclosed in an impermeable adiabatic wall: work is the only permissible type of energy transfer. Since we can quantify mechanical work, we can “measure” the internal energy change of the system. • In summary: There exists walls, called adiabatic, with the property that the work done in taking an adiabatically enclosed system between two given states is determined entirely by the states, independent of all external conditions. The work done is the di ff erence in the internal energy of the two states. Thus, changes in internal energy can be measured. 9
1845 Joule: "The Mechanical Equivalent of Heat" An experiment, in which he specified a numerical value for the amount of mechanical work required to produce a unit of heat … to raise the temperature of a pound of water by 1ºF and found a consistent value of 778.24 foot pound force (4.1550 J·cal − 1 ). Two states, A and B. A → B or B → A possible, but not both. Concept of irreversibility. 10
Quantitative definition of heat • ∆ U between any two equilibrium states is measurable: • The heat flux to a system in any process (at constant mole numbers) is simply the di ff erence in internal energy between the final and initial states, diminished by the work done in that process. • Quasi-static mechanical work: δ W M = − PdV … e.g., compression, displacing a piston in a cylinder … work on the system vs. by the system • Quasi-static heat in an infinitesimal quasi-static process (at constant mole N j ): δ Q = dU − δ W M = dU + PdV δ Q , δ W M … imperfect di ff erentials, depend on process Units of energy, heat, work: joule or J or kg ⋅ m 2 ⋅ s − 2 11
The basic problem of thermodynamics • One central problem that defines the core of thermodynamic theory: The single, all-encompassing problem of thermodynamics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system. Closed composite system Internal constraint Internal constraint removed Initial state Final equilibrium state Rigid free Adiabatic diathermal permeable 12
The entropy maximum postulate • The simplest conceivable formal solution to the basic problem : extremum principle — U , V , N k such that they maximize some function. Postulate 2 There exists a function (called the entropy S ) of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those that maximize the entropy over the manifold of constrained equilibrium states. • N.B.: • ∃ of S only postulated for equilibrium states • If S = S(U,V,N k ) (i.e., the fundamental relation ) is known, problem is solved. 13
Postulate 3 The entropy of a composite system is additive over the constituent subsystems. The entropy is continuous and di ff erentiable and is a monotonically increasing function of the energy. Consequences: S ( α ) • S = P α • S ( α ) = S ( α ) ( U ( α ) , V ( α ) , N ( α ) , . . . , N ( α ) ) r 1 • S of a simple system is a homogeneous first-order function of the extensive parameters: S ( λ U, λ V, λ N k ) = λ S ( U, V, N k ) ∂ S = 1 � � � V,N k > 0 • � ∂ U T • S can be inverted and solved for U : S ( U, V, N k ) → U ( S, V, N k ) . . . An alternative form of the fundamental relation (all info about the system) ⇣ ⌘ N , N k • λ = 1 N , V U N : S ( U, V, N k ) = NS = s ( u, v, X k ) . . . s, u, v, X k are per-mole quantities N (can be also made per-unit-mass aka specific quantites) Postulate 4 The entropy of any system vanishes in the state for which ∂ U/ ∂ S| V,Nk = 0 (that is, at zero temperature). Postulate 4 is an extension, due to Planck, of the so-called Nernst postulate or third law of thermodynamics . 14
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