...Thermodynamics. Entropy: The state function for the Second Law. Entropy ds = d Q. Central Equation du = TdS PdV

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1 ...Thermodynamics Entropy: The state function for the Second Law Entropy ds = d Q T Central Equation du = TdS PdV Ideal gas entropy s = c v ln T /T 0 + R ln v/v 0 Boltzmann entropy S = klogw Statistical Entropy S = k B i p i ln(p i ) October 13, / 25

2 Fewer entropy with chickens Counting Things You can t count a continuum. This includes states which may never be actually realised. This is where the ideal gas entropy breaks down Boltzmann s entropy requires Quantum Mechanics. October 13, / 25

3 Equilibrium and the thermodynamic potentials Second Law : entropy increases in any isolated system. Entropy of system+surroundings must increase. A system can reduce its entropy, provided the entropy of the surroundings increases by more. October 13, / 25

4 Equilibrium and boundary conditions Main postulate of thermodynamics: systems tend to equilibrium. Equilibrium depends on boundary conditions. October 13, / 25

5 A function for every boundary condition Another statement of 2nd Law... System+surroundings maximise S Compare with mechanical equilibrium System minimises energy In thermodynamic equilibrium System minimises... Free energy October 13, / 25

6 Free energy Willard Gibbs : A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Means of Surfaces, 1873 Equation of state plotted as a surface of energy vs T and P. Volume, entropy are slopes of this surface Heat capacity, compressibility are curvatures. October 13, / 25

7 Surface and Slopes: Not like this October 13, / 25

8 Surfaces and slope: More like something from Lovecraft H.P.Lovecraft 1928 The Call of Cthulhu PV or TS space is completely non-euclidean. Einstein 1928: [on general relativity] The fact that the metric tensor is denoted as geometrical is simply connected to the fact that this formal structure first appeared in the area of study denoted as geometry. However, this is by no means a justification for denoting as geometry every area of study in which this formal structure plays a role, October 13, / 25

9 Why Free energy Different axes: different function Unmeasurable entropy axis is awkward Equilibrium depends on the type of boundary conditions. Different Definition of Free energy for every boundary condition. Depends on the dependent variables. Maximum work available from a system when boundaries move. Free energies are all state variables. Also referred to as thermodynamic potentials. October 13, / 25

10 The thermodynamic potentials potential differential natural variables entropy S TdS = du + PdV U, V internal energy U du = TdS PdV S, V enthalpy H H = U + PV dh = TdS + VdP S, P Helmholtz free energy F F = U TS df = PdV SdT T, V the Gibbs free energy G G = H TS dg = VdP SdT T, P October 13, / 25

11 Hamiltonians, Lagrangians and all that Physicists often lazily think we can start with the Hamiltonian ( energy ). This makes a hidden assumption about boundary conditions In reality, careful thought is needed about what energy to use. e.g. Particle moving in a potential: S,V boundary, use U= PE + KE e.g. Air as sound wave passes: S,P boundary, use H e.g. Reagents dissolved in water: T, V boundary, use F e.g. Water exposed to atmosphere : T, P boundary, use G October 13, / 25

12 Maxwell Relations State functions have TWO independent variables. Must be relationships between P, V, T and S. Equation of state is one - material specific. Also general mathematical relationships: Maxwell relations. Essential for most thermodynamic derivations. October 13, / 25

13 Derivation of Maxwell s relations The four Maxwell s relations are identities involving P, V, T and S. They are most conveniently derived from U,H,F,G State functions: second derivatives do not depend on the order of differentiation. e.g. du = TdS -PdV etc. 2 U = 2 U = T = P V S S V S V V S V s S v October 13, / 25

14 Derivation of Maxwell s relations To recall the Maxwell relations from their derivation it can be seen that: 1 The independent (natural) variables of the potential from which each Maxwell relation is derived appear in the denominators of the relation. 2 Cross multiplication of numerators and denominators yields products of pairs of conjugate variables, S T and P V. 3 The sign can be deduced by recourse to the appropriate potential function. October 13, / 25

15 Potentials etc potential differential Maxwell relation internal energy U du = TdS PdV ( T ) V S = ( ) P S V enthalpy H H = U + PV Helmholtz free energy F F = U TS the Gibbs free energy G G = H TS dh = TdS + VdP df = PdV SdT dg = VdP SdT ( T ) P S = ( ) V S P ( P ) T V = ( ) S V T ( V ) T P = ( ) S P T Worth remembering! October 13, / 25

16 TdS equations To relate entropy changes to measurable quantities. ( ) P Tds = c V dt + T dv T v ( ) V Tds = c P dt T dp T P ( ) ( ) T T Tds = c P dv + c V dp V P P V such as Heat capacities, thermal expansion, compressibility and Gay-Lussac coefficient October 13, / 25

17 Gay Lussac Coefficient Gay-Lussac, or Amonton s Law greater pressure causes more hot air k GL = P T k GL = ( ) P T V = ( ) S V T = or maybe not - unspoken isothermal assumption. k GL = ( P T k GL = ) P T S = ( S V ) T Louis Joseph Gay-Lussac; These k GL are properties of the material. October 13, / 25

18 ISO iso- is a prefix from the Greek isos, meaning equal. Equal in all parts/directions isotropic Equal element isotopic Equal Temperature: isothermal Equal Pressure: isobaric Equal Volume: isochoric = isovolumetric Equal Enthalpic: isenthalpic Equal Energy: isoenergetic Equal Entropy: isentropic Equal Heat: adiabatic October 13, / 25

19 Internal Energy for Isochoric Isentropic processes If no work is done (V constant) du = TdS ( ) PdV U = ds + S Equate coefficients to get: 1 T = ( ) U S V 2 P = ( U V TdS equation V ) S, Tds = c V dt + T ( ) U dv V S ( ) P dv T v U = Q (isochoric, no work) note: d q R = TdS for any REVERSIBLE change with no restriction on dv Heat capacity For reversible, isochoric heat flows: C V = d Q V dt = du V dt = ( ) U T V and C V = T ds dt V = T ( ) S T V From ( partial derivatives T ) V S = ( ) P S V, a Maxwell s relation. October 13, / 25

20 Enthalpy for Isobaric Isentropic processes H=U+PV dh = du + PdV + VdP = TdS ( + ) VdP H dh = ds + S P ( ) H dp P S Equate coefficients to get: 1 T = ( ) H S P 2 V = ( ) H P S TdS equation ( ) V Tds = c P dt T dp T P Constant P 0, only P 0 V work: H = Q (P f = P i = P 0 ), dh = TdS for isobaric process. note d q R = T ds for any REVERSIBLE changes with no restriction on dp. Heat capacity C P = d Q P dt = dh P dt = ( ) H T P and C P = T ds P dt = T ( ) S T P From ( partial derivatives T ) P S = ( ) V S P, another Maxwell s relation. October 13, / 25

21 Helmholtz for Isobaric Isothermal processes F=U-TS df = du TdS SdT = PdV SdT F F dv + dt = V T T V Equate coefficients to get: F 1 P = V T F 2 S = T V e.g. Chemicals in a closed container, Finally, second derivatives of F. P S = reactants dispersed in water. T V V T, another Maxwell s relation. October 13, / 25

22 Gibbs for Isobaric isothermal processes. dg = dh TdS SdT = VdP SdT G G = dp + dt P T T P For systems open to pressure transmitting medium, no exchanging material. Equate coefficients to get: G 1 V = P T G 2 S = T P from partial derivatives V S T P = P T, another Maxwell s relation. October 13, / 25

23 System in contact with a T 0 & P 0 reservoir, dg=0 Consider a system held at T 0, P 0, initially out of equilibrium. Heat flows, and work is done as it equilibrates. Change in surroundings S 0 = Q/T 0, V 0 = V = W /P 0 Eventually system reaches equilibrium. free diathermal wall system Q TP 2nd law S sys + S st law Q = U + P 0 V Combined: U + P 0 V T 0 S 0 reservoir T 0 P 0 adiabatic wall October 13, / 25

24 System in contact with itself: dg=0 Combined Laws: U + P 0 V T 0 S 0 free diathermal wall Since P 0 = P i = P f and T 0 = T i = T f U + (P f V f P i V i ) (T f S f T i S i ) 0 U + (PV ) (TS) 0 (U + PV TS) 0 G 0 system Q TP reservoir T 0 P 0 adiabatic wall Any change moving towards equilibrium must reduce G sys Minimising G sys is equivalent to maximising S sys + S 0 : i.e. obeys 2nd law. October 13, / 25

25 Max s Maxwell Mnemonic Good Physicists Have Studied Under Very Fine Teachers Max Born ( , UoE ) P and S have negative signs. Potentials: coefficients in opposite corners, differentials adjacent. Sign goes with coefficient. e.g. df = SdT PdV Maxwell: Use corners, signs and constants from the bottom variables. e.g. ds dp T = dv dt P October 13, / 25