Lecture 7: Thermodynamic Potentials Chater II. Thermodynamic Quantities A.G. Petukhov, PHY 743 etember 27, 2017 Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 1 / 11
Equation of tate Parameters,, T (and additional external fields, if any) are thermodynamic arameters that secify a thermodynamic state. In equilibrium there is an equation f(,, T ) 0, which relates three thermodynamic arameters, and T P (, T ). This equation is called the equation of state. For examle: RT is the equation of state for one mole of an ideal gas. Energy (Natural variables and ). Consider the energy differential (first law of thermodynamics for quasi-static rocesses): E E de d + d T d d (1) The energy differential de is an exact differential and we can read off the artial derivatives: Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 2 / 11
Conjugate Thermodynamic ariables ( ) E T T (, ) (2) ( ) E (, ) (3) Therefore if we know E as a function of and we can use Eqs. (1)-(3) to relate four thermodynamic variables,,t, and by means of two equations (2) and (3). If we eliminate we can find the equation of state: (T, ) There are only two indeendent thermodynamic variables It is convenient to treat on the same footing with, and T. The airs (, T ) and (, ) are called conjugate thermodynamic arameters (variables). Each air contains one extensive and one intensive variable Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 3 / 11
Characteristic Functions (Thermodynamic Potentials) Knowledge of E(, ) allows us comletely describe thermodynamics of the system. That is why E is called a characteristic function (thermodynamic otential) and, are called its natural variables. Note that if we knew E as a function of T, we would not be able to obtain the equation of state. Examle: from E 3NT/2 for an ideal gas we cannot find, i.e. it would be imossible to obtain the equation of state. It is ossible to find three more thermodynamic otentials for which (T, ), (, ), and (, T ) are natural variables (arguments). It means that each thermodynamic otential is exressed as an exact differential of its natural arguments with artial derivatives equal to their conjugates. The rocedure which allows us to generate other thermodynamic otentials is called Legendre Transformation Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 4 / 11
Thermodynamic Potentials (TP) and Legendre Transformation Enthaly (Natural ariables and ). We need to erform Legendre transformation from (, ) to (, ): de T d d T d d d + d T d + d d( ) or d(e + ) T d + d W E + Enthaly Thus ( ) W dw T d + d, where T ; Using E and W we can calculate secific heats: E c T c T ) ( ) ( W ( ) W Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 5 / 11
Thermodynamic Potentials cont d Helmholtz Free Energy (Natural ariable T and ). We need Legendre transformation from (, ) to (T, ): de T d d T d d +dt dt dt +d +d(t ) or d(e T ) dt d F E T Helmholtz Free Energy Thus df dt d, where ( ) F T ; ( ) F Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 6 / 11
Thermodynamic Potentials cont d Gibbs Free Energy (Natural ariable T and ). We need Legendre transformation from (T, ) to (T, ): df dt d dt d d+ d dt + d d( ) or d(f + ) dt + d Φ F + Gibbs Free Energy Thus dφ dt + d, where ( ) Φ ; ( ) Φ T Φ F + E T + Remark: In the literature they often use G instead of Φ. Also sometimes they call Φ the Thermodynamic Potential while other three are called by the same name but in general sense. Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 7 / 11
General Diagram We can relate W and Φ as: W Φ + T Φ T Φ T 2 Φ T The following diagram is useful to generate eight Maxwell relations: F E T W Φ T E W T Φ F Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 8 / 11
mall Increment Theorem One can introduce other variables and forces such as de T d d i Λ i dλ i ince Legendre transformations for, T,, do not affect λ, Λ we can obtain Λ i using any thermodynamic otential: E F W Φ Λ λ, λ T, λ, λ T, Thus if λ changes slightly we obtain The small increment theorem: Λdλ (δe), (δf ) T, (δw ), (δφ) T,, Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY27, 743 2017 9 / 11
Minimal alues of TP in Equilibrium We know that T d δq de + d d(t ) de + d + dt,t const d(e T ) d dt df dt 0,T const Therefore at T, const this inequality always holds with sign corresonding to equilibrium. We can obtain similar inequalities E, W and Φ. Thus for all TP we have: de dt df dt 0,const 0,T const dw dt 0,const dφ dt 0,T const Under different external conditions the thermodynamic otentials have minima in equilibrium, i.e. behave analogously to the otential energy in mechanical systems Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY 27, 743 2017 10 / 11
Mixed Derivatives Using the roerties of mixed derivatives: or f(x, y) x y ( ( ) ) f(x, y) x y x y f(x, y) y x ( ( ) ) f(x, y) y x We can find four more Maxwell relations. For instance: ( ( ) ) ( ( ) ) E E Or ( ) ( ) In the same fashion we can roceed with other relations (Read L&L!) y x Chater II. Thermodynamic Quantities Lecture 7: Thermodynamic Potentials A.G. Petukhov, etember PHY 27, 743 2017 11 / 11