DOCSLIB.ORG

Maxwell Relations Maxwell Relations and R Id L Ti Residual Properties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Maxwell Relations Maxwell Relations and R Id L Ti Residual Properties Maxwell relations and RidlResidual properties 1 B.Srinivas ChEngg GVP CE Maxwell relations and Residual properties Why they have been defined and how do we use them in thermodynamics. Towards the end of the class I am going to show you with three examples how residual ppproperty can be used to obtain ∆U, ∆H and ∆S for a gas obeying the Van der Waals EOS. 2 B.Srinivas ChEngg GVP CE Till date we have seen non flow systems, flow systems and entropy calculations. You would have noticed that everytime we were given P and T and asked to calculate ∆U, ∆H and ∆S. You would start wondering why cant we measure ∆U, ∆H and ∆S just like we measure P and T and solve the ppyppgroblem by skipping these calculations of ∆U, ∆H and ∆S. 3 B.Srinivas ChEngg GVP CE The answer is simple: we just do not have measuring instruments to measure ∆U, ∆H and ∆S as of today. So we measure P and T with instruments and then thermodynamics gives us the relevant equations to estimate ∆U, ∆H and ∆S. 4 B.Srinivas ChEngg GVP CE Take the case of an ideal gas undergoing an isothermal reversible process. We measure P and T at the initial and final states and then thermodynamics gives us the necessary equations to calculate ∆U, ∆HHand and ∆S. UCTV H CTP TV22 SCV ln R ln TV11 constant Cp and Cv TP22 SCp ln R ln TP11 5 B.Srinivas ChEngg GVP CE What about more complicated cases of when the gas is not following the ideal gas behavior and say it is following the van der Waals EOS ? Then how do the equations look like ? How do we estimate ∆U, ∆H and ∆S. 6 B.Srinivas ChEngg GVP CE 7 B.Srinivas ChEngg GVP CE 8 B.Srinivas ChEngg GVP CE 9 B.Srinivas ChEngg GVP CE How entropy changes with P at const T, this ibliiis a problematic quantity since we cannot measure this quantity. But Maxwells reltilation rep laces this quantity with another relation which can be easily obtained . 10 B.Srinivas ChEngg GVP CE This integrand now involves all measurable properties !! 11 B.Srinivas ChEngg GVP CE 12 B.Srinivas ChEngg GVP CE 13 B.Srinivas ChEngg GVP CE 14 B.Srinivas ChEngg GVP CE Another application of Maxwells eqn in throttling process 15 B.Srinivas ChEngg GVP CE 16 B.Srinivas ChEngg GVP CE 17 B.Srinivas ChEngg GVP CE 18 B.Srinivas ChEngg GVP CE Cubic EOS like vdW, PR, SRK or any other EOS can predict the thermodynamic properties of liquids as well as gases if the molecules are non polar like hydrocarbons. But they predict very poorly for polar molecules like alcohols, acids etc. 19 B.Srinivas ChEngg GVP CE 20 B.Srinivas ChEngg GVP CE 21 B.Srinivas ChEngg GVP CE 22 B.Srinivas ChEngg GVP CE 23 B.Srinivas ChEngg GVP CE 24 B.Srinivas ChEngg GVP CE 25 B.Srinivas ChEngg GVP CE 26 B.Srinivas ChEngg GVP CE Since ∆H is a state function we calculate this in a two step process. Approach 1: 27 B.Srinivas ChEngg GVP CE 28 B.Srinivas ChEngg GVP CE 29 B.Srinivas ChEngg GVP CE Approach 2: 30 B.Srinivas ChEngg GVP CE 31 B.Srinivas ChEngg GVP CE 32 B.Srinivas ChEngg GVP CE 33 B.Srinivas ChEngg GVP CE In order to get over this problem we solve the problem in a three step process. Step 1 and 3 are isothermal Step 2 is iso bar ic btbut a t low pressure that the gas behaves like an ideal gas and we can use th e idea l gas Cp values !! 34 B.Srinivas ChEngg GVP CE 35 B.Srinivas ChEngg GVP CE 36 B.Srinivas ChEngg GVP CE 37 B.Srinivas ChEngg GVP CE 38 B.Srinivas ChEngg GVP CE 39 B.Srinivas ChEngg GVP CE 40 B.Srinivas ChEngg GVP CE For a gas obeying vanderWaal EOS 41 B.Srinivas ChEngg GVP CE Two moles of a gas are confined in a piston-cylinder. Initially, the tempera ture is 300 K an d the pressure is 1 bar. The gas is compresse d isothermally to 5 bar. The gas has Cp=(7/2)R. Find Q, W, ΔU, ΔH and ΔS for the gas if: a) The gas is an ideal gas b) The gas follows the van der Waals equation of state, with a=5*106 bar cm6/mol2 and b=30 cm3/mol RT a P 2 VV VbV 22RT a W n PdVn dV PV VbV 2 Z VV11 RT Vb2 aa a n RT ln HZRTR (1) VbV121 V V a U R V Vb SRZRR ln ln V 42 B.Srinivas ChEngg GVP CE 43 B.Srinivas ChEngg GVP CE 44 B.Srinivas ChEngg GVP CE 45 B.Srinivas ChEngg GVP CE 46 B.Srinivas ChEngg GVP CE 47 B.Srinivas ChEngg GVP CE 48 B.Srinivas ChEngg GVP CE 49 B.Srinivas ChEngg GVP CE 50 B.Srinivas ChEngg GVP CE 51 B.Srinivas ChEngg GVP CE 52 B.Srinivas ChEngg GVP CE Methane enters a turbine at T = 600 K and P = 10 bar and leaves at T = 400 K and P = 2 bar. How much work is produced for each mole of gas? Use the following data: van der Waals constants for methane are a = 0. 2303 Pa- m6/mol2 and b = 4.306*10-5 m3/mol, and -3 -5 2 -8 3 -11 4 CP/R = 4.568 - 8.975*10 T + 3.631*10 T - 3.407*10 T + 1.091*10 T Algorithm RT a P VbV 2 Calc V knowing P and T PV Z RT Calc Z R a HRTZ (1) R V ClCalc H 53 B.Srinivas ChEngg GVP CE At the inlet P = 10 bar and T = 600 K R V = 0.005, Z = 0.999 and H 1 = -48.926 At the outlet P = 2 bar and T = 400 K R V = 0.017, Z = 0.998 and H 2 = -19.096 400 C HHRR RP dTH 21 600 R = -19. 096 – 9289 – (-48. 926) = 9319 54 B.Srinivas ChEngg GVP CE 55 B.Srinivas ChEngg GVP CE 56 B.Srinivas ChEngg GVP CE 57 B.Srinivas ChEngg GVP CE 58 B.Srinivas ChEngg GVP CE 59 B.Srinivas ChEngg GVP CE For liquids which are poorly described a cubic EOS another approach is usually followed by defining isothermal compressibilty and volume expansivity. For moderate changes in T and P these can be considered constant and integrations can be carried out to determine changes in U, H and S. 60 B.Srinivas ChEngg GVP CE 61 B.Srinivas ChEngg GVP CE 62 B.Srinivas ChEngg GVP CE.
Load more

Recommended publications
  • ( ∂U ∂T ) = ∂CV ∂V = 0, Which Shows That CV Is Independent of V . 4
    so we have ∂ ∂U ∂ ∂U ∂C =0 = = V =0, ∂T ∂V ⇒ ∂V ∂T ∂V which shows that CV is independent of V . 4. Using Maxwell’s relations. Show that (∂H/∂p) = V T (∂V/∂T ) . T − p Start with dH = TdS+ Vdp. Now divide by dp, holding T constant: dH ∂H ∂S [at constant T ]= = T + V. dp ∂p ∂p T T Use the Maxwell relation (Table 9.1 of the text), ∂S ∂V = ∂p − ∂T T p to get the result ∂H ∂V = T + V. ∂p − ∂T T p 97 5. Pressure dependence of the heat capacity. (a) Show that, in general, for quasi-static processes, ∂C ∂2V p = T . ∂p − ∂T2 T p (b) Based on (a), show that (∂Cp/∂p)T = 0 for an ideal gas. (a) Begin with the definition of the heat capacity, δq dS C = = T , p dT dT for a quasi-static process. Take the derivative: ∂C ∂2S ∂2S p = T = T (1) ∂p ∂p∂T ∂T∂p T since S is a state function. Substitute the Maxwell relation ∂S ∂V = ∂p − ∂T T p into Equation (1) to get ∂C ∂2V p = T . ∂p − ∂T2 T p (b) For an ideal gas, V (T )=NkT/p,so ∂V Nk = , ∂T p p ∂2V =0, ∂T2 p and therefore, from part (a), ∂C p =0. ∂p T 98 (a) dU = TdS+ PdV + μdN + Fdx, dG = SdT + VdP+ Fdx, − ∂G F = , ∂x T,P 1 2 G(x)= (aT + b)xdx= 2 (aT + b)x . ∂S ∂F (b) = , ∂x − ∂T T,P x,P ∂S ∂F (c) = = ax, ∂x − ∂T − T,P x,P S(x)= ax dx = 1 ax2.
    [Show full text]
  • Thermodynamics
    ME346A Introduction to Statistical Mechanics { Wei Cai { Stanford University { Win 2011 Handout 6. Thermodynamics January 26, 2011 Contents 1 Laws of thermodynamics 2 1.1 The zeroth law . .3 1.2 The first law . .4 1.3 The second law . .5 1.3.1 Efficiency of Carnot engine . .5 1.3.2 Alternative statements of the second law . .7 1.4 The third law . .8 2 Mathematics of thermodynamics 9 2.1 Equation of state . .9 2.2 Gibbs-Duhem relation . 11 2.2.1 Homogeneous function . 11 2.2.2 Virial theorem / Euler theorem . 12 2.3 Maxwell relations . 13 2.4 Legendre transform . 15 2.5 Thermodynamic potentials . 16 3 Worked examples 21 3.1 Thermodynamic potentials and Maxwell's relation . 21 3.2 Properties of ideal gas . 24 3.3 Gas expansion . 28 4 Irreversible processes 32 4.1 Entropy and irreversibility . 32 4.2 Variational statement of second law . 32 1 In the 1st lecture, we will discuss the concepts of thermodynamics, namely its 4 laws. The most important concepts are the second law and the notion of Entropy. (reading assignment: Reif x 3.10, 3.11) In the 2nd lecture, We will discuss the mathematics of thermodynamics, i.e. the machinery to make quantitative predictions. We will deal with partial derivatives and Legendre transforms. (reading assignment: Reif x 4.1-4.7, 5.1-5.12) 1 Laws of thermodynamics Thermodynamics is a branch of science connected with the nature of heat and its conver- sion to mechanical, electrical and chemical energy. (The Webster pocket dictionary defines, Thermodynamics: physics of heat.) Historically, it grew out of efforts to construct more efficient heat engines | devices for ex- tracting useful work from expanding hot gases (http://www.answers.com/thermodynamics).
    [Show full text]
  • 7 Apr 2021 Thermodynamic Response Functions . L03–1 Review Of
    7 apr 2021 thermodynamic response functions . L03{1 Review of Thermodynamics. 3: Second-Order Quantities and Relationships Maxwell Relations • Idea: Each thermodynamic potential gives rise to several identities among its second derivatives, known as Maxwell relations, which express the integrability of the fundamental identity of thermodynamics for that potential, or equivalently the fact that the potential really is a thermodynamical state function. • Example: From the fundamental identity of thermodynamics written in terms of the Helmholtz free energy, dF = −S dT − p dV + :::, the fact that dF really is the differential of a state function implies that @ @F @ @F @S @p = ; or = : @V @T @T @V @V T;N @T V;N • Other Maxwell relations: From the same identity, if F = F (T; V; N) we get two more relations. Other potentials and/or pairs of variables can be used to obtain additional relations. For example, from the Gibbs free energy and the identity dG = −S dT + V dp + µ dN, we get three relations, including @ @G @ @G @S @V = ; or = − : @p @T @T @p @p T;N @T p;N • Applications: Some measurable quantities, response functions such as heat capacities and compressibilities, are second-order thermodynamical quantities (i.e., their definitions contain derivatives up to second order of thermodynamic potentials), and the Maxwell relations provide useful equations among them. Heat Capacities • Definitions: A heat capacity is a response function expressing how much a system's temperature changes when heat is transferred to it, or equivalently how much δQ is needed to obtain a given dT . The general definition is C = δQ=dT , where for any reversible transformation δQ = T dS, but the value of this quantity depends on the details of the transformation.
    [Show full text]
  • Maxwell Relations
    MAXWELLRELATIONS 7.I THE MAXWELL RELATIONS In Section 3.6 we observedthat quantities such as the isothermal compressibility, the coefficientof thermal expansion,and the molar heat capacities describe properties of physical interest. Each of these is essentiallya derivativeQx/0Y)r.r..,. in which the variablesare either extensiveor intensive thermodynaini'cparameters. with a wide range of extensive and intensive parametersfrom which to choose, in general systems,the numberof suchpossible derivatives is immense.But thire are azu azu ASAV AVAS (7.1) -tI aP\ | AT\ as),.*,.*,,...: \av)s.N,.&, (7.2) This relation is the prototype of a whole class of similar equalities known as the Maxwell relations. These relations arise from the equality of the mixed partial derivatives of the fundamental relation expressed in any of the various possible alternative representations. 181 182 Maxwell Relations Given a particular thermodynamicpotential, expressedin terms of its (l + 1) natural variables,there are t(t + I)/2 separatepairs of mixed second derivatives.Thus each potential yields t(t + l)/2 Maxwell rela- tions. For a single-componentsimple systemthe internal energyis a function : of three variables(t 2), and the three [: (2 . 3)/2] -arUTaSpurs of mixed second derivatives are A2U/ASAV : AzU/AV AS, AN : A2U / AN AS, and 02(J / AVA N : A2U / AN 0V. Thecompleteser of Maxwell relations for a single-componentsimple systemis given in the following listing, in which the first column states the potential from which the ar\ U S,V I -|r| aP\ \ M)',*: ,s/,,' (7.3) dU: TdS- PdV + p"dN ,S,N (!t\ (4\ (7.4) \ dNls.r, \ ds/r,r,o V,N _/ii\ (!L\ (7.5) \0Nls.v \0vls.x UITI: F T,V ('*),,.: (!t\ (7.6) \0Tlv.p dF: -SdT - PdV + p,dN T,N -\/ as\ a*) ,.,: (!"*),.
    [Show full text]
  • PHSX 446 EXAM 2 – SOLUTIONS Spring 2015 1 1. First, Some Basic
    PHSX 446 EXAM 2 – SOLUTIONS Spring 2015 1 1. First, some basic knowledge questions. You need not show work here; just give the answer. More than one answer might apply. Don’t waste time transcribing answers; just write on this exam sheet. (a) An inexact differential is: a) inexpressible as a function, b) of no use in mathematics or physics, c) only approximately correct. (b) Maxwell relations express: a) conservation laws, b) the conditions under which phase transitions occur, c) relations among thermodynamic variables, d) the properties of ideal gases only. (c) Does equipartition always hold for a gas in equilibrium? Briefly explain. No. Equiparition holds for an ideal gas, and generally not for a non-ideal gas. (d) Give an example of an adiabatic process that is not reversible. Thermal diffusion in a closed system, mixing, free expansion. (e) Give an example of a reversible process that is not adiabatic. Heating of a system from one temperature to another by an outside heat source. The heat can always be removed (by putting the system in contact with a reservoir at the original temperature), which will restore the original state with unchanged entropy. The pressure, or volume, or both, could change. (f) What property must a gas have for it to cool in the Joule-Thompson (throt- tling) process? The gas must be non-ideal. (g) Write down the expression for the heat current in thermal conduction. ∂T j = −k one dimension. ∂x (h) Why are engines usually cyclical? Because we want to repeat the process to get some amount of work at each cycle, rather than do it only once.
    [Show full text]
  • Thermodynamic Potentials and Maxwell's Relations
    Thermodynamic Potentials and Maxwell’s Relations (A lecture ​ notes). By : Chandan kumar Department of Physics. S N S College Jehanabad Dated: 04/09/2020 Introduction: In this lecture we introduce other thermodynamic potentials and Maxwell relations. The energy and entropy representations We have noted that both S(U,V,N) and U(S,V,N) contain complete thermodynamic ​ ​ ​ ​ ​ ​ ​ ​ information. We will use the fundamental thermodynamic identity dU = TdS − pdV + µdN ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ as an aid to memorizing the of temperature, pressure, and chemical potential from the consideration of equilibrium conditions. by calculating the appropriate partial derivatives we have and We can also write the fundamental thermodynamic identity in the entropy representation: d 1 from which we find and By calculating the second partial derivatives of these quantities we find the Maxwell relations. Maxwell relations can be used to relate partial derivatives that are easily measurable to those that are not. Starting from and we can calculate , and . ​ Now since under appropriate conditions = and then . This result is called a Maxwell relation. By considering the other second partial derivatives, we find two other Maxwell relations from the energy representation of the fundamental thermodynamic identity. These are: and . Similarly, in the entropy representation, starting from d and the results , and . ​ we find the Maxwell relations: 2 and . Enthalpy H(S,p,N): We have already defined enthalpy as H = U + pV . We can calculate its differential and ​ ​ ​ ​ ​ ​ combine it with the fundamental thermodynamic identity to show that the natural variables of H are S, p0, and N. ​ ​ ​ ​ ​ ​ ​ ​ H = U + pV ​ ​ ​ ​ we have dH = dU +d(pV ) = dU + pdV + V dp, ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ and so inserting dU = TdS − pdV + µdN ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ we have dH = TdS − pdV + µdN + pdV + V dp ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ ​ resulting in dH = TdS + V dp + µdN.
    [Show full text]
  • Maxwell Relations
    Maxwell relations Maxwell's relations are a set of equations in thermodynamics which are derivable from the symmetry of second derivatives and from the definitions of the thermodynamic potentials. ese relations are named for the nineteenth- century physicist James Clerk Maxwell. Contents Equations The four most common Maxwell relations Derivation Derivation based on Jacobians General Maxwell relationships See also Equations Flow chart showing the paths between the Maxwell relations. P: e structure of Maxwell relations is a pressure, T: temperature, V: volume, S: entropy, α: coefficient of statement of equality among the second thermal expansion, κ: compressibility, CV: heat capacity at derivatives for continuous functions. It constant volume, CP: heat capacity at constant pressure. follows directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant (Schwarz theorem). In the case of Maxwell relations the function considered is a thermodynamic potential and xi and xj are two different natural variables for that potential: Schwarz' theorem (general) where the partial derivatives are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are n(n − 1)/2 possible Maxwell relations where n is the number of natural variables for that potential. e substantial increase in the entropy will be verified according to the relations satisfied by the laws of thermodynamics e four most common Maxwell relations e four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature T; or entropy S) and their mechanical natural variable (pressure P; or volume V): Maxwell's relations (common) where the potentials as functions of their natural thermal and mechanical variables are the internal energy U(S, V), enthalpy H(S, P), Helmholtz free energy F(T, V) and Gibbs free energy G(T, P).
    [Show full text]
  • Intro-Propulsion-Lect-19
    19 1 Lect-19 In this lecture ... • Helmholtz and Gibb’s functions • Legendre transformations • Thermodynamic potentials • The Maxwell relations • The ideal gas equation of state • Compressibility factor • Other equations of state • Joule-Thomson coefficient 2 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-19 Helmholtz and Gibbs functions • We have already discussed about a combination property, enthalpy, h. • We now introduce two new combination properties, Helmholtz function, a and the Gibbs function, g. • Helmholtz function, a: indicates the maximum work that can be obtained from a system. It is expressed as: a = u – Ts 3 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-19 Helmholtz and Gibbs functions • It can be seen that this is less than the internal energy, u, and the product Ts is a measure of the unavailable energy. • Gibbs function, g: indicates the maximum useful work that can be obtained from a system. It is expressed as: g = h – Ts • This is less than the enthalpy. 4 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-19 Helmholtz and Gibbs functions • Two of the Gibbs equations that were derived earlier (Tds relations) are: du = Tds – Pdv dh = Tds + vdP • The other two Gibbs equations are: a = u – Ts g = h – Ts • Differentiating, da = du – Tds – sdT dg = dh – Tds – sdT 5 Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay Lect-19 Legendre transformations • A simple compressible system is characterized completely by its energy, u (or entropy, s) and volume, v: u = u(s,v) ⇒ du = Tds − Pdv ∂u ∂u such that T = P = ∂s v ∂v s Alternatively, in the entropy representation, s = s(u,v) ⇒ Tds = du + Pdv 1 ∂s P ∂s such that = = T ∂u v T ∂v u 6 Prof.
    [Show full text]
  • On Maxwell's Relations of Thermodynamics for Polymeric
    640 Macromolecules 2011, 44, 640–646 DOI: 10.1021/ma101813q On Maxwell’s Relations of Thermodynamics for Polymeric Liquids away from Equilibrium Chunggi Baig,*,† Vlasis G. Mavrantzas,*,† and Hans Christian Ottinger€ ‡ †Department of Chemical Engineering, University of Patras & FORTH-ICE/HT, Patras, GR 26504, Greece, and ‡Department of Materials, Polymer Physics, ETH Zurich,€ HCI H 543, CH-8093 Zurich,€ Switzerland Received August 9, 2010; Revised Manuscript Received December 9, 2010 ABSTRACT: To describe complex systems deeply in the nonlinear regime, advanced formulations of nonequilibrium thermodynamics such as the extended irreversible thermodynamics (EIT), the matrix model, the generalized bracket formalism, and the GENERIC (=general equation for the nonequilibrium reversible-irreversible coupling) formalism consider generalized versions of thermodynamic potentials in terms of a few, well-defined position-dependent state variables (defining the system at a coarse-grained level). Straightforward statistical mechanics considerations then imply a set of equalities for its second derivatives with respect to the corresponding state variables, typically known as Maxwell’s relations. We provide here direct numerical estimates of these relations from detailed atomistic Monte Carlo (MC) simulations of an unentangled polymeric melt coarse-grained to the level of the chain conformation tensor, under both weak and strong flows. We also report results for the nonequilbrium (i.e., relative to the quiescent fluid) internal energy, entropy, and free energy functions of the simulated melt, which indicate a strong coupling of the second derivatives of the corresponding thermodynamic potential at high flow fields. 1. Introduction the set x of the proper state variables, a step of paramount Since Onsager’s pioneering work1 on irreversible phenomena importance requiring deep physical insight and experience.
    [Show full text]
  • The Maxwell Relations
    The Maxwell relations A number of second derivatives of the fundamental relation have clear physical significance and can be measured experimentally. For example: 1 ∂2G 1 ∂V Isothermal compressibility κ = − = − T V ∂P2 V ∂P The property of the energy (or entropy) as being a differential function of its variables gives rise to a number of relations between the second derivatives, e. g. : ∂2U ∂ 2U = ∂S∂V ∂V∂S The Maxwell relations ∂2U ∂ 2U = ∂S∂V ∂V∂S is equivalent to: ∂P ∂T − = ∂S ∂V V ,N1 ,N2 ,... S ,N1 ,N2 ,... Since this relation is not intuitively obvious, it may be regarded as a success of thermodynamics. The Maxwell relations Given the fact that we can write down the fundamental relation employing various thermodynamic potentials such as F, H, G, … the number of second derivative is large. However, the Maxwell relations reduce the number of independent second derivatives. Our goal is to learn how to obtain the relations among the second derivatives without memorizing a lot of information. For a simple one-component system described ∂2 ∂2 ∂2 U U U by the fundamental relation U(S,V,N), there are 2 ∂S ∂S∂V ∂S∂N nine second derivatives, only six of them are ∂2 ∂2 ∂2 independent from each other. U U U 2 ∂V∂S ∂V ∂V∂N In general, if a thermodynamic system has n 2 2 2 ∂ U ∂ U ∂ U independent coordinates, there are n(n+1)/2 2 independent second derivatives. ∂N∂S ∂N∂V ∂N The Maxwell relations: a single-component system For a single-component system with conserved mole number, N, there are only three independent second derivatives: ∂2 ∂2 ∂2 G G G ∂ 2G ∂ 2G ∂2G ∂T2 ∂T∂P ∂T∂N 2 2 2 2 2 ∂ G ∂ G ∂ G ∂ ∂ ∂ ∂ 2 T P T P ∂P∂T ∂P ∂P∂N 2 2 2 ∂2U ∂ G ∂ G ∂ G Other second derivatives, such as e.
    [Show full text]
  • Redalyc.Thermodynamic Response Functions and Maxwell Relations for a Kerr Black Hole
    Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México Escamilla, L.; Torres-Arenas, J. Thermodynamic response functions and Maxwell relations for a Kerr black hole Revista Mexicana de Física, vol. 60, núm. 1, enero-febrero, 2014, pp. 59-68 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57029680010 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative RESEARCH Revista Mexicana de F´ısica 60 (2014) 59–68 JANUARY-FEBRUARY 2014 Thermodynamic response functions and Maxwell relations for a Kerr black hole L. Escamilla and J. Torres-Arenas Division´ de Ciencias e Ingenier´ıas Campus Leon,´ Universidad de Guanajuato, Loma del Bosque 103, 37150 Leon´ Guanajuato, Mexico,´ e-mail: lescamilla@fisica.ugto.mx; jtorres@fisica.ugto.mx Received 18 February 2013; accepted 4 October 2013 Assuming the existence of a fundamental thermodynamic relation, the classical thermodynamics of a black hole with mass and angular momentum is given. New definitions of the response functions and T dS equations are introduced and mathematical analogous of the Euler equation and Gibbs-Duhem relation are founded. Thermodynamic stability is studied from concavity conditions, resulting in an unstable equilibrium at all the domain except for a region of local stable equilibrium. The Maxwell relations are written, allowing to build the thermodynamic squares. Our results shown an interesting analogy between thermodynamics of gravitational and magnetic systems.
    [Show full text]
  • Thermodynamic Potentials D
    Thermodynamic Potentials D. Jaksch1 Goals • Understand the importance and relevance of different thermodynamic potentials and natural variables. • Learn how to derive and use thermodynamic relations like e.g. Maxwell relations. • Learn how to calculate averages for a given probability distribution. Thermodynamic Potentials The natural variables for the internal energy U are the system volume V and the entropy S, i.e. dU = T dS−pdV . If U is given as a function of S and V then it determines all thermodynamic properties of the system; the temperature and pressure are then given by its partial derivatives. By using a Legendre transformation we can define the Helmholtz free energy F = U − TS whose derivative is dF = dU − d(TS) = −SdT − pdV ; its natural variables are T and V . Similarly we have the Enthalpy H = U + pV with dH = T dS + V dp and the Gibbs free energy G = U + pV − TS with dG = −SdT + V dp. The variable pairs p; V and T;S are called conjugate variables. We will later introduce the chemical potential µ and number of particles N as another conjugate pair. In some systems the magnetic field B and the magnetization M will take the role of p and V . In low dimensional systems area or length take the role of the volume. Maxwell relations The Maxwell relations follow from the equality of mixed derivatives of the thermodynamic potentials with respect to their natural variables. The most common relations are @T @p @2U + = − = ; @V S @S V @S@V @T @V @2H + = + = ; @p S @S p @S@p @S @p @2F + = + = ; @V T @T V @T @V @S @V @2G − = + = : @p T @T p @T @p Additional Maxwell relations arise when µ and N are also considered.
    [Show full text]