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Thermodynamic Potentials D

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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. Thermodynamic quantities and equations You might find the following websites useful • https://en.wikipedia.org/wiki/List of thermodynamic properties • https://en.wikipedia.org/wiki/Table of thermodynamic equations Please note that the list of symbols appearing in thermodynamics is vast and sometimes used inconsistently. Therefore only those equations whose meaning is unambiguously defined on those websites should be used. As always, all the symbols you use need to be defined. 1Several of these problems are based on problem sets by Profs A. Boothroyd, A.A. Schekochihin, S.J. Blundell, and previous Oxford exam questions Problems 1. State which of the properties quasi-static, reversible, irreversible and isothermal applies to each of the following processes (in each case list all those properties which apply): (a) Slow expansion of an ideal gas in a thermally-isolated cylinder, as a piston is withdrawn. (b) The JouleKelvin process applied to a non-ideal gas. (c) A volcanic eruption. (d) The process of oxidation (rusting) of a steel nail immersed in a beaker of water maintained at standard temperature and pressure. 2. Show that the change in internal energy of a simple system between states (V1;T1) and (V2;T2) is given by Z T2 Z V2 @p ∆U = CV dT + T − p dV: T1 V1 @T V Boyle observed that the pressure of an ideal gas varies inversely with its volume at constant temperature. Give the most general possible equation of state which is consistent with this finding. Explain why the @T Joule expansion experiment implies that @V U = 0. Show further that these results taken together imply that the equation of state is pV = RT where R is a constant of proportionality. 3. Derive the following general relations @T 1 h @p i (i) @V = − C T @T − p U V V @T 1 @p (ii) @V = − C T @T S V V @T 1 h @V i (iii) @p = C T @T − V H p p In each case the quantity on the left hand side is the appropriate thing to consider for a particular type of expansion. State what type of expansion each refers to. Using these relations, verify that for an ideal gas @T @T = 0 and = 0 @V U @p H @T γ and that @V S leads to the familiar relation pV =const along an isentrope. 4. Use the First Law of Thermodynamics to show that @U C − C = p V − p ; @V T V βp where βp is the coefficient of volume expansivity. 5. For a stretched rubber band, it is observed experimentally that the tension f is proportional to the temperature T if the length L is held constant. Prove that: (a) the internal energy U is a function of temperature only; (b) adiabatic stretching of the band results in an increase in temperature; (c) the band will contract if warmed while kept under constant tension. You may assume that @L > 0 : @f T 6. For a fixed surface area, the surface tension of water varies linearly with temperature from 75×10−4N/m at 5◦C to 70×10−3N/m at 35◦C. Calculate the surface contributions to the entropy per unit area and the ◦ @S −3 2 @U −3 2 internal energy per unit area at 5 C. Solution: @A T = 0:167×10 J=Km , @A T = 121:3×10 J=m . An atomizer produces water droplets of diameter 0:1µm. A cloud of droplets at 35◦C coalesces to form a single drop of water of mass 1g. Estimate the temperature of the drop assuming no heat exchange with the surroundings. What is the increase in entropy in this process? The specific heat capacity of water is −3 Cp = 4200J/(K kg). Solution: ∆T = 1:73K, ∆S = 13:6 × 10 J/K. 7. The magnetization M of a paramagnetic material is given by M = χB/µ0, where B is the magnetic flux density and the susceptibility χ follows Curies law χ = C=T with C a constant. 2 If the heat capacity per unit volume at constant M is cM = a=T , show that the heat capacity per unit volume at constant B is a B2C cB = 2 1 + : T µ0a If a sample is initially at temperature T1 in an applied field of flux density B1, show that the temperature after adiabatic reduction of the field to zero is T T = 1 : 2 1=2 B2C 1 + 1 µ0a Class Problems 8. If θ is a continuous random variable which is uniformly distributed between 0 and π, write down an expression for p(θ). Hence find the value of the following averages: (i) hθi (vi) hsin(θ)i (ii) hθ − π=2i (vii) hj cos(θ)ji (iii) θ2 (viii) cos2(θ) (iv) hθni for n ≥ 0 (ix) sin2(θ) (v) hcos(θ)i (x) sin2(θ) + cos2(θ) Check that your answers are what you expect. 9. If particle velocities are distributed isotropically, how are their angles distributed? Is the angle between the velocity vector and a fixed axis (chosen by you) distributed uniformly? Why? Answer these questions for the case of a 2- and 3-dimensional world. 10. Consider an isotropic distribution of particle velocities: f(b) = f(v), where v = jbj is the particle speed. In 3D, what is the distribution of the speeds, f~(v)? 11. Calculate the following averages of velocity components in 3D in terms of averages of speed (hvi, v2 , etc.). Here the indices take on values i; j; k; l = x; y; z. 2 (i) hvii, (ii) hjviji, (iii) vi , (iv) hvivji, (v) hvivjvki, (vi) hvivjvkvli..
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