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