The Form of Magnetic Work in a Fundamental Thermodynamic Equation for a Paramagnet

Magnetic work takes two forms in the thermodynamics of a paramagnet as developed in many textbooks. We observe that in the case when the lattice energy is excluded, the form δW = BdM cannot be used in a fundamental thermodynamic equation. This shows that there are thermodynamic systems with no fundamental thermodynamic equation. The thermal physics of magnetic systems has been the source of continuing confusion. Mandl writes in the second edition of his text, “As is well known, the thermodynamic discussion of magnetic systems easily leads to misleading or even wrong statements, and I fear that the first edition was not free from these.” And according to Kittel, “A great deal of unnecessary confusion exists as to how to write the First Law of Thermodynamics for a magnetic system.” For a paramagnetic crystal in a uniform magnetic field B, with total magnetic dipole moment M , there are two forms for the work done when B and M change: δWms = BdM and δWs = −MdB. (1) The form δWms applies when the mutual field energy is included in the system, the form δWs when it is not. 3 The forms δWms and δWs are readily derived also by means of statistical mechanics; we include these derivations in the appendix. The thermodynamic derivations of these forms given by Mandl, Kittel, F. Mandl, Statistical Physics (Wiley, Chichester, 1988), 2nd ed., p. vii. Charles Kittel, Elementary Statistical Physics (Wiley, New York, 1958), pp. 77-82. We take the paramagnetic spin system to be ideal : interaction among the dipole spins is essentially absent. The model which results from this assumption is widely used, is an excellent approximation to real systems within its domain of application, and does not obey the third law of thermodynamics. It shares all these attributes with the ideal gas model. Reference 1, pp. 21-28 and 336-339. Reference 2, pp. 77-82.