Question

06) Suppose a paramagnetic system that is under the action of an auxiliary (magnetic) field H. Thus, the work involved, it has to be that the work involved in a reversible process is described as đW = 10Hdm. where m is the total magnetic dipole moment of the system and u0 is the magnetic permeability of the vacuum. a) Determine the expression of the differential du of the internal energy U and indicate the two independent natural variables associated with internal energy. b) From the differential dU of internal energy U and using appropriate Legendre transforms, determine the differentials dE, dF and dG associated with enthalpy E and free energies of Helmholtz F and Gibbs G, respectively. Indicate the two ndependent natural variables associated with each thermodynamic potential. Use the analogy with the hydrostatic system to identify the independent variables. c) From the relations obtained in items (a) and (b), determine the 8 equations of state, that is, the expressions of T, S, H and m in terms of partial derivatives of the thermodynamic potentials, and the 4 Maxwell relations associated with the system. d) Determine one of the possible TdS equations for this system. Remember that TDS equations must be written in terms of appropriate thermal capacities and partial derivatives involving the thermodynamic variables T, m and H. XmH Xm a'/T e) A certain paramagnetic material obeys Curie's law, i.e., Xm where is the magnetic susceptibility of the material and a' is a constant. Since the magnetization M and volume V of the system is constant, we have that m = aH/T , where a is a constant. Determine the relationship between temperatures T1 and T2 of the material in an adiabatic process of changing the applied magnetic field from H1 to H2. Consider that the thermal capacity involved is constant.

Answer 1

6.a au= Tas- 10H dm (work done on magnetic system is considered negative bu convention) E = ut 10Hm (Replace pu by 10 Hm in degulee expression dE= dut 10(dH 7m + 10H dm Tas -lo Hdm + 10(dH) m+ loham Tastlomdh F = U-TS dF = du - (dT) S- Set (ast Tds 1.-104 am slari-set (ds)T -saT - 10 Ham. - S = E- TS =de (dt) S-8Tds) Tdst lomdh - (ans- Tag -s dT + lomdh b natural variable marw ell eq. potential 2T) du T ds-10H am sim ams as m lat om dE=Tastlomoh Š, H 10 (GH 'as 124 dF=-sat-100am lam LOT = -10 OH Osim 3). = Ti .) یا ح =-SATT lomah T, H DS Ән т 100)

(25) - HE (1 54 2 S 102H lag blon De S Sa - -lag әТ 2T 1st Tds ea T asa cu dT + T (OP) du nlow du = Tas- with. du= 10H dm comparing Tds - p du 10H V cm + T!D. T ds = d'T '2H y dm. latim Heat Capecity at constant magnetisation is experimental quantity lat controled le corresponding T measureel given H. Cm 104 because H can be can be at e. ан df=0 -sdT - 10 Ham. =10.149 saT = 10H adh T -TGOT = 10 H.adh (s is some unknown function of s) t2 -TOT + Tos). dt = Joronen TY a 10. (H₂ - H,? 7 2 2 ht S as ƏT (+2-71) = a ntivo on managing Urological ailments