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. WWWWWWWWWWWWWW

Answer 1

(a) The combined form of first and second laws of time dynamics gives us

$dU=TdS-dW$

Hence in this case

.

dU = Tds - HoHdm

The two independent variables associated with internal energy are the magnetic dipole moment and the entropy .

(b) By Legendre transformation, we can write the enthalpy as

$E=U+\mu_0 H m$.

Hence

.

dE = du t lo CH dm +mdH) - Tds Molt dm + Mo (Hdm tmdH) = Tds tllomd H

The two independent variables associated with E are S and H.

Similarly, the Helmholtz free energy and Gibbs free energy are given by respectively

.

F = U - TS and G=E-TS. (Tdst + sdt) du dF = (Tds - Mohdm]-(Tds+SdT) s dT . - =-Mot d m and dG = dE - (Tds 75dT) - (TAS =(Tds tuo mdH) + S&T) Mom dH -sat

The independent variables associated with F are m and T and those with G are H and T.

(c) The 8 equations of states can be obtained as

du= Tds - Motam as t с ( 23 ), ( 2 ) Ass н= - 2 ) ги as 2 = ( 2 ) m ан dE = Tds thom dH ds t - Н ( 2 ) (3), а утг . (), 1 Т. (3) и ās, H s IT df но на от ат + ( ) 4 dT ar dmt am/ T .. H2 - - - -), , S = -(а) m dG= nomdh-sat (2), dT — ditt T / та До с S = - ( ( 2 ), Ә, ) ат. Н.

The 4 Maxwell's relations can be obtained from the above red marked equations of states as

2S /т H= ӘU am s Since To (35) ล่น а) ан 16 S as am Since m = 2E 2S A. C3). T= әH/s Н E ( 2 ), ги» (2), — Әтән Since На себе 2 мо am T S = - 5) » a E с - ио (SH) = - 2 ) T әт әТ - с. - (Е), Since ən 2G с 7 та иб S = - 20 (2) ант Ən. .. — Иб а), -- (25) T ӘНӘТ

(d) The equation is

Tas = dut aw =eu dT + Mo H dm Since du=Cv dT cari specific heat at constant volume) and dw = Molt dm.