8.7 Problems for This Chapter
479
mine whether the temperature increases or decreases for such paramagnetic
crystals when they are demagnetized adiabatically.
12. The thermodynamic Helmholtz energy A spin (T , V , B 0 ) appropriate to a canonical ensemble of N distinguishable spin1
2 nuclei is related to the internal energy
U(S, V , M) via Eq. (8.5.11a). Evaluate U(S, V , M) explicitly for this system,
and show that U ≡ 0. Comment upon why this result is actually appropriate for
this spin system.
13. Obtain expressions for the variance, ((M) 2 , and the standard deviation, σ M ,
associated with magnetic moment fluctuations to which a solid macroscopic
sample containing N independent spin1
2 nuclei in a magnetic induction field
of strength B 0 is subject.
14. The standard deviation σ M for (also referred to as the fluctuation in) the
macroscopic magnetic moment M of a thermodynamic magnetic system is
defined as the square root of its variance M 2 − M
2 . Obtain an expression for
σ M /M in terms of the magnetic susceptibility χ )m) . Use your result to justify
the statement that the thermodynamic magnetic moment M is the same as M.
15. Expressions (8.4.13a) were obtained by appealing to the observation that −M
corresponds to V and B 0 corresponds to P in terms of their roles in thermodynamics. By starting from Eq. (8.4.6) for P and V constant, show that the same
final expression for the heat capacity difference C B − C M is attained directly.
Consider a homogeneous magnetically isotropic thermodynamic system, so
that M and B 0 may be replaced by scalars M and B 0 .
16. When a sample of a magnetizable substance is magnetized by bringing it
into a magnetic field of strength B while maintaining thermal contact with
its surroundings at temperature T , the magnetization is hence inherently a
function of both temperature and applied field strength. It thus makes sense
to think in terms of energy exchange, δQ, between a (thermodynamic) system
and its surroundings under the condition of constant applied magnetic field
strength, B, rather than at constant magnetization. Obtain an expression for
the difference between C B (T ) and C M (T ), and discuss its analogy with the
difference between C P (T ) and C V (T ) obtained in Chap. 2.
17. The combined 1st and 2nd laws expression for a magnetizable substance may
be written as
dU = T dS + BdM − P dV ,
with B the magnetic induction and M the magnetic moment of the substance,
so that BdM represents the work involved in changing the sample magnetic
moment by dM. Obtain an expression for the heat, δQ rev , exchanged between
the sample and its surroundings, and show that for U ≡ U(T , M), δQ rev can
be written as
δQ rev = C M (T ) dT +
∂U
∂M
T
− B
dM,
with C M (T ) the heat capacity of the solid sample at constant magnetization.
479
mine whether the temperature increases or decreases for such paramagnetic
crystals when they are demagnetized adiabatically.
12. The thermodynamic Helmholtz energy A spin (T , V , B 0 ) appropriate to a canonical ensemble of N distinguishable spin1
2 nuclei is related to the internal energy
U(S, V , M) via Eq. (8.5.11a). Evaluate U(S, V , M) explicitly for this system,
and show that U ≡ 0. Comment upon why this result is actually appropriate for
this spin system.
13. Obtain expressions for the variance, ((M) 2 , and the standard deviation, σ M ,
associated with magnetic moment fluctuations to which a solid macroscopic
sample containing N independent spin1
2 nuclei in a magnetic induction field
of strength B 0 is subject.
14. The standard deviation σ M for (also referred to as the fluctuation in) the
macroscopic magnetic moment M of a thermodynamic magnetic system is
defined as the square root of its variance M 2 − M
2 . Obtain an expression for
σ M /M in terms of the magnetic susceptibility χ )m) . Use your result to justify
the statement that the thermodynamic magnetic moment M is the same as M.
15. Expressions (8.4.13a) were obtained by appealing to the observation that −M
corresponds to V and B 0 corresponds to P in terms of their roles in thermodynamics. By starting from Eq. (8.4.6) for P and V constant, show that the same
final expression for the heat capacity difference C B − C M is attained directly.
Consider a homogeneous magnetically isotropic thermodynamic system, so
that M and B 0 may be replaced by scalars M and B 0 .
16. When a sample of a magnetizable substance is magnetized by bringing it
into a magnetic field of strength B while maintaining thermal contact with
its surroundings at temperature T , the magnetization is hence inherently a
function of both temperature and applied field strength. It thus makes sense
to think in terms of energy exchange, δQ, between a (thermodynamic) system
and its surroundings under the condition of constant applied magnetic field
strength, B, rather than at constant magnetization. Obtain an expression for
the difference between C B (T ) and C M (T ), and discuss its analogy with the
difference between C P (T ) and C V (T ) obtained in Chap. 2.
17. The combined 1st and 2nd laws expression for a magnetizable substance may
be written as
dU = T dS + BdM − P dV ,
with B the magnetic induction and M the magnetic moment of the substance,
so that BdM represents the work involved in changing the sample magnetic
moment by dM. Obtain an expression for the heat, δQ rev , exchanged between
the sample and its surroundings, and show that for U ≡ U(T , M), δQ rev can
be written as
δQ rev = C M (T ) dT +
∂U
∂M
T
− B
dM,
with C M (T ) the heat capacity of the solid sample at constant magnetization.