Modern Engineering Thermodynamics

122 CHAPTER 4: The First Law of Thermodynamics and Energy Transport Mechanisms
EXAMPLE 4.10
The magnetic susceptibility of the diamond in the gold engagement ring shown in
Figure 4.17 is −2.20 × 10 −5 at 20.0°C. Determine the (a) total magnetic and (b) material
magnetic work required to change the magnetic field of a 1 carat diamond having a
volume of 5.00 × 10 −6 m 3 from 0.00 to 1.00 × 10 3 A/m.
Solution
a. The total magnetic work required is given by Eq. (4.62) as
ð1W 2 Þ magnetic
= − μ 0 Vð1 + χ m Þ H2 2 −
H2 1
2
FIGURE 4.17
Example 4.10.
a) W total magnetic = ?
b) W material magnetic = ?
1 carat diamond
where μ 0 = 4π × 10 −7 V·s/A·m and χ m = −2.20 × 10 −5 . Then,
ð1W 2
Þ magnetic = − 4π × 10 −7 V ⋅ s
ð5:00 × 10 − 6 m 3 Þð1 − 2:20 × 10 −5 Þ 1:00 × 106 − 0A 2 /m 2
A ⋅ m
2
= −3:14 × 10 − 6 J
b. The magnetic work required to change the magnetic field strength inside the diamond alone is given by Eq. (4.63) as
and, using the values from part a, we get
2 W 2
H2 2 ð1W 2 Þ materia1
= − μ 0 Vχ −
H2 1
m
2
magnetization
= − 4π × 10 − 7 V ⋅ s
ð5:00 × 10 − 6 m 3 Þð−2:20 × 10 − 5 Þ 1:00 × 106 − 0A 2 /m 2
magnetic A ⋅ m
2
= 6:91 × 10 − 11 J
Exercises
22. The magnetic susceptibility of gold is −3.60 × 10 −5 . If the gold in the ring of Example 4.10 has a volume of 1.00 × 10 −5 m 3 ,
determine the total magnetic work required to change the magnetic field strength of the ring (the gold plus the diamond)
from 0 to 1.00 × 10 3 A/m. Answer: ( 1 W 2 ) magnetic = −9.42 × 10 −6 J.
23. The magnetic susceptibility of a ferromagnetic material such as iron varies with the applied magnetic field. However, if we
assume it is constant over a small range of field strength at a value of 1800, then determine the (a) total work and (b) the
material work required to magnetize a rectangular iron bar 0.500 inches square by 6.00 inches long from an initial magnetic
field strength of zero to a magnetic field strength of 100. A/m. Answer: ( 1 W 2 ) total = ( 1 W 2 ) iron = −2.78 × 10 −4 J.
4.7.4 Chemical Work
Chemical work occurs whenever a specific chemical species is added to or removed from a system. Here, the
generalized force is the intensive property μ i , the Gibbs chemical potential of chemical species i, and the generalized
displacement is the extensive property m i , the mass of the chemical species added or removed. 8 Since any
number of chemical species may be involved in a process, we write the chemical work as the sum over all k of
the i species that are moved from the system to the surroundings as
and so
ðdWÞ chemical = −∑ k
i=1
μ i dm i (4.64)
Z 2
ð 1
W 2 Þ chemical
= −
1
∑ k
i=1
μ i dm i (4.65)
8 In chemistry texts, the chemical potential is usually defined on a molar (i.e., per unit gram mole) basis. In this text, we define it as a
standard intensive (per unit mass) property.