9.5 The total differential 261
(ii) u 1 = 1 (x
21 + 1 y
21 + 1 z
2)
122(iii) x 1 = 1 r 1 sin 1 θ 1 cos 1 φ
0 Exercises 31–35
EXAMPLE 9.11Differential volume
The volume of a one-component thermodynamic system is a function of pressure
p, temperature T, and amount of substance n:V 1 = 1 V(p, T, n). The total differential
volume is
where
the thermal expansivity (coefficient of thermal expansion)
the isothermal compressibility
the molar volume
0 Exercise 36
V
V
n
mpT
=
∂
∂
,
κ=−
∂
∂
,
1
V
V
p
Tn
α=
∂
∂
,
1
V
V
T
pn
=−+ακVdT Vdp V dn
mdV
V
T
dT
V
p
dp
V
n
pn Tn pT
=
∂
∂
∂
∂
∂
∂
,,,
dn
=+sin cosθθθθφφφdr rcos cos d r−sin sin dφ
dx
x
r
dr
x
d
x
rr
=
∂
∂
∂
∂
∂
∂
θθ,, ,φφ
θ
θ
φ
dφ
∂
∂
=,
∂
∂
=,
∂
∂
=−
x
r
x
r
x
sin cosθ cos cos rsin sin
θ
φφθθ
φ
φφ
du
u
x
dx
u
y
dy
u
z
yz zx xy
=
∂
∂
∂
∂
∂
∂
,, ,
dz=++
u
xdx ydy zdz
1
()
∂
∂
=++ =,
∂
∂
=,
∂
∂
=
−u
x
xx y z
x
u
u
y
y
u
u
z
z
u
()
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