Physical Chemistry (I)
Quiz 02
Oct. 11, 2007
1.
(30%) The change of pressure (
Δ
P
) of a liquid with increasing temperature can be
estimated
if
one
knows
the
volumetric
thermal
expansion
coefficients
[
1
(
)
P
V
V
T
∂
=
∂
β
] and the isothermal compressibility [
1
(
)
T
V
V
P
∂
= 
∂
κ
] of the system.
(a)
Assuming
P
=
P
(
T
,
V
), using the mathematics of partial derivatives to derive
d
P
in terms of
β
and
κ
. (5%)
(b)
Using the chain rule for differentiation, show that the isobaric expansion
coefficient expressed in terms of density is given by
1
.
P
T
ρ
β
ρ
∂
= 
∂
(5%)
(c)
Because
V
is a state function and d
V
is an exact differential, show that the
isothermal compressibility and isobaric expansion coefficient are related by
.
T
P
P
T
β
κ
∂
∂
= 
∂
∂
(5%)
(d)
A vessel is filled completely with liquid water and sealed at 25.0ºC and a
pressure of 1.00 bar.
What is the pressure if the temperature of the system
is raised to 60.0ºC?
Assuming
β
water
= 2.04
×
10
–4
K
–1
,
β
vessel
= 1.02
×
10
–4
K
–1
, and
κ
water
= 4.59
×
10
–5
bar
–1
. (15%)
2.
(25%) The total differential of the internal energy can be written as
V
V
T
V
U
U
P
dU
dT
dV
C dT
T
P dV
T
V
T
∂
∂
∂
=
+
=
+

∂
∂
∂
.
(a)
Express the internal pressure
T
U
V
∂
∂
in terms of
P
,
β
,
T
, and
κ
. (5%)
(b)
Determine the value of
T
U
V
∂
∂
for an ideal gas. (5%)
(c)
Derive Joule coefficient (
μ
J
) in relation to
T
U
V
∂
∂
and
C
V
. (5%)
(d)
Derive the following expression for calculating the isothermal change in the
constant volume heat capacity:
2
2
.
V
T
V
C
P
T
V
T
∂
∂
=
∂
∂
(10%)
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3.
(25%)
The
total
differential
of
the
enthalpy
can
be
written
as
P
V
T
P
H
H
V
dH
dT
dP
C dT
V
T
dP
T
P
T
∂
∂
∂
=
+
=
+

∂
∂
∂
.
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 Winter '06
 pathievan
 Physical chemistry, Thermodynamics, pH, dt, P T T P

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