PHYSICS 218
SOLUTION TO HW 8
Created: November 20, 2004 7:15 pm
Last updated: November 21, 2004
1. Schroeder 1.16
(a) The three forces acting on the slab of thickness
dz
are gravity and the pressure from above and
below. To achieve equilibrium they must add to zero.
0=
−
mgρAdz
−
[
P
(
z
+
dz
)
−
P
(
z
)]
A
(1)
Expanding P(z) in a Taylor series yields
P
(
z
+
dz
)=
P
(
z
)+
∂P/∂z
×
dz
=
P
(
z
dP
(
z
) and after
division by
Adz
we thus Fnd:
dP
dz
=
−
mgρ
(2)
(b) The ideal gas law is
PV
=
Nk
B
T
⇒
ρ
=(
mP
)
/
(
k
B
T
) inserting gives the barometric equation
dP
dz
=
−
mg
k
B
T
P
(3)
(c) ±or a height independent temperature we use an exponential ansatz
P
(
z
P
(0)
e
λz
that yields
dP
dz
=
λP
(0)
e
λz
=
−
mg
k
B
T
P
(0)
e
λz
(4)
This again gives
λ
=
−
(
mg
)
/
(
k
B
T
) and thus
P
(
z
P
(0)
e
−
mgz/kT
(5)
Substituting this function for
P
(
z
) into the formula for
ρ
=
mP/kT
gives us
ρ
(
z
mP
(0)
kT
e
−
mgz/kT
=
ρ
(0)
e
−
mgz/kT
(6)
(d) The numerical value for
mg/kT
is equal to 1
.
18
×
10
−
4
/m
for an average molecule mass 28
.
8
u
and
a temperature of 293
K
.ThusweFnd
Ogden, Utah
Leadville, Colorado
Mt. Whitney, California
Mt. Everest, Nepal/Tibet
P / atm
0.84
0.69
0.59
0.35
However we have to bear in mind that these values are approximate. Especially for the Mt. Everest
we would Fnd a very di²erent value if we took the temperature decrease into account.
2. Schroeder 1.22
(a) Each molecule hitting the Area
A
will on average exert a force of 2
m
v
x
, dividing the total force by
this amount we Fnd an approximation for the average number of molecules hitting the wall.
∆
N
=
PA
2
m
v
x
∆
t
(7)
(b) Using
³
v
2
x
´
1
/
2
as an approximation to
v
x
we use
v
2
=
v
2
x
+
v
2
y
+
v
2
z
=3
v
2
x
kT/m
to Fnd
v
x
≈
q
.
1
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View Full DocumentPHYSICS 218
SOLUTION TO HW 8
(c) Using the ideal gas law to substitute
P
and (b) we fnd
dN
dt
=
−
PA
2
m
v
x
=
−
A
2
V
s
kT
m
N
(8)
Again an exponential ansatz
N
(
t
)=
N
(0)
e
−
t/τ
will lead us to the result
−
1
τ
N
(
t
−
A
2
V
s
m
N
(
t
)
⇒
τ
=
2
V
A
r
m
(9)
(d) Assuming the same data For air as in (1) we fnd the characteristic time
τ
=6
.
88
s
.
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 Fall '04
 POLLACK
 Thermodynamics, Magnetism, Force, Gravity, Schroeder

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