Environmental Engineering Reference
In-Depth Information
m
3
67
10
27
kg
¼
3
106
10
31
m
3
N
p
¼
0
:
3397
1
:
527
105 kg
=
=
1
:
:
:
ð
2
:
2
Þ
527
10
5
kg
m
3
68
10
27
kg
¼
1
46
10
31
m
3
N
He
¼
0
:
6405
1
:
=
=
6
:
:
:
ð
2
:
3
Þ
The electron concentration, then is
N
e
¼
(3.106
þ
2 (1.46))
10
31
m
3
¼
6.026
10
31
m
3
. (For comparison, the free electron and positive ion densities in a metal
are on the order of 5
10
28
m
3
, about 1000 times smaller.) From
N
p
¼
3.106
10
31
m
3
, we can infer that the interproton spacing is
N
p
1/3
¼
3.18
10
11
m.
Compared to the hydrogen atom radius
a
o
¼
0.0529 nm
¼
5.29
10
11
m, this
spacing is about 0.6
a
o
, but on a femtometer basis it is large, 31 800 f. From an
atomic point of view, the spacing less than the Bohr radius would mean that the
Mott transition (Chapter 3) has occurred, electrons are free to roamaway from their
protons, even at low temperature. The protons, however, are a dilute system
because their spacing greatly exceeds their charge radius. This means that only
two-particle collisions will be at all likely to occur. (We will see in a moment that the
classical approach distance at the available energy is 1113 f, too great a spacing for a
nuclear reaction to occur).
We will need to estimate the total number of protons,
N
cp
in the suns core, dened
as 0
r R
S
/4. Since the sun is not a solid but a dense gas, its density strongly varies
with radius. It is reported (www.nasa.gov/worldbook/sun_worldbook.html dated 11/
27/2007.) that the density at
R
S
/4 is 20 g/cc, about 0.133 relative to the density at
r ¼
0.
It is also reported [19] that the density
r
(
r
) decays exponentially as
r
(
r
)
¼
exp(
ar
).
With radius in units of
R
s
we have
r
(0.25)/
r
(0)
¼
0.133
¼
exp(
0.25
a
) that gives
a¼
8.06
R
S
. This function will apply to the proton density,
N
p
, with value at
r ¼
0,
N
p
(0)
¼
3.106
10
31
m
3
.
Using this information, we can write
ð
0
:
25
4
pr
2
exp
ð
8
51
10
56
N
cp
¼ N
p
ð
0
Þ
:
06
rÞ
d
r ¼
2
:
:
0
This estimate may be high, since it is also reported [19, 20] that the
total
number of
protons in the sun is 8.9
10
56
.
Recall fromChapter 1 that the contact distance for two protons is about 2.4 f, so the
protons in the suns core are far apart from this point of view. Since the nuclear force
has a range of only about 1 f, the particles have to approach each other within a few
femtometers to react. We can reasonably apply the classical speed distribution
function (given in Chapter 1) to the protons at the suns core. Making use of the
core temperature 15 million K, the most probable speed is (2
kT
/
m
)
1/2
¼
0.498
10
6
m/s, and the corresponding kinetic energy is 1293 eV. Again, taking the core
temperature of 15 million K, the closest approach, call it
r
2
, requires
k
B
T¼ k
c
e
2
/
r
2
¼
1293 eV, which gives
r
2
¼
1113 f. Since this ismuch larger than
r ¼
2.4 f, twice the
charge radius of a proton, this thermally available approach distance is far too large
for any reaction to occur.
So the classical particle picture is inadequate to explain the
