The Pb-Pb isochron is slightly different type of isochron
than what I discussed in the other thread. Lets look at
how this one works.
As before we will assume nothing being gained or lost and see with
that condition a line will form and that if something is gained or
lossed the proof of linearity becomes invalid and a line would not
be expected.
P[sub]now[/sub] = P[sub]orig[/sub] e[sup]-kt[/sup]
P[sub]now[/sub] e[sup]kt[/sup] = P[sub]now[/sub] + D[sub]formed[/sub]
P[sub]now[/sub] e[sup]kt[/sup] = P[sub]now[/sub] + D[sub]now[/sub] - D[sub]orig[/sub]
P[sub]now[/sub] ( e[sup]kt[/sup] - 1 ) = D[sub]now[/sub] - D[sub]orig[/sub]
D[sub]now[/sub] = D[sub]orig[/sub] + P[sub]now[/sub] ( e[sup]kt[/sup] - 1 )
Pb-Pb actually uses two distinct decay series: [sup]235[/sup]U to [sup]207[/sup]Pb and
[sup]238[/sup]U to [sup]206[/sup]Pb. There are no intermediates in common in these
two series and hense the two series are independent of each other. Lets plug in
to the last equation for each of the two above series:
[sup]207[/sup]Pb[sub]now[/sub] = [sup]207[/sup]Pb[sub]orig[/sub] + [sup]235[/sup]U[sub]now[/sub] ( e[sup]k[sub]2[/sub]t[/sup] - 1 )
[sup]206[/sup]Pb[sub]now[/sub] = [sup]206[/sup]Pb[sub]orig[/sub] + [sup]238[/sup]U[sub]now[/sub] ( e[sup]k[sub]1[/sub]t[/sup] - 1 )
The subscripts on the decay constants are because each radioactive decay has a different
decay constant. The above two equations can be combined into a single equation:
([sup]207[/sup]Pb[sub]now[/sub]-[sup]207[/sup]Pb[sub]orig[/sub])/([sup]206[/sup]Pb[sub]now[/sub]-[sup]206[/sup]Pb[sub]orig[/sub]) = ([sup]235[/sup]U[sub]now[/sub]/[sup]238[/sup]U[sub]now[/sub]) ( e[sup]k[sub]2[/sub]t[/sup] -1 )/( e[sup]k[sub]1[/sub]t[/sup] - 1)
[sup]235[/sup]U[sub]now[/sub]/[sup]238[/sup]U[sub]now[/sub] is the ratio of the two isotopes
in the here and now. This ratio is constant since this isotope ratio is known from
to be the same on the Earth, on the Moon and on meteorites. This can be replaced by
the measured ratio. [sup]204[/sup]Pb is form a lead that is not involved with either
radiometric series. Given a closed system its amount will not change. It will perform
the role of D[sub]i[/sub] did in the previous method of isochron mentioned.
( ([sup]207[/sup]Pb/[sup]204[/sup]Pb)[sub]now[/sub] - ([sup]207[/sup]Pb/[sup]204[/sup]Pb)[sub]orig[/sub] ) ) / ( ([sup]206[/sup]Pb/[sup]204[/sup]Pb)[sub]now[/sub] - ([sup]206[/sup]Pb/[sup]204[/sup]Pb)[sub]orig[/sub] ) )
= (1/137.88) ( e[sup]k[sub]2[/sub]t[/sup] - 1)/( e[sup]k[sub]1[/sub]t[/sup] -1 )
A plot [sup]206[/sup]Pb/[sup]204[/sup]Pb vs. [sup]207[/sup]Pb/[sup]204[/sup]Pb will
give a straight line. The age can be determined by the slope. Unlike the
previous type of isochron method I discussed the y-intercept has no physical
interpretation thus the initial daughter cannot be calculated though the method
works with any amount of intial daughter.
This is probably not immediately obvious that the equation is a line so lets
show it. The right hand side is a constant since we are doing this
at a specific point in time (i.e. right now). The initial isotope ratios are
obviously constants as well. So we have:
([sup]207[/sup]Pb/[sup]204[/sup]Pb - c[sub]1[/sub])/([sup]206[/sup]Pb/[sup]204[/sup]Pb - c[sub]2[/sub]) = c[sub]3[/sub]
The c's are constants. Lets rearrange:
[sup]207[/sup]Pb/[sup]204[/sup]Pb = c[sub]3[/sub][sup]206[/sup]Pb/[sup]204[/sup]Pb + c[sub]4[/sub]
This is in the form of y=mx+b and thus the plot is a line.
According to modern ideas of how the solar system formed, the solar system was once
a uniform nebula. If this is true then measuring the isotope ratios of various
objects in the solar system and ploting them as described above should degenerate
a straight line (the isochron) whose slope reflects the amount of time since the
solar system's isotope ratios were uniform. If this hypothesis is false the odds
are overwhelming that no line will form since there is no reason for a line
to form.
Since the system says I have to have a hundred posts to post an image I will have to have everyone click to see the image.
Image of the isochron diagram discussed
Again this is a self-testing method since if the assumptions of this
method are false, the isochron will fail to form and hense the geologist
will know that he has to try something else.
than what I discussed in the other thread. Lets look at
how this one works.
As before we will assume nothing being gained or lost and see with
that condition a line will form and that if something is gained or
lossed the proof of linearity becomes invalid and a line would not
be expected.
P[sub]now[/sub] = P[sub]orig[/sub] e[sup]-kt[/sup]
P[sub]now[/sub] e[sup]kt[/sup] = P[sub]now[/sub] + D[sub]formed[/sub]
P[sub]now[/sub] e[sup]kt[/sup] = P[sub]now[/sub] + D[sub]now[/sub] - D[sub]orig[/sub]
P[sub]now[/sub] ( e[sup]kt[/sup] - 1 ) = D[sub]now[/sub] - D[sub]orig[/sub]
D[sub]now[/sub] = D[sub]orig[/sub] + P[sub]now[/sub] ( e[sup]kt[/sup] - 1 )
Pb-Pb actually uses two distinct decay series: [sup]235[/sup]U to [sup]207[/sup]Pb and
[sup]238[/sup]U to [sup]206[/sup]Pb. There are no intermediates in common in these
two series and hense the two series are independent of each other. Lets plug in
to the last equation for each of the two above series:
[sup]207[/sup]Pb[sub]now[/sub] = [sup]207[/sup]Pb[sub]orig[/sub] + [sup]235[/sup]U[sub]now[/sub] ( e[sup]k[sub]2[/sub]t[/sup] - 1 )
[sup]206[/sup]Pb[sub]now[/sub] = [sup]206[/sup]Pb[sub]orig[/sub] + [sup]238[/sup]U[sub]now[/sub] ( e[sup]k[sub]1[/sub]t[/sup] - 1 )
The subscripts on the decay constants are because each radioactive decay has a different
decay constant. The above two equations can be combined into a single equation:
([sup]207[/sup]Pb[sub]now[/sub]-[sup]207[/sup]Pb[sub]orig[/sub])/([sup]206[/sup]Pb[sub]now[/sub]-[sup]206[/sup]Pb[sub]orig[/sub]) = ([sup]235[/sup]U[sub]now[/sub]/[sup]238[/sup]U[sub]now[/sub]) ( e[sup]k[sub]2[/sub]t[/sup] -1 )/( e[sup]k[sub]1[/sub]t[/sup] - 1)
[sup]235[/sup]U[sub]now[/sub]/[sup]238[/sup]U[sub]now[/sub] is the ratio of the two isotopes
in the here and now. This ratio is constant since this isotope ratio is known from
to be the same on the Earth, on the Moon and on meteorites. This can be replaced by
the measured ratio. [sup]204[/sup]Pb is form a lead that is not involved with either
radiometric series. Given a closed system its amount will not change. It will perform
the role of D[sub]i[/sub] did in the previous method of isochron mentioned.
( ([sup]207[/sup]Pb/[sup]204[/sup]Pb)[sub]now[/sub] - ([sup]207[/sup]Pb/[sup]204[/sup]Pb)[sub]orig[/sub] ) ) / ( ([sup]206[/sup]Pb/[sup]204[/sup]Pb)[sub]now[/sub] - ([sup]206[/sup]Pb/[sup]204[/sup]Pb)[sub]orig[/sub] ) )
= (1/137.88) ( e[sup]k[sub]2[/sub]t[/sup] - 1)/( e[sup]k[sub]1[/sub]t[/sup] -1 )
A plot [sup]206[/sup]Pb/[sup]204[/sup]Pb vs. [sup]207[/sup]Pb/[sup]204[/sup]Pb will
give a straight line. The age can be determined by the slope. Unlike the
previous type of isochron method I discussed the y-intercept has no physical
interpretation thus the initial daughter cannot be calculated though the method
works with any amount of intial daughter.
This is probably not immediately obvious that the equation is a line so lets
show it. The right hand side is a constant since we are doing this
at a specific point in time (i.e. right now). The initial isotope ratios are
obviously constants as well. So we have:
([sup]207[/sup]Pb/[sup]204[/sup]Pb - c[sub]1[/sub])/([sup]206[/sup]Pb/[sup]204[/sup]Pb - c[sub]2[/sub]) = c[sub]3[/sub]
The c's are constants. Lets rearrange:
[sup]207[/sup]Pb/[sup]204[/sup]Pb = c[sub]3[/sub][sup]206[/sup]Pb/[sup]204[/sup]Pb + c[sub]4[/sub]
This is in the form of y=mx+b and thus the plot is a line.
According to modern ideas of how the solar system formed, the solar system was once
a uniform nebula. If this is true then measuring the isotope ratios of various
objects in the solar system and ploting them as described above should degenerate
a straight line (the isochron) whose slope reflects the amount of time since the
solar system's isotope ratios were uniform. If this hypothesis is false the odds
are overwhelming that no line will form since there is no reason for a line
to form.
Since the system says I have to have a hundred posts to post an image I will have to have everyone click to see the image.
Image of the isochron diagram discussed
Again this is a self-testing method since if the assumptions of this
method are false, the isochron will fail to form and hense the geologist
will know that he has to try something else.
