Science > Physics > Define proportionality constant in radioactive decay
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Science > Physics |
| User: |
"" |
| Date: |
20 Sep 2007 06:28:25 AM |
| Object: |
Define proportionality constant in radioactive decay |
Hello all, I am prepering something for my students on first order
differential equations.
The decay-rate of radioactive material is proportional to the number
of atoms in the material:
Amont of plutonium: P(t)
decay rate dP/dt.
The equation becomes dP/dt =-k*P, where k is around 3E-5 according to
my book.
Looking at the dimensions k must be per seconds, so my explanation of
k would be that k= the inverse of the average time one atom needs to
decompose into Neptunium (Is it Neptunium?)
Am I right? And how do they measure k?
Greatful for some help
Kindest regards,
Lasse Karagiannis
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| User: "Robert S" |
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| Title: Re: Define proportionality constant in radioactive decay |
20 Sep 2007 07:47:20 AM |
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On 20 Sep, 12:28, "Lasse.Karagian...@swipnet.se"
<Lasse.Karagian...@swipnet.se> wrote:
Hello all, I am prepering something for my students on first order
differential equations.
The decay-rate of radioactive material is proportional to the number
of atoms in the material:
Amont of plutonium: P(t)
decay rate dP/dt.
The equation becomes dP/dt =-k*P, where k is around 3E-5 according to
my book.
Looking at the dimensions k must be per seconds,
s^-1
so my explanation of
k would be that k= the inverse of the average time one atom needs to
decompose into Neptunium (Is it Neptunium?)
When time = 1/k, the amount of plutonium left is the amount of
plutonium at the start divided by e.
P(t) = P(0)exp(-kt)
Am I right? And how do they measure k?
By measuring the decay event rate.
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| User: "PD" |
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| Title: Re: Define proportionality constant in radioactive decay |
20 Sep 2007 01:06:43 PM |
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On Sep 20, 6:28 am, "Lasse.Karagian...@swipnet.se"
<Lasse.Karagian...@swipnet.se> wrote:
Hello all, I am prepering something for my students on first order
differential equations.
The decay-rate of radioactive material is proportional to the number
of atoms in the material:
Amont of plutonium: P(t)
decay rate dP/dt.
The equation becomes dP/dt =-k*P, where k is around 3E-5 according to
my book.
Looking at the dimensions k must be per seconds, so my explanation of
k would be that k= the inverse of the average time one atom needs to
decompose into Neptunium (Is it Neptunium?)
Am I right? And how do they measure k?
By counting the time it takes for dP/dt (the hit rate in a Geiger
counter, for example) to decline to (dP/dt)/e, which is 1/k.
Greatful for some help
Kindest regards,
Lasse Karagiannis
.
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| User: "Puppet_Sock" |
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| Title: Re: Define proportionality constant in radioactive decay |
20 Sep 2007 10:38:36 AM |
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On Sep 20, 7:28 am, "Lasse.Karagian...@swipnet.se"
<Lasse.Karagian...@swipnet.se> wrote:
Hello all, I am prepering something for my students on first order
differential equations.
The decay-rate of radioactive material is proportional to the number
of atoms in the material:
Amont of plutonium: P(t)
decay rate dP/dt.
The equation becomes dP/dt =-k*P, where k is around 3E-5 according to
my book.
Your book better give units for this, or it's not a very good book.
Looking at the dimensions k must be per seconds, so my explanation of
k would be that k= the inverse of the average time one atom needs to
decompose into Neptunium (Is it Neptunium?)
Am I right? And how do they measure k?
Usually the letter used for this parameter is the greek letter lambda,
and it's called a time constant. Since it's annoying to type lambda
all the time, I'll just put L, and you should remember it's lambda.
When it's done the way you did it, L is called the decay constant.
The solution to the differential equation, as Robert S has said, is
this.
P(t) = P(t=0) exp(- L t)
dP/dt = - L P(t)
Here, P(t=0) is the amount of stuff at time 0. What this means is,
the rate of decays, the number of decays per unit time, is L P.
You can measure L by measuring the decay rate, and then
dividing by the amount of P that produces this rate.
From this you could work out the averge life time of each Pu atom.
It's the integral of the number at each time divided by the starting
number.
integral_0^infinity P0 exp(-Lt) dt / P0
And this is just 1/L. So your explanation is correct.
Another way to express this is in terms of a half life. That's the
time it would take for half the material to decay. You can work
out the half life as follows. Work out the time, T2, required
for this to be true.
exp(- L T2) = 1/2
Take the natural log of boths sides.
L T2 = ln(2)
And so T2 = ln(2) / L is the half life.
Yet another way to express things is in terms of a time constant.
That's just 1/L and is usually given the greek letter tau.
tau = 1/L
By the way, the decay constant, and the decay mode, depends on
the isotope of Plutonium. Some of them decay by Alpha decay into
a Uranium isotope. One decays by Beta decay into Americurium.
Each decay has a different time constant.
http://en.wikipedia.org/wiki/Plutonium
Note that what is given there are half lives, and they are in years.
So you'd have to work out the decay constant, and be careful
about units.
Socks
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| User: "John C. Polasek" |
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| Title: Re: Define proportionality constant in radioactive decay |
20 Sep 2007 12:53:34 PM |
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On Thu, 20 Sep 2007 08:38:36 -0700, Puppet_Sock
<puppet_sock@hotmail.com> wrote:
On Sep 20, 7:28 am, "Lasse.Karagian...@swipnet.se"
<Lasse.Karagian...@swipnet.se> wrote:
Hello all, I am prepering something for my students on first order
differential equations.
The decay-rate of radioactive material is proportional to the number
of atoms in the material:
Amont of plutonium: P(t)
decay rate dP/dt.
The equation becomes dP/dt =-k*P, where k is around 3E-5 according to
my book.
snip
Do this:
(dP/P)/dt = -k units per second
since dP/P is the fractional change
ln(P) = -kt
P/P0 = exp(-kt) = exp(-t/T)
The characteristic time is T = 1/k = 1/3e-5 = 1/.00003 = 33,333
seconds or 9.26hrs to decay entirely IF it continued at that rate.
P/P0 = exp(-t/33,333)
John Polasek
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