- #1
SReinhardt
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What lead them to use e and the natural log of 2 in the decay formula? A much simpler (to me at least) method would is:
N=No*.5^(time/half life)
N=No*.5^(time/half life)
rock.freak667 said:Well since
[tex]N=N_0 e^{- \lambda t}[/tex]
when t=half-life(T); N=[itex]\frac{N_0}{2}[/itex]
[tex]\frac{N_0}{2}=N_0 e^{- \lambda T}[/tex]
simplify that by canceling the N_0 and then take logs and you'll eventually get
[tex]T=\frac{ln2}{\lambda}[/tex]
Vanadium 50 said:It's the same question as "why log base e and not log base 2"? It happens to make some calculations easier. (Note that your calculator has a ln(x) button but probably not a log2(x) button)
malawi_glenn said:the differential equation is:
[tex] \frac{dN}{dt} = \lambda N [/tex]
Solve it.
SReinhardt said:Wouldn't there have to be a negative sign in there somewhere >_>
I believe you have to use integrals to solve that, which I haven't done yet.
malawi_glenn said:yeah it should have a minus sign, good! :-)
Solving this:
[tex] \int N ^{-1}dN = - \int \lambda dt [/tex]
[tex] \ln(N(t)) - \ln(N(0)) = -\lambda t [/tex]
[tex] \ln(N(t)/N(0)) = -\lambda t [/tex]
[tex] N(t)/N(0) = e^{-\lambda t } [/tex]
[tex] N(t) = N(0) e^{-\lambda t } [/tex]
Lambda is the number of decays per unit time, is related to half life by:
[tex] \lambda = \frac{\ln 2}{T_{1/2}} [/tex]
The radioactive decay formula, also known as the decay law, describes the rate at which a radioactive substance decays over time. It states that the rate of decay is proportional to the amount of the substance present. This means that as the amount of the substance decreases, so does the rate of decay.
The mathematical representation of the radioactive decay formula is N(t) = N0 * e^(-λt), where N(t) is the amount of substance remaining at time t, N0 is the initial amount of substance, e is the mathematical constant 2.71828, and λ is the decay constant.
The decay constant, λ, is determined by the half-life of the radioactive substance. The half-life is the amount of time it takes for half of the initial amount of substance to decay. The decay constant is calculated using the equation λ = ln(2)/t1/2, where t1/2 is the half-life.
Yes, the radioactive decay formula applies to all radioactive substances. This is because the rate of decay is dependent on the amount of substance present, not the specific type of substance.
The radioactive decay formula is used in a variety of scientific research and applications, including carbon dating, radiometric dating, and nuclear medicine. It allows scientists to determine the age of objects, track the movement of materials through different systems, and diagnose and treat medical conditions.