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Chemistry Forums for Students => High School Chemistry Forum => Topic started by: jacobcolbert on February 17, 2007, 01:18:49 PM

Title: 2 Step Nuclear Decay
Post by: jacobcolbert on February 17, 2007, 01:18:49 PM

I recently have been trying to model 2 step nuclear decay. I said that X decays into Y with a rate constant k_x. And Y decays into Z with rate constant k_x. I then set up the differential equation dy/dt=k_xx-k_yY. I then substituted in X=X₀e^-k_xt for X in the differential equation. The result when I solve the equation however has the term (k_y-k_x) in the demominator of a fraction which makes no sense to me because because what if they are equal? And what if k_x is greater than k_y? There shouldnt be a negative amount of Y. Once i find a place to host my work as a picture I can post that. Anyways if anyone knows of a correct equation to model this or could confirm or reject my original differential equation it would be appreciated. Thanks.

Title: Re: 2 Step Nuclear Decay
Post by: Mitch on February 21, 2007, 12:08:12 AM

The equation doesn't work when ky equals kx. Great catch! Are you sure you are in high school though?

Title: Re: 2 Step Nuclear Decay
Post by: Yggdrasil on February 21, 2007, 01:30:20 AM

In the case where k_x = k_y = k you can derive a different model for Y by choosing a particular solution of the form te^kt.

In the case where k_x > k_y the equation will still be positive since this implies that e^k_xt < e^k_yt for all t > 0 and two negative expressions will multiply to give a positive value.