Decay contant of xenon and strontium

[cowboi123]

Homework Statement

The decay constant of xenon-142 is 0.462(s^-1), and the decay constant of strontium-90 is 7.85*10^-10 (s^-1). Initially, a sample of radioactive waste material contains equal numbers of strontium-90 and xenon-142 nuclei.

Calculate the time taken for the ratio: number of strontium-90 nuclei/number of xenon-142 nuclei to become equal to 1.20*10^6.

Homework Equations

Not sure

The Attempt at a Solution

Not sure, please help!

 

[ideasrule]

Have you seen the formula N=N0*e^(-kt)? Write out a formula for xenon, a formula for strontium, and a formula for the strontium:xenon ratio. The reason for doing this is that you know this ratio's numerical value and can set it equal to the formula.

 

[kerimek]

I would first identify the relevant equations and principles related to radioactive decay. The first equation that comes to mind is the decay law, which states that the rate of decay (dN/dt) of a radioactive material is proportional to the amount of material present (N) and the decay constant (λ). This can be expressed as dN/dt = -λN.

In this case, we have two different decay constants for xenon-142 and strontium-90. We also know that initially, the sample contains equal numbers of both nuclei. Using this information, we can set up the following equations:

dN_xe/dt = -λ_xe * N_xe
dN_sr/dt = -λ_sr * N_sr

Where dN_xe/dt and dN_sr/dt represent the rates of decay for xenon-142 and strontium-90 respectively, and N_xe and N_sr represent the number of nuclei for each isotope.

Since we are interested in the ratio of strontium-90 to xenon-142 nuclei, we can divide the two equations to eliminate the decay constants:

(dN_sr/dt) / (dN_xe/dt) = (N_sr/N_xe)

We also know that initially, N_sr = N_xe, so we can substitute that into the equation:

(dN_sr/dt) / (dN_xe/dt) = 1

Now, we can integrate both sides of the equation to solve for the time (t) when the ratio of nuclei becomes equal to 1.20*10^6:

∫(dN_sr/dt) / (dN_xe/dt) dt = ∫1 dt
∫(N_sr/N_xe) dt = t

Since we want the ratio to be equal to 1.20*10^6, we can rewrite the equation as:

∫(1.20*10^6) dt = t

Solving for t, we get t = 1.20*10^6 seconds. This is the time it takes for the ratio of strontium-90 to xenon-142 nuclei to become equal to 1.20*10^6.