El decaimiento radioactivo de un elemento químico respecto al tiempo esta dado por la funciónN ( t ) = No e k t ?

Dianita es sencillo,

La ecuación para el decaimiento radiactivo es :

<img src="https://tex.z-dn.net/?f=N%28t%29%3DN_%7B0%7De%5E%7B-kt%7D" /> Ec.

1

Para la primera pregunta te informan que No = 36grs, k = 0, 026 y t = 100, sustituyendo todo en la Ec.

1 tenemos :

<img src="https://tex.z-dn.net/?f=N%28100%29%3D36%5Bgrs%5De%5E%7B-0%2C026%28100%29%7D%3D36%5Bgrs%5De%5E%7B-2%2C6%7D%5Capprox2%2C67grs" /> Ec.

1

Para la segunda pregunta hay que despejar t de la Ec.

1, así :

<img src="https://tex.z-dn.net/?f=N%28t%29%3DN_%7B0%7De%5E%7B-kt%7D%20%3D%3E%5Cfrac%7BN%28t%29%7D%7BN_%7B0%7D%7D%3De%5E%7B-kt%7D%3D%3Eln%28%5Cfrac%7BN%28t%29%7D%7BN_%7B0%7D%7D%29%3D-kt" /> \ frac{N(t)}{N_{0}} = e ^ { - kt} = >ln( \ frac{N(t)}{N_{0}}) = - kt" alt = "N(t) = N_{0}e ^ { - kt} = > \ frac{N(t)}{N_{0}} = e ^ { - kt} = >ln( \ frac{N(t)}{N_{0}}) = - kt" align = "absmiddle" class = "latex - formula">

<img src="https://tex.z-dn.net/?f=ln%28%5Cfrac%7BN_%7B0%7D%7BN%28t%29%7D%29%3Dkt%3D%3Et%3D%5Cfrac%7B1%7D%7Bk%7Dln%28%5Cfrac%7BN_%7B0%7D%7BN%28t%29%7D%29" />t = \ frac{1}{k}ln( \ frac{N_{0}{N(t)})" alt = "ln( \ frac{N_{0}{N(t)}) = kt = >t = \ frac{1}{k}ln( \ frac{N_{0}{N(t)})" align = "absmiddle" class = "latex - formula"> Ec.

2

Luego, sustitututendo en la Ec.

2 : N(t) = 1gr, k = 0, 026 y No = 36grs, tenemos :

<img src="https://tex.z-dn.net/?f=t%3D%5Cfrac%7B1%7D%7B0%2C026%7Dln%28%5Cfrac%7B36%5Bgrs%5D%7D%7B1%5Bgrs%5D%7D%29%3D%3Et%5Capprox137%2C8" />t \ approx137, 8" alt = "t = \ frac{1}{0, 026}ln( \ frac{36[grs]}{1[grs]}) = >t \ approx137, 8" align = "absmiddle" class = "latex - formula"> años.