Respuesta :
Answer:
It would take 27.5 minutes the element to decay to 154 grams.
Step-by-step explanation:
The decay equation:
[tex]\frac {dN}{dt}\propto -N[/tex]
[tex]\Rightarrow \R\frac {dN}{dt}=-\lambda N[/tex]
[tex]\Rightarrow \frac {dN}N=-\lambda dt[/tex]
Integrating both sides
[tex]\Rightarrow \int \frac {dN}N=\int-\lambda dt[/tex]
[tex]\Rightarrow ln|N|=-\lambda t+c[/tex]
When t=0, N=[tex]N_0[/tex] = initial amount
[tex]ln|N_0|=-\lambda .0+c[/tex]
[tex]\Rightarrow c=ln|N_0|[/tex]
[tex]ln|N|=-\lambda t+ln|N_0|[/tex]
[tex]\Rightarrow ln|N|-ln|N_0|=-\lambda t[/tex]
[tex]\Rightarrow ln|\frac{N}{N_0}|=-\lambda t[/tex]
Decay equation:
[tex]ln|\frac{N}{N_0}|=-\lambda t[/tex]
Given that, the half life of of element X is 11 minutes.
For half life, [tex]N=\frac12 N_0[/tex], t= 11 min.
[tex]ln|\frac{N}{N_0}|=-\lambda t[/tex]
[tex]\Rightarrow ln|\frac{\frac12N_0}{N_0}|=-\lambda . 11[/tex]
[tex]\Rightarrow ln|\frac12}|=-\lambda . 11[/tex]
[tex]\Rightarrow -\lambda . 11=ln|\frac12}|[/tex]
[tex]\Rightarrow \lambda =\frac{ln|\frac12|}{-11}[/tex]
[tex]\Rightarrow \lambda =\frac{ln|2|}{11}[/tex] [ [tex]ln|\frac12|=ln|1|-ln|2|=-ln|2|[/tex] , since ln|1|=0]
N=154 grams, [tex]N_0[/tex] = 870 grams, t=?
[tex]ln|\frac{N}{N_0}|=-\lambda t[/tex]
[tex]\Rightarrow ln|\frac{154}{870}|=-\frac{ln|2|}{11}.t[/tex]
[tex]\Rightarrow t= \frac{ln|\frac{154}{870}|\times 11}{-ln|2|}[/tex]
=27.5 minutes
It would take 27.5 minutes the element to decay to 154 grams.
Answer:
60.6 minutes
Step-by-step explanation:
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