Answer :
Step-by-step explanation:
xf = xo (.5)^n where x o = 737.8
xf = 92.23
n = # half lives
92.23 = 737.8 (.5)^n
92.23 / 737.8 = .5 ^n 'LOG' both sides , solve for 'n'
n = 2.999 = ~ 3 half lives
Answer:
3 half lives occurred
Step-by-step explanation:
Half-Life Equation
An equation can be derived to find the half life of a substance.
[tex]N(t)=N_0\left(\dfrac{1}{2} \right)^\frac{t}{t_H}[/tex],
where N(t) is the amount of the substance that remains after t time, [tex]N_0[/tex] is the initial amount, t is the time that elapses and [tex]t_H[/tex] is the amount of time it takes the substance to half its size.
Understanding the Exponent
The [tex]\dfrac{t}{t_H}[/tex] represents how many times the substance's quantity is halved or how many half-lives it experiences
Both t and [tex]t_H[/tex] must have the same metric of time, either than that, it can be in terms of,
- seconds
- minutes
- hours
- days
- etc.
For example,
if element A has a half life of 6 years and 12 years elapses then, element A halves itself twice or experiences two half-lives.
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Solving the Problem
The problem asks for the number of half-lives the given sample had in a course of 47 days, meaning that we must solve for the exponent in the equation the word problem makes.
[tex]92.23=737.8\left(\dfrac{1}{2} \right)^T[/tex],
let T be the number of half-lives that this substance experiences, writing out the exponent is unnecessary since the problem doesn't ask for the value of [tex]t_H[/tex] nor each element in the exponent's fraction.
Now we rearrange and isolate the T variable.
[tex]\dfrac{92.23}{737.8} =\left(\dfrac{1}{2}\right)^T[/tex]
[tex]0.125=\dfrac{1}{8} =\left(\dfrac{1}{2}\right)^T[/tex]
Knowing that,
[tex]log_{\dfrac{1}{2} } \left(\dfrac{1}{8} \right)=3[/tex]
or that,
[tex]\left(\dfrac{1}{2} \right)^3=\dfrac{1^3}{2^3} =\dfrac{1}{8}\\[/tex]
T = 3.
So, after 47 days the substances experiences 3 half-lives.
