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if 6% of the original radioactive isotope x remains in a sample of your fossil, approximately how old is the fossil?

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xero099

The approximate age of the fossil is 507.36 million years.

Determine the age of a fossil by radioactive isotope decay

The amount of an isotope ([tex]n[/tex]) decays exponentially in time ([tex]t[/tex]), in years, whose formula is presented below:

[tex]n = n_{o}\cdot e^{-\frac{t}{\tau} }[/tex]   (1)

Where:

  • [tex]n_{o}[/tex] - Initial amount of the isotope.
  • [tex]n[/tex] - Current amount of the isotope.
  • [tex]t[/tex] - Time, in years
  • [tex]\tau[/tex] - Time constant, in years

The time constant ([tex]\tau[/tex]), in years, is calculated in terms of half-life ([tex]t_{1/2}[/tex]), in years:

[tex]\tau = \frac{t_{1/2}}{\ln 2}[/tex]   (2)

If we know that [tex]\frac{n}{n_{o}} = 0.06[/tex] and [tex]t_{1/2} = 125000000\,yr[/tex], then the age of the fossil is:

[tex]\tau = \frac{125000000\,yr}{\ln 2}[/tex]

[tex]\tau \approx 180336880.1\,yr[/tex]

[tex]t = -\tau \cdot \ln \frac{n}{n_{o}}[/tex]  

[tex]t = -180336880.1\cdot \ln 0.06[/tex]

[tex]t \approx 507361711.1\,yr[/tex]

The approximate age of the fossil is 507.36 million years. [tex]\blacksquare[/tex]

Remark

The statement is incomplete, complete form is shown below:

A radioactive isotope has a half-life of 125,000,000 years. If 6 % of the original radioactive isotope X remains in a sample of your fossil, approximately how old is the fossil?

To learn more on isotopes, we kindly invite to check this verified question: https://brainly.com/question/13214440

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