Is everyone having a good thanksgiving?

Suppose you find a rock that contains 10 micrograms of radioactive potassium-40, which has a half-life of 1.25 billion years. By

attashe74 [19]

Answer:

The rock is 3.75 billion years ago.

Explanation:

Given that:

Half life = 1.25 billion years

1 billion years = 10⁹ years

So,

Half life = 1.25 × 10⁹ years

$t_{1/2}=\frac {ln\ 2}{k}$

Where, k is rate constant

So,

$k=\frac {ln\ 2}{t_{1/2}}$

$k=\frac {ln\ 2}{1.25\times 10^9}\ year^{-1}$

The rate constant, k = 5.5452 × 10⁻¹⁰ years⁻¹

Initial amount [A₀] = 80 micrograms

Final amount $[A_t]$ = 10 micrograms

Using integrated rate law for first order kinetics as:

$[A_t]=[A_0]e^{-kt}$

So,

$10=80\times e^{-5.5452\times 10^{-10}\times t}$

$80e^{-5.5452\times \:10^{-10}x}=10$

$\frac{80e^{-5.5452\times \:10^{-10}x}}{80}=\frac{10}{80}$

$\ln \left(e^{-5.5452\times \:10^{-10}x}\right)=\ln \left(\frac{1}{8}\right)$

$x=\frac{10^{10}\times \:3\ln \left(2\right)}{5.5452}=3.75\times 10^9$

So,

<u>The rock is 3.75 billion years ago.</u>

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