For each carbon-14 atom, there are approximately 10^12 carbon-12 atoms. This ratio is constant in each organism. The carbon-12 remains const

Question

For each carbon-14 atom, there are approximately 10^12 carbon-12 atoms. This ratio is constant in each organism. The carbon-12 remains constant but the carbon-14 decays. Suppose a piece of wood is analysed, and it contains 10^14 carbon-12 atoms and 40 carbon-14 atoms.

1) Determine how many carbon-14 atoms were present in the wood when it died.

2) Use the half life of carbon-14 atoms to obtain a function to model the number N of carbon-14 atoms present in the wood (t) years after it died

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Diễm Thu 3 years 2021-08-14T16:26:03+00:00 1 Answers 6 views 0

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  1. minhkhang
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    2021-08-14T16:27:36+00:00
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    Answer:

    1) There are 100 carbon-14 atoms in the wood when it died

    2) A function that models the number ‘N’ of carbon-14 atoms present in the wood (t) years after it died is  presented as follows;

    N = 100 \times \left (\dfrac{1}{2} \right )^{\dfrac{t}{5,730} }

    Step-by-step explanation:

    The given parameters are;

    The number of carbon-12 for each carbon-14 atom ≈ 10¹² carbon-12 atoms

    The number of carbon-12 in the piece of wood = 10¹⁴ carbon-12 atoms

    The number of carbon-14 in the piece of wood = 40 carbon-14 atoms

    The number of carbon-12 in an organism = Constant

    The number of carbon-14 in an organism = Decays

    1) Given that the number of carbon-12 in an organism is constant, and there are 10¹² carbon-12 atoms per each carbon-14 atom, therefore, we have;

    The number of carbon-12 atoms in the wood when it died = 10¹⁴ carbon-12 atoms

    The number of carbon-14 atoms in the wood when it died = (10¹⁴ carbon-12 atoms)/(10¹² carbon-12 atoms/(carbon-14 atom)) = 100 carbon-14 atoms

    The number of carbon-14 atoms in the wood when it died = 100 carbon-14 atoms

    2) The half life of a radioactive isotope is given by the following formula;

    N(t) = N_0 \cdot \left (\dfrac{1}{2} \right )^{\dfrac{t}{t_{\frac{1}{2} }} }

    The half life of carbon-14 atoms, t_{\frac{1}{2} } ≈ 5,730 years

    N₀ = The amount of carbon-14 present in the wood when it died = 100 carbon-14 atoms

    Therefore, we have;

    The number, N, of carbon-14 atoms present in the wood (t) years after it died is given as follows;

    N = 100 \times \left (\dfrac{1}{2} \right )^{\dfrac{t}{5,730} }

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