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An unknown radioactive element decays with a half life of 0.440 days. What is the decay constant for the element? Express your answer in 1/day.

Sagot :

We are asked to determine the decay constant of a radioactive element. To do that we will use the following formula:

[tex]A=A_0e^{-kt}[/tex]

Where:

[tex]\begin{gathered} A=\text{ quantity of the element} \\ A_0=\text{ initial quantity} \\ k=\text{ decay constant} \\ t=\text{ time} \end{gathered}[/tex]

The half time is the time when the quantity of the element is half the initial quantity. Therefore, we have:

[tex]\frac{A_0}{2}=A_0e^{-kt}[/tex]

Now, we cancel out the initial quantitu:

[tex]\frac{1}{2}=e^{-kt}[/tex]

Now, we solve for "t". First, we take the natural logarithm to both sides:

[tex]\ln(\frac{1}{2})=-kt[/tex]

Now, we divide both sides by -t:

[tex]-\frac{1}{t}\ln(\frac{1}{2})=k[/tex]

Now, we plug in the value of the time:

[tex]-\frac{1}{0.44day}\ln(\frac{1}{2})=k[/tex]

Solving the operations:

[tex]1.575\frac{1}{day}=k[/tex]

Therefore, the decay constant is 1.575 1/day.