Question

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.

Answer 1

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

`A=A_0e^(-kt)`

Where:

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

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

`(A_0)/(2)=A_0e^(-kt)`

Now, we cancel out the initial quantitu:

`(1)/(2)=e^(-kt)`

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

`\ln((1)/(2))=-kt`

Now, we divide both sides by -t:

`-(1)/(t)\ln((1)/(2))=k`

Now, we plug in the value of the time:

`-(1)/(0.44day)\ln((1)/(2))=k`

Solving the operations:

`1.575(1)/(day)=k`

Therefore, the decay constant is 1.575 1/day.