Question

After 30 days, 5.0 grams of a radioactive isotope remains from an original 40.-gram sample. what is the half-life of this element?

Answer 1

The half-life of the radioactive isotope is approximately 10 days, calculated by applying the exponential decay formula to the given values of the remaining amount of substance after a certain period.

Step-by-step explanation:

To calculate the half-life of the radioactive isotope in question, we use the concept that the amount of substance left over time follows an exponential decay model. For a substance with a half-life, the quantity remaining after one half-life will be half the original amount. After two half-lives, it will be one-fourth the original amount, and so on.

According to the question, 5.0 grams remain from an original 40-gram sample after 30 days. With this information, we need to determine how many half-lives have passed to go from 40 grams to 5 grams.

Let's use the formula for exponential decay: remaining amount = original amount * (1/2)^(time/half-life). Plugging in the values gives us: 5 = 40 * (1/2)^(30/half-life).

Solving for the half-life gives us: (30/half-life) = log2(40/5) or half-life = 30 / log2(8). Finally, we find the half-life of the radioactive isotope is approximately 10 days.

Answer 2

m₀ = 40 g, original mass
m₁ = 5.0 g, mass remaining after 30 days.

The decay equation is of the form

`m(t) = m_(0) e^(-kt)`
where
t = time, days
k = constant

Therefore

`40e^(-30k) = 5 \\ e^(-30k) = 5/40 = 0.125 \\ -30k = ln(0.125) \\ k = ln(0.125)/-30 = 0.0693`

At half-life, m = 20 g.
The time for half-life is

`e^(-0.0693t) = 1/2 \\ -0.0693t = ln(0.5) \\ t = ln(0.5)/-0.0693 = 10 \, days`

Answer: 10 days