857 views
5 votes
a chemist, running tests on an unknown sample from an illegal waste dump, isolates 71 grams of what she suspects is a radioactive element. in order to help identify the element, she would like to know the half-life. she determines that after 46 days only 54 grams of the original element remains. what is the half-life of this mystery element? round your answer to two decimal places, if necessary.

User Creris
by
8.2k points

1 Answer

3 votes

Final answer:

The half-life of the radioactive element in question is approximately 65.66 days, calculated using the half-life formula and the given data of weight reduction from 71 to 54 grams over the span of 46 days.

Step-by-step explanation:

The student is asking for the half-life of a radioactive element that was found in an illegal waste dump. The sample started with 71 grams and after 46 days, only 54 grams were left. To find the half-life of this element, we can use the concept that after each half-life, half of the original amount of radioactive material remains.

Let's use the formula:

N = N0 * (1/2)^(t/T)

Where:

  • N is the remaining amount of substance after time t,
  • N0 is the original amount of substance,
  • T is the half-life of the substance,
  • t is the time elapsed.

We have N = 54 grams, N0 = 71 grams, and t = 46 days. We need to find T.

54 = 71 * (1/2)^(46/T)

To isolate T, we must use logarithms. Taking the natural log (ln) of both sides:
ln(54) = ln(71) + (46/T) * ln(1/2)

Solving for T gives us:
T = -46 / [(ln(54) - ln(71)) / ln(1/2)]

After calculating, we round to two decimal places:

T ≈ 65.66 days

So, the half-life of this radioactive element is approximately 65.66 days.

User Ed Fine
by
7.7k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.