A biochemist used 10.000 grams of P-32 in a test on plant growth but forgot to record the date of his experiment. When he measured the amount of phosphorus-32 on another day, he found only 1.250 grams of the isotope remaining. If the half-life of P-32 is fourteen days, when did he start his experiment? And could you please explain how you did it!? Thanks! :)

Question

A biochemist used 10.000 grams of P-32 in a test on plant growth but forgot to record the date of his experiment. When he measured the amount of phosphorus-32 on another day, he found only 1.250 grams of the isotope remaining. If the half-life of P-32 is fourteen days, when did he start his experiment? And could you please explain how you did it!? Thanks! :)

Answer 1

To determine the date when the experiment started, we can use the concept of half-life in radioactive decay. The half-life is the amount of time it takes for half of a radioactive substance to decay.

In this case, we know that the half-life of P-32 is fourteen days. So, every fourteen days, half of the P-32 will decay, and the remaining half will remain.

We can set up an equation to solve for the number of half-lives that have occurred:

Remaining P-32 = Initial P-32 * (1/2)^(number of half-lives)

We are given that the remaining P-32 is 1.250 grams, and the initial P-32 was 10.000 grams. Let's substitute these values into the equation:

1.250 grams = 10.000 grams * (1/2)^(number of half-lives)

To find the number of half-lives, we need to isolate it on one side of the equation. Divide both sides of the equation by 10.000 grams:

(1.250 grams) / (10.000 grams) = (1/2)^(number of half-lives)

Now, take the natural logarithm (ln) of both sides of the equation. This will help us solve for the number of half-lives:

ln[(1.250 grams) / (10.000 grams)] = ln[(1/2)^(number of half-lives)]

Using the logarithmic property log base b(a^c) = c * log base b(a), the equation simplifies to:

ln(1.250) - ln(10.000) = number of half-lives * ln(1/2)

To solve for the number of half-lives, divide both sides of the equation by ln(1/2):

(number of half-lives) = [ln(1.250) - ln(10.000)] / [ln(1/2)]

By plugging these values into a calculator or using a math software, we can determine the number of half-lives. In this case, the number of half-lives is approximately 4.426 half-lives.

Now, to find the number of days, we multiply the number of half-lives by the half-life of P-32:

Number of days = (number of half-lives) * (half-life)

Number of days = 4.426 half-lives * 14 days/half-life

Number of days = approximately 61.9 days

Therefore, the biochemist started his experiment approximately 61.9 days ago.