You have just isolated a new radioactive element. If you can determine its half-life, you will win the Nobel Prize in physics. You purify a sample of 2 grams. One of your colleagues steals half of it, and eight days later you find that 0.1 gram of the radioactive material is still left. What is the half-life? (Round your answer to three significant digits.)

Question

You have just isolated a new radioactive element. If you can determine its half-life, you will win the Nobel Prize in physics. You purify a sample of 2 grams. One of your colleagues steals half of it, and eight days later you find that 0.1 gram of the radioactive material is still left. What is the half-life? (Round your answer to three significant digits.)

Answer 1

after the theft, there was 1 g left.

If the half-life is n days, then the fraction left after t days is

(1/2)^(t/n)

so, solve this for n:

0.1 = (1/2)^(8/n)
ln 0.1 = (8/n) ln 0.5
ln .1/ln .5 = 8/n
n = 8 * ln .5/ln .1
n = 2.4 days

Answer 2

To determine the half-life of the radioactive element, we can use the equation for exponential decay:

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

Where:
N is the final amount of radioactive material
N0 is the initial amount of radioactive material
t is the time that has passed
T is the half-life of the radioactive element

In this case, we have an initial amount of 2 grams and a final amount of 0.1 grams. Let's plug in these values into the equation:

0.1 = 2 * (1/2)^(t / T)

To simplify, divide both sides of the equation by 2:

0.05 = (1/2)^(t / T)

Now, take the logarithm of both sides to solve for t / T:

log(0.05) = log((1/2)^(t / T))

Using logarithmic properties, we can bring down the exponent:

log(0.05) = (t / T) * log(1/2)

Now, we need to solve for t / T, so divide both sides of the equation by log(1/2):

(t / T) = log(0.05) / log(1/2)

Evaluate the right side of the equation using a calculator:

(t / T) ≈ -2.995

Now, we can solve for T by multiplying both sides of the equation by T:

t ≈ -2.995 * T

Since the half-life cannot be negative, we take the absolute value:

t ≈ 2.995 * T

Now, we know that after eight days, 0.1 grams of the radioactive material is left. We need to convert this time into the units of the half-life (T) to find the value of T:

8 days ≈ 2.995 * T

Divide both sides of the equation by 2.995 to solve for T:

T ≈ 8 / 2.995

Evaluate the right side of the equation to get the value of T:

T ≈ 2.672 days

Rounding to three significant digits, the half-life of the radioactive element is approximately 2.67 days.