If a 40 gram sample of radioactive isotope was present at 4pm, but a 5 gram sample remained at 4:30 pm, what is the half-life in minutes?

ln(No/N) = kt

No = 40g
N = 5
k = ?
t = 90 min
Solve for k

Then k = 0.693/t1/2
Solve for half life in minutes.

Another way to do it is
(40/2*x) = 5
solve for x
Then 90 minutes/x = time for 1 half life.
You should get the same answer both ways.

To determine the half-life, we need to calculate the amount of time it takes for half of the radioactive isotope to decay. In this case, we know that the sample decreased from 40 grams to 5 grams in 30 minutes.

First, let's calculate the difference in grams between the initial and final amounts: 40 grams - 5 grams = 35 grams.

Next, let's find out how many half-lives occurred during this time by dividing the change in grams by the initial amount: 35 grams / 40 grams = 0.875.

Since each half-life represents a reduction to half the initial amount, we get the following equation: 1/2^n = 0.875, where n is the number of half-lives.

To solve this, we need to find the value of n, which represents the number of half-lives that have occurred. Taking the logarithm base 2 of both sides of the equation, we have: log base 2 (1/2^n) = log base 2 (0.875).

Simplifying further, we get: -n = log base 2 (0.875).

To find the value of n, we need to find the logarithm of 0.875 base 2: n = -log base 2 (0.875).

Using a scientific calculator or a computer program capable of computing logarithms, we can find that log base 2 (0.875) ≈ -0.135.

Therefore, n ≈ 0.135.

Since n represents the number of half-lives that have occurred, we know that each half-life occurred in 30 minutes. Thus, the half-life of this radioactive isotope is approximately 30 minutes.