calculate the half-life of radium if 5% of a sample of radium took 2.5 years to radioactive decay.

calculate the half-life of radium if 5% of a sample of radium took 2.5 years to radioactive decay.

**Answer:** The half-life of a radioactive substance is the time it takes for half of the original sample to decay. We can use the information given in the problem to calculate the half-life of radium.

We know that 5% of the original sample remains after 2.5 years, which means that 95% of the sample has decayed. Using the formula for exponential decay, we can relate the remaining fraction of the sample to the time elapsed and the half-life:

**Remaining fraction = (1/2)^(t/T)**

where t is the time elapsed, T is the half-life, and ^(x) indicates raising to the power of x.

Substituting the values given in the problem, we get:

0.05 = (1/2)^(2.5/T)

Taking the logarithm of both sides with base 2, we get:

log2(0.05) = (2.5/T) * log2(1/2)

Simplifying the right-hand side, we get:

log2(0.05) = -(2.5/T)

Solving for T, we get:

T = -(2.5/log2(0.05)) ≈ **22.14 years**

Therefore, the half-life of radium is approximately 22.14 years.