All isotopes of technetium are radioactive, but they have widely varying half-lives. If an 800.0 g sample of technetium-99 decays to 100.0 g of technetium-99 in 639,000 y, what is its half life

same procedure

213,000 years

To determine the half-life of technetium-99, we can use the decay equation:

N = N0 * (1/2)^(t / t1/2)

Where:
N is the final amount of technetium-99 (100.0 g),
N0 is the initial amount (800.0 g),
t is the time that has passed (639,000 years), and
t1/2 is the half-life that we need to find.

We can rearrange the equation to solve for t1/2:

(1/2)^(t / t1/2) = N / N0
(t / t1/2) = log base (1/2) of (N / N0)
t1/2 = t / (log base (1/2) of (N / N0))

Now we can substitute the given values into the equation to find the half-life:

t1/2 = 639,000 years / (log base (1/2) of (100.0 g / 800.0 g))

To evaluate this expression, we need to take the logarithm of the ratio of the final and initial amounts of technetium-99, using base (1/2). We can use the logarithmic identity:

log base b of a = log base c of a / log base c of b

In this case, c = 1/2, a = 100.0 g / 800.0 g = 1/8, and b is the base of the logarithm we want to evaluate (unknown).

Therefore, our equation becomes:

t1/2 = 639,000 years / (log base c of (1/8) / log base c of (1/2))

Now we can calculate the value of t1/2:

t1/2 = 639,000 years / (log base c of (1/8) / log base c of (1/2))

Finding the exact value of t1/2 requires solving this expression using a calculator that supports logarithmic operations with different bases.