Answer:
Half-life = 63 years
Mystery element is Titanium-44
There will be 1 gram left after 419 years have passed
Step-by-step explanation:
In general the equation describing half-life is:
[tex]m=m_0 (0.5)^{\frac {t}{t_{1/2}}[/tex]
Where m is the final mass, m0 is the starting mass, t is the total elapsed time, and t(1/2) is the length of the half life. t / t(1/2) is actually the number of half-lives that have passed.
Anyways - all your questions can be answered using the above equation.
What is the half-life?
I subbed in the values we know from the question. In solving this I used the natural logarithm (but a log of any base will do - the goal is to bring down the exponent):
[tex]96.43=100(0.5)^\frac{3.3}{t_{1/2}}\\0.9643=(0.5)^\frac{3.3}{t_{1/2}}\\ln(0.9643)=ln(0.5)^\frac{3.3}{t_{1/2}}\\ln(0.9643)=\frac{3.3}{t_{1/2}} \times ln(0.5)\\t_{1/2} =\frac{3.3}{ln(0.9643)} \times ln(0.5)\\t_{1/2} = 63[/tex]
What is the mystery element?
Titanium-44, because it's half-life in the table is closest to the calculated half-life.
When will there be 1 gram left?
Again use the equation. This time the total time, t, is unknown. We can use the previously calculated half-life:
[tex]1=100(0.5)^\frac{t}{63}\\0.01=(0.5)^\frac{t}{63}\\ln(0.01)=ln(0.5)^\frac{t}{63}\\ln(0.01)=\frac{t}{63} \times ln(0.5)\\t=\frac{63 \times ln(0.01)}{ln(0.5)}\\t=419[/tex]
