The half-life of a radioactive material is $4.5$ days. How many days are required for a sample, with an initial activity of $1.0 \times 10 ^ { 5 } \mathrm {~Bq}$, to decay to an activity of $100 \mathrm {~Bq}$ ?

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The half-life formula is used to calculate the time required for a quantity to reduce to half its initial value. The formula is $t = \frac{\ln(\frac{N_0}{N})}{\lambda}$, where $t$ is the time, $N_0$ is the initial quantity, $N$ is the final quantity, and $\lambda$ is the decay constant. The decay constant $\lambda$ can be found from the half-life $T_{1/2}$ using $\lambda = \frac{\ln(2)}{T_{1/2}}$. However, for this question, we can use a simpler approach since we're dealing with half-lives directly.Given:- Initial activity $N_0 = 1.0 \times 10^5 \mathrm{~Bq}$- Final activity $N = 100 \mathrm{~Bq}$- Half-life $T_{1/2} = 4.5$ daysWe want to find how many half-lives it takes to go from $1.0 \times 10^5 \mathrm{~Bq}$ to $100 \mathrm{~Bq}$. First, find the ratio of the initial to final activity: $\frac{1.0 \times 10^5}{100} = 1000$.Next, determine how many times the material needs to halve to reach from $1.0 \times 10^5 \mathrm{~Bq}$ to $100 \mathrm{~Bq}$. $1000 = 2^{10}$, meaning it takes 10 half-lives to decay from $1.0 \times 10^5 \mathrm{~Bq}$ to $100 \mathrm{~Bq}$.Finally, calculate the total time: $10 \times 4.5 \mathrm{~days} = 45 \mathrm{~days}$.

The Half-Life of a Radioactive Material Is 4.54.54.5 Days How Many Days Are Required for a Sample, with Many

The Half-Life of a Radioactive Material Is $4.5$ Days How Many Days Are Required for a Sample, with Many