Animated Solution for Chemistry - Chemical Kinetics: During the nuclear explosion, one of the products is ${}^{90}\text{Sr}$ with half-life of $6.93\text{ years}$. If $1\mu \text{g}$ of ${}^{90}\text{Sr}$ was absorbed in the bones of a newly born baby in place of Ca, how much time, in years, is required to reduce it by $90\%$ if it is not lost metabolically ......... .

Strontium-90

Imagine the aftermath of a nuclear explosion. Among the invisible, silent threats released into the atmosphere is Strontium-90 (${}^{90}\text{Sr}$). Because Strontium and Calcium belong to the same group in the periodic table, the human body often mistakes Strontium for Calcium. If a newborn baby is exposed to it, the body eagerly absorbs the ${}^{90}\text{Sr}$ and deposits it directly into their growing bones.

Once lodged in the skeletal system, it begins a slow, relentless process of radioactive decay. This decay perfectly follows first-order kinetics, meaning the rate at which it breaks down depends entirely on how much of it is currently present. The question asks us to find out how long it will take for this dangerous isotope to be reduced by $90\%$.

For any process that follows first-order kinetics, we rely on two master equations. The first relates the decay constant ($\lambda$) to the half-life ($t_{1/2}$):

The second is the integrated rate law, which allows us to calculate the time ($t$) required for the initial amount ($N_0$) to decay to a final amount ($N_t$):

Here is where many students make a critical error. The problem states that the Strontium-90 is reduced by $90\%$. This does not mean that $90\%$ is left. If you lose $90\%$ of something, you only have $10\%$ remaining.

Therefore, the final amount $N_t$ is $10\%$ of the initial amount $N_0$. This gives us a very clean ratio to work with:

Notice that the initial mass of $1\mu \text{g}$ is completely irrelevant to our calculation! In first-order kinetics, the time taken to reach a specific fraction of the original amount is independent of the starting mass.

Let's first find the decay constant $\lambda$. We plug in the given half-life of $6.93\text{ years}$:

Now, we substitute $\lambda$ and our ratio into the time equation:

Since $\log (10)=1$, the math simplifies beautifully:

It takes over 23 years just to eliminate $90\%$ of the Strontium-90 from the baby's bones. This stark mathematical reality highlights exactly why radioactive fallout poses such a severe, multi-generational hazard to human health.