an 80.0 g sample of iodine - 131 was placed in a sealed vessel forty days ago. only 2.5 g of this isotope is…

Explanation:

Step1: Identify decay formula

The radioactive - decay formula is $N = N_0(\frac{1}{2})^{\frac{t}{T_{1/2}}}$, where $N$ is the final amount, $N_0$ is the initial amount, $t$ is the time elapsed, and $T_{1/2}$ is the half - life. We know that $N_0=80.0\ g$, $N = 2.5\ g$, and $t = 40$ days.

Step2: Substitute values into formula

Substitute the values into the formula: $2.5=80.0\times(\frac{1}{2})^{\frac{40}{T_{1/2}}}$. First, divide both sides by 80.0: $\frac{2.5}{80.0}=(\frac{1}{2})^{\frac{40}{T_{1/2}}}$. Since $\frac{2.5}{80.0}=\frac{25}{800}=\frac{1}{32}$, the equation becomes $\frac{1}{32}=(\frac{1}{2})^{\frac{40}{T_{1/2}}}$. We know that $\frac{1}{32}=(\frac{1}{2})^5$, so $(\frac{1}{2})^5=(\frac{1}{2})^{\frac{40}{T_{1/2}}}$.

Step3: Solve for half - life

Since the bases are the same ($\frac{1}{2}$), we can set the exponents equal to each other: $5=\frac{40}{T_{1/2}}$. Cross - multiply to get $5T_{1/2}=40$. Then solve for $T_{1/2}$: $T_{1/2}=\frac{40}{5}=8$ days.

Answer:

8