Question

1. isotope a 84 po→2he + 82 pb isotope b half - life = t1/2 = __________ how much remains at 25 seconds ________ time that has a 50 - atom sample = ________ half - life = t1/2 = ________ how much remains at 25 seconds ________ time that has a 50 - atom sample = __________

Explanation:

Step1: Determine half - life of Isotope A

The half - life is the time when the number of radioactive atoms is half of the initial amount. For Isotope A, if we assume the initial number of atoms is 120 (from the y - axis at t = 0), half of it is 60. Looking at the graph, when the number of atoms is 60, the time is 25 seconds. So the half - life of Isotope A ($t_{1/2}$) is 25 seconds.

Step2: Find the amount remaining at 25 seconds for Isotope A

Since 25 seconds is the half - life of Isotope A, if the initial amount is 120 atoms, the amount remaining at 25 seconds is 60 atoms.

Step3: Find the time for 50 - atom sample of Isotope A

By looking at the graph of Isotope A, when the number of atoms is 50, the time is approximately 35 seconds.

Step4: Determine half - life of Isotope B

Assume the initial number of atoms of Isotope B is 120 (from the y - axis at t = 0), half of it is 60. Looking at the graph of Isotope B, when the number of atoms is 60, the time is 50 seconds. So the half - life of Isotope B ($t_{1/2}$) is 50 seconds.

Step5: Find the amount remaining at 25 seconds for Isotope B

The decay formula is $N = N_0(\frac{1}{2})^{\frac{t}{t_{1/2}}}$, where $N_0$ is the initial amount, $t$ is the time elapsed, and $t_{1/2}$ is the half - life. For Isotope B, $N_0 = 120$, $t = 25$ seconds, and $t_{1/2}=50$ seconds. Then $N = 120\times(\frac{1}{2})^{\frac{25}{50}}=120\times\frac{\sqrt{2}}{2}\approx84.85\approx85$ atoms.

Step6: Find the time for 50 - atom sample of Isotope B

By looking at the graph of Isotope B, when the number of atoms is 50, the time is approximately 75 seconds.

Answer:

Isotope A:
Half - Life ($t_{1/2}$) = 25 seconds
Amount remaining at 25 seconds = 60 atoms
Time for 50 - atom sample = 35 seconds
Isotope B:
Half - Life ($t_{1/2}$) = 50 seconds
Amount remaining at 25 seconds = 85 atoms
Time for 50 - atom sample = 75 seconds