The initial number of atoms in a radioactive element is 6×10^(20) and its half life is 10hrs. Determine the number of atoms that will have decayed after 30he's.

To determine the number of atoms that have decayed after 30 hours for a radioactive element with an initial quantity of 6×10²⁰ atoms and a half-life of 10 hours, we can follow a systematic approach using the concept of half-lives.

Understanding Half-Life

The half-life of a radioactive substance is the time it takes for half of the radioactive atoms to decay. In this case, the half-life is 10 hours, meaning that every 10 hours, half of the remaining atoms will have decayed.

Calculating the Number of Half-Lives

First, we need to find out how many half-lives fit into the 30 hours we are considering:

• 30 hours ÷ 10 hours/half-life = 3 half-lives

Applying the Half-Life Concept

Next, we can calculate how many atoms remain after each half-life:

• After the first half-life (10 hours):
  • Remaining atoms = 6×10²⁰ / 2 = 3×10²⁰
• After the second half-life (20 hours):
  • Remaining atoms = 3×10²⁰ / 2 = 1.5×10²⁰
• After the third half-life (30 hours):
  • Remaining atoms = 1.5×10²⁰ / 2 = 0.75×10²⁰

Finding the Number of Atoms Decayed

Now that we know the number of atoms remaining after 30 hours, we can find out how many have decayed:

• Initial atoms = 6×10²⁰
• Remaining atoms after 30 hours = 0.75×10²⁰
• Atoms decayed = Initial atoms - Remaining atoms
• Atoms decayed = 6×10²⁰ - 0.75×10²⁰ = 5.25×10²⁰

Final Result

In summary, after 30 hours, the number of atoms that have decayed is 5.25×10²⁰ atoms. This illustrates how the concept of half-life allows us to predict the decay of radioactive substances over time effectively.