(b) In 30 hours , a sample of plutonium - 243 decays to (1)/(64) of its original amount.Determine the half- life of the isotope. [TC3]

Here's how to determine the half-life of Plutonium-243:<br /><br />**Understanding Half-Life**<br /><br />Half-life is the time it takes for half of a radioactive substance to decay. After one half-life, 50% remains. After two half-lives, 25% remains (half of the remaining 50%), and so on.<br /><br />**Setting up the Equation**<br /><br />We can model radioactive decay with the following equation:<br /><br />N(t) = N₀ * (1/2)^(t/T)<br /><br />Where:<br /><br />* N(t) is the amount of the substance remaining after time t<br />* N₀ is the initial amount of the substance<br />* t is the elapsed time<br />* T is the half-life<br /><br />**Solving for the Half-Life**<br /><br />1. **Fractional Amount Remaining:** The problem states that after 30 hours, 1/64 of the original amount remains. So, N(t) = (1/64) * N₀.<br /><br />2. **Substitute into the Equation:** Substitute the given values into the decay equation:<br /><br /> (1/64) * N₀ = N₀ * (1/2)^(30/T)<br /><br />3. **Simplify:** Divide both sides by N₀:<br /><br /> 1/64 = (1/2)^(30/T)<br /><br />4. **Rewrite with a Common Base:** Notice that 1/64 can be expressed as (1/2)^6:<br /><br /> (1/2)^6 = (1/2)^(30/T)<br /><br />5. **Equate the Exponents:** Since the bases are the same, the exponents must be equal:<br /><br /> 6 = 30/T<br /><br />6. **Solve for T:** Multiply both sides by T and then divide by 6:<br /><br /> T = 30/6<br /> T = 5 hours<br /><br />**Answer:** The half-life of Plutonium-243 is 5 hours.<br />