To tackle the problem of determining the time it takes for a radioactive element to decay to one-sixth of its original amount, we can use the concept of half-life effectively. Let's break this down step by step to clarify the approach and ensure you arrive at the correct answer.

Understanding Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In your case, the half-life is given as 100 seconds. This means that every 100 seconds, the amount of the substance reduces to half of its previous amount.

Decay to One-Sixth

To find out how long it takes for the substance to decay to one-sixth of its original amount, we can use the relationship between the half-life and the remaining quantity of the substance.

Step-by-Step Calculation

• Let’s denote the original amount of the substance as N₀.
• After one half-life (100 seconds), the amount remaining is N₁ = N₀ / 2.
• After two half-lives (200 seconds), the amount remaining is N₂ = N₀ / 4.
• After three half-lives (300 seconds), the amount remaining is N₃ = N₀ / 8.
• After four half-lives (400 seconds), the amount remaining is N₄ = N₀ / 16.

Now, we need to find out when the amount is equal to N₀ / 6. To do this, we can express the remaining quantity in terms of half-lives:

Using the Half-Life Formula

The relationship can be expressed as:

N(t) = N₀ * (1/2)^(t/T_1/2)

Where:

• N(t) is the remaining quantity after time t.
• T_1/2 is the half-life (100 seconds in this case).

We want to find t when:

N(t) = N₀ / 6

Setting Up the Equation

Substituting into the equation gives:

N₀ / 6 = N₀ * (1/2)^(t/100)

We can simplify this by dividing both sides by N₀ (assuming it's not zero):

1/6 = (1/2)^(t/100)

Solving for Time

Now, we take the logarithm of both sides to solve for t:

log(1/6) = (t/100) * log(1/2)

Rearranging gives:

t = 100 * (log(1/6) / log(1/2))

Calculating the Values

Now, you can use a calculator to find the logarithmic values:

• log(1/6) ≈ -0.7781
• log(1/2) ≈ -0.3010

Substituting these values into the equation:

t ≈ 100 * (-0.7781 / -0.3010) ≈ 258.5 seconds

Final Thoughts

Thus, it takes approximately 258.5 seconds for the radioactive element to decay to one-sixth of its original amount. This method effectively utilizes the half-life concept and logarithmic properties to arrive at the solution. If you have any further questions or need clarification on any step, feel free to ask!