Question

Do you think the shape of the curve on your graph would change if you increased the half-life to 20 seconds? What does this reveal about radioactive decay?

Answer 1

Answer:

The curve is still exponential but decreases at a lower rate for a greater half-life;

The greater the half-life, the slower radioactive decay is.

Explanation:

From the context of the actual problem, it looks like you plotted the number of non-decayed atoms against time. Since you are analyzing a radioactive decay in this problem, the number of the atoms remaining for the first-order rate law can be represented by the following equation:

$m_t = m_o e^{-kt}$

Here k is the rate constant. It is defined in terms of half-life by the following relationship:

$k = \frac{ln(2)}{T_{\frac{1}{2}}}$

That said, in terms of half-life, our equation becomes:

$m_t = m_o e^{-kt}=m_o e^{-\frac{ln(2)}{T_{\frac{1}{2}}}t}$

Notice that the greater the half-life is, the less negative the coefficient in front of the time variable in the exponent.

As a result, the decay for a greater half-life would occur at a lower rate. The curve would still be exponential in terms of shape but would decrease at a lower rate.

We may conclude that the greater the half-life, the slower radioactive decay is.