Rate of Radioactive Decay: Videos & Practice Problems

For the radioactive decay of lead-202 the decay constant is 1.32 x 10^-5 yr^-1. How long will it take in hours to decrease to 53% of its initial amount?

• A. 5.72 x 10⁴ hrs
• B. 5.01 x 10⁹ hrs
• C. 4.81 x 10⁴ hrs
• D. 4.21 x 10⁸ hrs

Answer: (D)

During World War I radium-226 was used in the manufacturing of luminous paint. If it takes 2.12 x 10⁴ days for its degradation to be 2.49% complete, what is its decay constant?

• A. 1.74 x 10^–4 days^–1
• B. 1.08 x 10^–5 days^–1
• C. 1.19 x 10^–6 days^–1
• D. 2.14 x 10^–5 days^–1

Answer: (C)

If the decay constant for polonium-209 is 6.80 x 10^-3 ys^-1, what fraction of it remains after 1.1 x 10⁴ years?

• A. 9.00 x 10^–30
• B. 3.27 x 10^–33
• C. 8.24 x 10^–42
• D. 9.13 x 10^–43

Answer: (B)

Here's what students ask on this topic:

What is the integrated rate law for radioactive decay and how is it derived?

The integrated rate law for radioactive decay is given by the equation:

$\ln \left(N\right)=-kt+\ln \left(N{0}_{}\right)$

Here, $N$ is the amount of radioactive nuclei at time $t$, $N{0}_{}$ is the initial amount, and $k$ is the decay constant. This equation is derived from the first-order rate law, which states that the rate of decay is proportional to the number of undecayed nuclei. By integrating the differential rate equation $\mathrm{dN}/\mathrm{dt}=-kN$, we obtain the integrated form. This law helps us calculate the remaining radioactive nuclei after a certain time and is fundamental in nuclear chemistry and physics.

How does the decay constant (k) affect the rate of radioactive decay?

The decay constant $k$ is a crucial parameter in radioactive decay that determines the rate at which a radioactive substance decays. It has units of inverse time (e.g., s⁻¹, day⁻¹) and represents the probability per unit time that a nucleus will decay. A larger $k$ means a faster decay rate, so the substance loses its radioactivity more quickly. Conversely, a smaller $k$ indicates a slower decay process. In the integrated rate law $\ln \left(N\right)=-kt+\ln \left(N{0}_{}\right)$, the slope of the plot of $\ln \left(N\right)$ versus time is $-k$, showing how $k$ controls the rate of decrease in radioactive nuclei.

Why is radioactive decay considered a first-order reaction?

Radioactive decay is considered a first-order reaction because the rate of decay depends linearly on the number of radioactive nuclei present at any given time. Mathematically, the rate law is expressed as:

$\mathrm{rate}=-\mathrm{dN}/\mathrm{dt}=kN$

This means the decay rate is proportional to $N$, the number of undecayed nuclei. The constant $k$ is the decay constant. Because the rate depends on only one reactant concentration (the radioactive nuclei), it fits the definition of a first-order process. This leads to an exponential decrease in the number of nuclei over time, which is characteristic of first-order kinetics.

How can the half-life of a radioactive substance be calculated from the decay constant?

The half-life ($t{\mathrm{1/2}}_{}$) of a radioactive substance is the time required for half of the radioactive nuclei to decay. It is related to the decay constant $k$ by the formula:

$t{\mathrm{1/2}}_{}=\frac{\ln }{2}/k$

This equation comes from setting $N=\frac{N}{0_{}}$ in the integrated rate law and solving for $t$. The half-life is constant for a given isotope and does not depend on the initial amount of substance. Knowing the decay constant allows you to easily calculate the half-life, which is a key parameter in understanding the stability and decay behavior of radioactive materials.

What does the graph of ln(N) versus time look like for radioactive decay?

The graph of $\ln \left(N\right)$ versus time for radioactive decay is a straight line with a negative slope. This is because the integrated rate law can be written as:

$\ln \left(N\right)=-kt+\ln \left(N{0}_{}\right)$

Here, $\ln \left(N\right)$ is the y-axis, time $t$ is the x-axis, the slope is $-k$, and the y-intercept is $\ln \left(N{0}_{}\right)$. Because the slope is negative, the line decreases over time, reflecting the exponential decay of the radioactive nuclei. This linear relationship is useful for determining the decay constant experimentally by plotting data and calculating the slope.