Rate of Radioactive Decay - Online Tutor, Practice Problems & Exam Prep

Radioactive decay follows a first-order rate law, characterized by the radioactive integrated rate law equation:

$\mathrm{Ln}\left(N\right)=-kt+\mathrm{Ln}\left(N_0\right)$

where N is the final amount of radioactive nuclei, N₀ is the initial amount, k is the decay constant (in time^-1), and t is time. This equation parallels the equation of a straight line,

$Y=MX+B$

with the natural log of concentration as Y, the decay constant as the slope (M), time as X, and the natural log of the initial concentration as B. The negative slope indicates a decrease in radioactive nuclei over time. This relationship can be graphically represented with Ln(N) on the Y-axis and time on the X-axis, showing a descending slope equal to -k. Understanding this equation is crucial for analyzing radioactive processes, which may involve different units such as disintegrations per second, not just molarity.

Radioactive Integrated Rate Law

Now recall under chemical kinetics that all radioactive processes or reactions follow a first order rate law. Here we're going to say for these reactions we use the following equation. So here we're going to have our radioactive integrated rate law. Here we have our final reactant concentration of a radioactive nuclei. Here instead of using our typical A value, we're using NN. Sub zero or subnot equals our initial reacting concentration of a radioactive nuclei.

K is no longer the rate constant, it now becomes the decay constant, but it isn't. Still it's still in times. Inverse time here could be in days, years, months, et cetera. But remember the units that time uses is based on the units of K. If K were in days inverse that mean that time would have to be in days always. Make sure they match. Together these variables give us our radioactive integrated rate law, which is ln⁡(N)=-Kt+ln⁡(N₀) which is our natural log of the final concentration equals negative KT plus ln of our initial concentration.

Now here the word concentration doesn't only mean molarity in our earlier sections. Remember, when it comes to radioactive processes, we could talk about disintegrations per second. We could talk about a lot of different terminology. Doesn't only mean molarity. Now this equation is also related to the equation of a straight line. It's y=mx+b, where y is ln of our final concentration. M here is our slope, which is equal to K, x is equal to time and b is equal to ln of our initial concentration.

So remember, because it's negative K, that means our slope is decreasing over time. It's negative. With this, we can also think of the plot of y versus x, which again is ln of our final concentration versus time. Here on our y axis we have ln of our final concentration of our reactant and then on our x axis we have time. Our initial starts here and the K our slope is negative so it's descending overtime. Slope is equal to negative K and remember our slope is equal to change and rise over change and run which is delta y over delta x.

This translates to the change in the natural log of the concentration of a reactant divided by change in time. This is just a way of us plotting the information in terms of a graph, but the most important part of this entire section is remembering this radioactive integrated rate law. You'll be employing this equation more often than not compared to everything else we're seeing here, so just remember the variables involved with each one and what they stand for.

Rate of Radioactive Decay Example

Here we're told that the radioactive element of Acetene 210 has a decay constant of 0.086 hours-1. Inverse, how many minutes would it take for its concentration to go from 9.3 * 105 disintegrations per second to 2.7 * 104 disintegrations per second? All right, so they wanted to find minutes, so they're asking us to find time. Our variable T is the missing variable.

All right, so we're going to have our radioactive integrated rate law, which remember would be lane of our final concentration. Remember, concentration here doesn't necessarily mean molarity in this case, it's a disintegration per second and that's fine, minus KT plus lane of our initial concentration. Remember, your final concentration should be the smaller total because we're decreasing our amount of reactant over time as it's decaying away.

So lane of 2.7 * 104 disintegrations per second are the K constant. K we're told is -0.086 hours-1. Inverse, we don't know what T is. That's what we need to find plus lane of our initial, which is the larger amount. Subtract Ln of 9.3 * 105 from both sides. When we do that, we're going to get Initially 353934772 equals negative 086 hours-1 times T.

Divide both sides by your K value and we it's important to keep the units around so we can see what our units will be at the end. At the end time is in hours, but remember I want the answer in minutes, so we have to do one last conversion. Remember that one hour is equal to 60 minutes. So time here would be equal to 2.5 * 103 minutes. This would be our final answer.

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

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

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

Here’s what students ask on this topic:

What is the rate constant, k, (in days), for this radioactive decay?

To determine the rate constant (k) for radioactive decay, you would use the first-order decay equation:

$k=\frac{\ln \left(2\right)}{t_1/2}$

Here, ln(2) is the natural logarithm of 2 (approximately 0.693), and t_1/2 is the half-life of the radioactive substance, expressed in days.

The half-life is the time required for half of the radioactive atoms in a sample to decay. Once you know the half-life of the substance, you can plug it into the equation to calculate the rate constant. The units of k will be day^-1 since it's the reciprocal of time (in days).

For example, if a radioactive substance has a half-life of 4 days, the rate constant k would be calculated as follows:

$k=\frac{\ln \left(2\right)}{4}\approx \frac{0.693}{4}\approx 0.173\text{day}⁻$

The constant rate at which a radioactive element decays is its what?

The constant rate at which a radioactive element decays is known as its half-life. The half-life of a radioactive element is the time it takes for half of the radioactive atoms in a sample to decay into a different, more stable element. This is a characteristic property of each radioactive isotope and remains constant over time regardless of the initial amount of the substance or its external conditions such as temperature or pressure. For example, if a radioactive isotope has a half-life of one year, then after one year, only half of the original amount of the isotope will remain; after two years, only a quarter will be left, and so on. This concept is crucial in fields such as geology, archaeology, and medicine, where it is used for dating artifacts, rocks, and for medical treatments, respectively.