Main content

## MCAT

### Course: MCAT > Unit 8

Lesson 22: Atomic nucleus- Atomic nucleus questions
- Radioactive decay types article
- Decay graphs and half lives article
- Atomic number, mass number, and isotopes
- Atomic mass
- Mass defect and binding energy
- Nuclear stability and nuclear equations
- Writing nuclear equations for alpha, beta, and gamma decay
- Types of decay
- Half-life and carbon dating
- Half-life plot
- Exponential decay formula proof (can skip, involves calculus)
- Introduction to exponential decay
- Exponential decay and semi-log plots
- More exponential decay examples
- Mass spectrometer

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# Decay graphs and half lives article

## How can scientists tell when something is releasing radiation?

We know that radiation is more than just the spooky, silent threat that we see in movies. Healthcare providers can actually harness the unique properties of radiation to look inside the human body and diagnose diseases in new ways. We also know that all radiation occurs when an unstable nucleus releases energy to become more stable. This happens when the nucleus changes into a different nucleus This happens in three different ways:

**Alpha decay**: The nucleus splits into two chunks, a little chunk called an “alpha particle” (which is just two protons and two neutrons) and a daughter nucleus with a lower atomic number than the initial nucleus. The “radiation” here is the small chunk, which generally moves away from the nucleus at a pretty high speed.**Beta decay**: There are two types of beta decay: In beta-minus decay, a neutron in an atom changes into a proton, an electron, and an antineutrino, creating and releasing an electron along the way (since the total charge has to stay the same!). The radiation here is the super fast-moving electron released. In beta-positive decay, a proton turns into a neutron, causing the nucleus to shoot out an exotic positive particle called a “positron” or “anti-electron.”**Gamma decay**: The number of protons, neutrons, and electrons stays the same, but they rearrange themselves within the atom, giving off energy in the form of high-energy photons (gamma radiation), in order to have lower overall energy.

For all of these cases, the total amount of the radioactive element decreases over time. So if a scientist takes a chunk of carbon-10 (which undergoes beta decay), counts the number of carbon-10 atoms inside it, goes to make coffee, and then comes back to count the number of atoms again, she’ll find that the total number of atoms of carbon-10 is now smaller! In their place she’ll find the beta decay product of carbon-10, which is the element boron.

Although the decay of individual nuclei happens randomly, it turns out that large numbers of nuclei can be modelled by a mathematical function that predicts the amount of radioactive nuclei remaining at a given time:

N(t) = Nstart subscript, 0, end subscriptestart superscript, minus, k, t, end superscript

This states that the number of carbon-10 nuclei (N(t)) left in a sample that started out with N0 atoms decreases

**exponentially**in time. The constant k is called the**decay constant**, which controls how quickly the total number of nuclei decreases. The value of the decay constant is specific to the type of decay (alpha, beta, gamma) and isotope being studied, and so unknown isotopes can be identified based on how quickly they decay.Because the mass in an isotope sample is directly related to the total number of atoms in the sample, the total mass of an isotope also decays exponentially with the same decay constant, M(t) = Mstart subscript, 0, end subscriptestart superscript, minus, k, t, end superscript. Because of conservation of mass, as the total amount of the isotope decreases the total mass of produced decay products increases - like boron or radiation particles. A plot of the mass of a sample of carbon-10 versus time looks something like this:

One funny property of exponential decay is that the total mass of radioactive isotopes never actually reaches zero. The mass just keeps getting closer and closer to zero as the amount of time for the isotope to decay gets larger and larger. Realistically, there are only a fixed number of atoms in a radioactive sample, and so the mass of an isotope will eventually reach zero as all the nuclei decay into another element.

## How do you read a decay graph?

If a scientist comes across a funky new radioactive rock while exploring an excavation site, she can identify what radioactive isotope is present in it by measuring the the decay constant of the isotopes inside the rock.

An exponential decay graph like the one shown above can be generated by taking a sample of an unknown radioactive isotope and repeatedly measuring the total mass of radioactive material within it. At the start of the experiment, a piece of the sample is run through a mass spectrometer or similar device that can identify the relative mass of various isotopes currently present. Then, at several later times, the procedure is repeated and the new fraction of various isotopes is recorded.

The fraction of radioactive isotopes observed in the spectrometer will decrease exponentially in time, while the mass of decay products (like boron for carbon-10) will gradually increase. The scientist can use this information to draw an exponential decay plot like the one above and estimate the decay constant. She can then look her value up in a glossary of known radioactive decay constants to figure out which isotope is in her sample.

Another type of graph that scientists like to use to show nuclear decay data is a

*semilog*plot (shown below)Semilog plots are pretty tricky because the vertical axis has funny spacing. In the plot above, 100 appears to come halfway between 10 and 1000. This is because when we take logarithms of these numbers we get log(1000)=6.9, log(100)=4.6, log(10)=2.3, which are evenly spaced. So when we read the slope on a semilog plot, we need to remember to always take the logarithm of whatever values we read off the vertical axis. The slope of the line on the semilog plot corresponds to the same decay constant k, that we can identify in a normal exponential decay plot.

But, when decay data is plotted as it is in our exponential decay graph, the decay constant is much harder to figure out because it’s not that easy to compare the “sharpness” of different exponential decay curves. Finding the slope of straight lines, however, is generally much easier. By plotting data on semi-log plots, the scientist can better compare and identify different isotopes.

Further information about an unknown radioactive isotope can be identified simply by analyzing the radiation that it shoots out of the isotope. Gamma radiation produces photons, beta decay produces electrons or positrons, and alpha decay releases entire alpha particles (helium nuclei).

## What is a half-life?

Half-life is defined as the amount of time it takes for half of an isotope to change into another isotope.

Like the decay constant, the half-life tells us everything we need to know to guess what kind of isotope we might have. It even turns out that the two numbers are equivalent if you correctly solve the radioactive decay equation. This means that, like the decay constant, the half-life gives an estimate of the stability of a particular radioactive substance, and it can thus be used to identify unknown isotopes. The primary reason that scientists use half-lives instead of decay constants is because half-lives have a more intuitive immediate meaning: if a scientist collects 20 kg of a radioactive isotope with a half-life of 30 min, and she leaves her lab to meet with a grad student for 30 minutes, when she comes back she will have 10 kg of the isotope remaining. If she then goes to check on another experiment for 30 minutes, when she gets back she will have 5 kg remaining… in other words, for every 30 minutes that passes, she’ll lose half of her sample!

## Consider the following… carbon dating

Most living things contain carbon-14, an unstable isotope of carbon that has a half-life of around 5,000 years. That means that when scientists dig up fossil bones, they can figure out how old they are by measuring the amount of carbon-14 remaining in the bones. For example, if a fossil bone has half as many of carbon-14 nuclei as a new, non-fossilized bone, then scientists can guess that the fossil is roughly 5,000 years old. This technique of

**carbon dating**has been used to estimate the ages of fossils from many different periods in Earth’s history, and at its core it simply relies on scientists drawing decay graphs and counting the number of half-lives that have passed. Carbon dating was recently used to study one of the oldest human-like fossils ever found, and it determined that it was nearly 100,000 years old!## Want to join the conversation?

- In the paragraph below the semilog figure, how did you get the logarithms of 1000 to be 6.9 and other logarithms. Shouldn't the log of 1000 be 3?

Thank you (Reference, article 2)(5 votes)- They are taking the natural log. Ln(1000) = 6.9(11 votes)

- Can someone explain to me the semilog plot?(2 votes)
- Decay is exponential, so the amount of radioactive stuff you have left is dependent on the exponent of the equation (here, time is what changes the exponent; as time gets bigger, the exponent gets smaller, and it does this linearly because exponent=-kt [<-- that's a linear y=mx+b equation, where exponent=y, m=-k, x=t, & b=0]).

The log of a number equals its exponent (with a given, constant/non-changing base). So if you plot decay using a log plot, you're plotting the exponent vs. time, and this is a linear relationship (see above).

We can get back to seeing how the amount of radioactive stuff gets exponentially smaller as more time passes by plugging the exponent number (plotted on the y-axis of the semilog plot) into the original equation, so you're solving for the actual amount of radioactive stuff left.(2 votes)

- Looking at the semilog plot the slope is said to be the decay constant, but does the y intercept have any meaning?(2 votes)
- what are the symbols for each decay(1 vote)
- The "Radioactive decay types article" said beta decay releases an electron and a neutrino, but this article says beta decay releases an electron and an antineutrino.(1 vote)
- It varies for beta-plus and beta-minus decay.

"In beta minus decay, a neutron decays into a proton, an electron, and an antineutrino... In beta plus decay, a proton decays into a neutron, a positron, and a neutrino"

http://www2.lbl.gov/abc/wallchart/chapters/03/2.html(1 vote)

- what is the relation between rate of decay of radioactive element with time?(1 vote)
- does anyone know how long does it take Bismuth-209 to completely decay?(1 vote)
- Isn't the half life for Carbon-14 much closer to 6000 years?(1 vote)
- Isn't the half life for Carbon-14 much closer to 6000 years?(1 vote)
- If you have only 2 radioactive nuclei of an isotope left, does half life still apply? If so, how?(0 votes)