# What is the rate of decay

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### Law of decay

For a given unstable nucleus, there is no way to predict when it will decay. However, it has been shown that the probability (decay constant with unit) for the decay within the next time period can be specified very well. A statement about the decays per unit of time (decay rate) can therefore be made for the entirety of a large number of similar nuclei.

The change in the number of particles () per unit of time () is negative in sign (there are fewer) and just as large in magnitude as the decay rate (each decay corresponds to a radioactive process), which is, however, counted as positive. The decay rate itself is the product of the number of nuclei and the decay constant. So we can write:

The remarkable thing about this statement is that it is completely independent of time. Completely different to living beings, whose probability of death depends crucially on the age reached - 100-year-olds have an extremely higher death rate than 20-year-olds - the probability of decay of atomic nuclei does not depend at all on their previous age.

So the above equation is a differential equation that describes radioactive decay. If we want to know the number of nuclei still present at the point in time - based on nuclei at the point in time - we have to integrate both sides: After the integration we get: For the equation simplifies with the calculation rules of the logarithm function: If we apply the exponential function on both sides, the result is: We still bring to the other side and finally get the law of decay.

- Law of decay
- In the case of a radioactive substance with the decay constant, nuclei are still present from the initial nuclei after time.

The time until the number of atoms decreases to the -th part is called the mean lifetime. For you applies

The half-life is the time that elapses until the number of atoms initially present has decayed by half. The following applies: or after taking the logarithm on both sides With the help of the half-life, the law of decay can also be written in the following form:

As already mentioned, the decay rate indicates the number of cores that decay per unit of time. It is also known as the activity of a sample. Your SI unit is the Becquerel Bq (decay per second). The activity is the change in the number of nuclei still present over time - we get it by deriving the law of decay with respect to time. or formulated with the original activity results

The exponential dependence of radioactivity on time is fulfilled so well that it is suitable for determining the age of minerals or also for determining the age of organic material.

- Tab. 1
- Half-lives of some isotopes

element | Half-life |
---|---|

Uranium-238 | 4,470,000,000 a |

Uranium-235 | 704,000,000 a |

Radium-226 | 1 600 a |

Hydrogen-3 | 12.3 a |

Radon-222 | 3.8 d |

Po-214 | 0.16 ms |

- Work order
The (fantasy) element Schwerium (mother) decays into the element Leichterium (daughter) with a decay constant of 0.6 per second, which in turn decays into the element Stabilium with a decay constant of 0.35 1 / s. Stabilium itself is not radioactive.

- Calculate the half-lives of the elements from the decay constants. On a piece of paper, sketch the expected course of the atomic numbers of the elements involved over time.
- Observe the decay rates using the arrows and the graph and compare this with your prediction.
- At what point in time is the maximum number of Leichterium atoms available?
- Also consider cases where the daughter element decays faster than the parent element by changing its decay constant in the model to 1.2 per second. After what time is the maximum number of child elements available?
- It should now be assumed that Stabilium also decays into the element Stabilium2 with a decay constant of 0.95 per second. Extend the model by this further process and insert an additional curve in the graph of the animation, which indicates the time course of the number of Stabilium2 cores.

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