Question

Radioactive decay is first order in the decaying isotope. For example, strontium-90 contained in fallout from nuclear explosions decays to yttrium-90 and a beta particle. Write the rate law for the decay of strontium-90.

Short answer

The rate law for the decay of strontium-90 (Sr-90) can be written as: Rate = k [Sr-90] Here, 'Rate' represents the decay rate, 'k' is the first-order rate constant, and [Sr-90] represents the concentration of Sr-90. This expression follows first-order kinetics, as the rate of decay is directly proportional to the concentration of Sr-90.

Step-by-step solution

Identify the reactant and products

In this decay process, strontium-90 (Sr-90) is the reactant, and yttrium-90 (Y-90) and a beta particle are the products. The decay reaction can be written as: Sr-90 → Y-90 + β

Recognize that radioactive decay follows first-order kinetics

As mentioned in the problem, radioactive decay follows first-order kinetics. This means that the rate of decay is directly proportional to the concentration of the decaying isotope (in this case, Sr-90).

Write the rate law

For a first-order decay process, the rate law can be written as: Rate = k [Sr-90] Here, 'Rate' represents the decay rate and 'k' is the first-order rate constant specific to the decay process of Sr-90. [Sr-90] represents the concentration of the strontium-90 isotope. And that's the final rate law for the decay of strontium-90.

Key concepts

First-Order Kinetics

First-order kinetics is a fundamental concept in chemical kinetics, particularly relevant to processes like radioactive decay. As observed in the decay of strontium-90, the rate at which the isotope decays is directly proportional to its concentration at any given time. What does this mean in simpler terms? Imagine you have a pile of strontium-90 atoms. The more atoms you have at the start, the faster they will decay, but in such a way that, as the pile gets smaller, the rate of decay decreases proportionately.

In mathematical terms, first-order kinetics is described using the rate law equation:
Rate = k [Sr-90]
Here, 'Rate' is the speed of the decay, 'k' is the decay constant unique to strontium-90, and [Sr-90] is a symbol for the concentration of our strontium-90 atoms.

Because it is first-order, if you were to plot the natural logarithm of the strontium concentration against time, you would get a straight line. This straight-line relationship is a hallmark of first-order reactions and is extremely useful when determining the half-life of an isotope, which is the time it takes for half of the radioactive material to decay.

Nuclear Chemistry

Nuclear chemistry delves into the transformations and reactions within the nuclei of atoms. Among these reactions, radioactive decay is a spontaneous process and a quintessential topic in nuclear chemistry. Strontium-90's decay into yttrium-90 and a beta particle is a great example of how unstable isotopes seek stability through radioactive decay. This natural transmutation alters the number of protons and consequently transforms one element into another.

In nuclear chemistry, understanding the stability of atomic nuclei is crucial. This not only explains why certain isotopes like strontium-90 decay but also allows predictions about the energies involved and the types of radiation emitted. For instance, beta decay, as seen with strontium-90, involves the release of beta particles which are high-energy, high-speed electrons or positrons. This emission changes the atomic structure, leading to a different element, and is a cornerstone of the changes observed in nuclear chemistry.

Radioactive Isotopes

Radioactive isotopes, also known as radionuclides, are variants of elements with an unstable combination of neutrons and protons in their atoms' nuclei, prompting them to release energy to reach a more stable state. Strontium-90 is an example of such an isotope, having additional neutrons that make it unstable. These isotopes are everywhere, from the soil we walk on to cosmic material. They have practical uses in medicine, industry, and research but also play a role in nuclear fallout, as mentioned with strontium-90 from nuclear explosions.

The period of time over which radioactive isotopes decay can vary widely, from fractions of a second to millions of years, depending on their stability and the type of radioactive decay they undergo. This decay is a natural 'clock' that scientists utilize to date archaeological artifacts or study geological formations. It is purely statistical in nature; we cannot predict exactly when a particular atom will decay, but we can predict the behavior of a large population of said atoms with remarkable precision. As such, radioactive isotopes serve as windows into the past and key tools in many technological applications.