Question

Nuclide \(X\) has a half-life of 2.6 years and decays by electron capture to produce \(\mathrm{Mn}-55\) and a gamma ray. Write a nuclear equation for the reaction.

Short answer

The nuclear equation is: \( \mathrm{Fe} + e^- \rightarrow \mathrm{Mn} + \gamma \).

Step-by-step solution

Understanding Electron Capture

Electron capture is a process where an atomic nucleus captures one of the atom's inner electrons and combines it with a proton to form a neutron. This changes the element into a new element while reducing the atomic number by 1.

Identifying Initial and Final Elements

The provided nuclide is referred to as nuclide \(X\) which decays into \(\mathrm{Mn}-55\). Manganese (\(\mathrm{Mn}\)) has an atomic number of 25. Given that the atomic number decreases by 1 during electron capture, nuclide \(X\) must have an atomic number of 26 (which corresponds to Iron, \(\mathrm{Fe}\)).

Determining Mass Number

Since the decay product is \(\mathrm{Mn}-55\), the mass number of the starting nuclide \(X\) is also 55 because the total nucleon number does not change during electron capture.

Writing the Nuclear Equation

Now use all the information to write the nuclear equation. The initial element is \(\mathrm{Fe}\) with mass number 55 (\(\mathrm{Fe}-55\)). The nuclear equation for electron capture is: \[ \mathrm{Fe} + e^- \rightarrow \mathrm{Mn} + \gamma \] where \( e^- \) represents the captured electron and \( \gamma \) represents the emitted gamma ray.

Key concepts

Electron Capture

Electron capture is a fascinating nuclear chemistry process where an atomic nucleus swallows one of its own electrons. This usually involves an inner electron, such as one from the K-shell. When this electron gets ingested by a proton within the nucleus, something interesting happens: the proton turns into a neutron. As a result, the atomic number of the element decreases by one. In other words, the element transforms into a different element, just one spot to the left on the periodic table. For instance, if an Iron nuclide ( Fe ) undergoes electron capture, it turns into Manganese ( Mn ).
In electron capture:

• One proton becomes a neutron.
• The atomic number goes down by one.
• The mass number remains unchanged.

Understanding electron capture helps us identify how different elements undergo transformations at the nuclear level.

Nuclear Equations

Nuclear equations are the mathematical sentences of nuclear reactions. They help us track changes in an atom's nucleus during various nuclear processes, including electron capture. These equations must be balanced, meaning the sum of mass numbers (top numbers) and atomic numbers (bottom numbers) must be equal on both sides of the equation.
In the case of electron capture, the equation for Iron ( Fe-55 ) transforming into Manganese ( Mn-55 ) through electron capture can be written as:

• Fe + e^- → Mn + γ
• Where Fe is Iron, e^- represents the captured electron, Mn is Manganese, and γ is a gamma ray emitted during the process.

Balancing these equations confirms that both the mass number and atomic number are preserved, maintaining the law of conservation of nucleons and charge.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to undergo decay. It’s a crucial concept in nuclear chemistry, as it helps determine the rate at which a radioactive sample transforms. Each nuclide has a unique half-life, which makes it predictable in terms of decay timing. For instance, in the original exercise, the nuclide X has a half-life of 2.6 years. This means after 2.6 years, only half of the original X nuclides remain.

• Simplifies calculations for determining remaining nuclide quantities over time.
• Essential for radiocarbon dating and nuclear medicine.

Being familiar with half-life concepts allows scientists and students alike to understand the longevity and transformation rates of radioactive elements.

Nuclide Transformation

Nuclide transformation involves changing one nuclide into another through nuclear reactions. For example, in electron capture, an initial nuclide loses an electron which leads to the transformation of a proton into a neutron. As a result, the nuclide changes into a different element. In the given exercise, nuclide X transforms into Mn-55 .

• Transmutation results in the creation of a new element.
• Essential for the production of stable isotopes and understanding decay series.

Understanding nuclide transformation aids in grasping the dynamic nature of atoms and their capability for change at the nuclear level.