Problem 54 - Fundamentals of Physics

Question

Some radionuclides decay by capturing one of their own atomic electrons, a \(K\) -shell electron, say. An example is $${ }^{49} \mathrm{~V}+\mathrm{e}^{-} \rightarrow{ }^{49} \mathrm{Ti}+\nu, \quad T_{1 / 2}=331 \mathrm{~d}$$ Show that the disintegration energy \(Q\) for this process is given by $$Q=\left(m_{\mathrm{V}}-m_{\mathrm{Ti}}\right) c^{2}-E_{K}$$ where \(m_{\mathrm{v}}\) and \(m_{\mathrm{Ti}}\) are the atomic masses of \({ }^{49} \mathrm{~V}\) and \({ }^{49} \mathrm{Ti},\) respectively, and \(E_{K}\) is the binding energy of the vanadium \(K\) -shell electron. (Hint: Put \(\mathbf{m}_{\mathrm{V}}\) and \(\mathbf{m}_{\mathrm{Ti}}\) as the corresponding nuclear masses and then add in enough electrons to use the atomic masses.)

Step-by-Step Solution

Answer

The disintegration energy is \(Q = (m_V - m_{Ti})c^2 - E_K\).

Step 1: Understanding the Reaction

The process involves capturing an atomic electron by \({ }^{49} \text{V}\) leading to the formation of \({ }^{49} \text{Ti}\) and releasing a neutrino \(u\). This electron capture changes the number of protons, transforming vanadium into titanium.

Step 2: Define Masses Involved

The reaction can be written as \( { }^{49} \text{V} + e^− \to { }^{49} \text{Ti} + u \). Here, \(m_V\) is the atomic mass of vanadium, and \(m_{Ti}\) is the atomic mass of titanium. We need to express the disintegration energy \( Q \) in terms of these atomic masses.

Step 3: Relate Nuclear to Atomic Masses

The nuclear masses are \( m_V' \) and \( m_{Ti}' \), the real atomic masses are related to these by \( m_{V} = m_V' + Z_Vm_e - E_K/c^2 \) and \( m_{Ti} = m_{Ti}' + Z_{Ti}m_e \). Here \( m_e \) is the electron mass, \( E_K \) is the binding energy of the K-shell electron, and \( Z_{V}, Z_{Ti} \) represent the atomic numbers.

Step 4: Calculating Disintegration Energy \(Q\)

The energy released \(Q\) is the mass difference, \( Q = \left(m_V' + Z_Vm_e - E_K/c^2 \right) - \left(m_{Ti}' + Z_{Ti}m_e\right) \) times \(c^2\). Simplicity shows that \( Z_V = Z_{Ti} + 1 \), leading to \( Q = \left(m_V - m_{Ti}\right) c^2 - E_K.\) This aligns with how the atomic electron mass and nuclear potential reduce.

Key Concepts

Binding Energy

Binding energy is a key concept when discussing atomic structure and nuclear reactions, like electron capture. In an atom, electrons are arranged in shells around the nucleus, with the innermost shell being the K-shell. Each electron is bound to the nucleus with a particular energy known as the binding energy.
The binding energy plays a crucial role in nuclear decay processes. In electron capture, a K-shell electron is captured by the nucleus. This particular process absorbs the binding energy of the electron.

• This energy must be considered when calculating the disintegration energy.
• In our example, the binding energy is subtracted from the total energy release, as the K-shell electron is assumed to be absorbed.

Understanding binding energy helps in predicting the stability of a nucleus and the likelihood of specific decay modes, like electron capture in certain radionuclides. It contributes to the overall energy balance in nuclear reactions.

Disintegration Energy

Disintegration energy, often denoted by the symbol \(Q\), refers to the energy released during a nuclear decay process such as electron capture. It is the difference in mass-energy between the original and resulting nuclei, adjusted by any associated electron binding energies.

• The disintegration energy is a crucial factor in determining the behavior of nuclear decays, as it reflects the changed binding energy from before and after the reaction.
• In the exercise, the equation \(Q = \left(m_{\text{V}} - m_{\text{Ti}}\right)c^2 - E_{K}\) illustrates how to account for the captured electron's binding energy when calculating the energy released.

To find the disintegration energy, one must consider the transformed atomic masses including any changes through electron capture. The mass difference is multiplied by the speed of light squared \(c^2\) to convert into an energy equivalent (from mass to energy - Einstein's famous \(E=mc^2\)). This calculated energy tells us how much energy will be liberated or absorbed during the decay.

Nuclear Decay

Nuclear decay, also known as radioactive decay, is a natural process where an unstable atomic nucleus loses energy by emitting radiation. The important concepts linked to nuclear decay include:

• Transmutation of elements: In electron capture, as detailed in this example, a K-shell electron is captured by the nucleus of an atom, transforming it into a different element.
• Stability factors: This process often occurs when the original atom is more unstable compared to the resulting atom. The electron capture process stabilizes the atom by converting a proton into a neutron, usually when additional neutrons can lead to a more stable nucleus.
• Released particles: In electron capture, a neutrino (often denoted \(u\)) is released. This process is one form of nuclear decay, differing from others like alpha or beta decay where particles are expelled from the atom.

Nuclear decay like electron capture is a key way in which elements change form over time, releasing energy and forming new elements. Understanding it helps scientists in fields ranging from nuclear physics to astrophysics as they study the lifespan of stars and the formation of elements.