Question

Write balanced nuclear equations for the following processes: (a) Beta emission of \({ }^{157} \mathrm{Eu}\) (b) Electron capture of \({ }^{126} \mathrm{Ba}\) (c) Alpha emission of \(^{146 \mathrm{sm}}\) (d) Positron emission of \({ }^{125} \mathrm{Ba}\)

Short answer

(a) \( ^{157}_{63}\mathrm{Eu} \rightarrow ^{157}_{64}\mathrm{Gd} + \beta^- \); (b) \( ^{126}_{56}\mathrm{Ba} + e^- \rightarrow ^{126}_{55}\mathrm{Cs} \); (c) \( ^{146}_{62}\mathrm{Sm} \rightarrow ^{142}_{60}\mathrm{Nd} + ^{4}_{2}\mathrm{He} \); (d) \( ^{125}_{56}\mathrm{Ba} \rightarrow ^{125}_{55}\mathrm{Cs} + \beta^+ \).

Step-by-step solution

Understanding Beta Emission

Beta emission occurs when a neutron in an unstable nucleus is transformed into a proton while emitting a beta particle \((\beta^-\) or electron \(e^-\)). The atomic number increases by 1, and the mass number remains unchanged.

Writing the Equation for Beta Emission

For \(^{157} \mathrm{Eu}\), beta emission will result in an increment of 1 in the atomic number (Europium, atomic number 63, changes to Gadolinium, atomic number 64). The balanced nuclear equation is:\[ ^{157}_{63}\mathrm{Eu} \rightarrow ^{157}_{64}\mathrm{Gd} + \beta^- \]

Understanding Electron Capture

Electron capture occurs when an inner orbital electron is captured by the nucleus, causing a proton to transform into a neutron. The atomic number decreases by 1, and the mass number remains unchanged.

Writing the Equation for Electron Capture

For \(^{126} \mathrm{Ba}\), the atomic number 56 decreases to 55 (becomes Cesium). The balanced nuclear equation is:\[ ^{126}_{56}\mathrm{Ba} + e^- \rightarrow ^{126}_{55}\mathrm{Cs} \]

Understanding Alpha Emission

Alpha emission involves the release of an alpha particle \((\alpha = ^{4}_{2}\mathrm{He})\) from the nucleus, causing a decrease in atomic number by 2 and mass number by 4.

Writing the Equation for Alpha Emission

For \(^{146} \mathrm{Sm}\), the atomic number 62 decreases to 60 (becomes Neodymium), and the mass number decreases to 142. The balanced nuclear equation is:\[ ^{146}_{62}\mathrm{Sm} \rightarrow ^{142}_{60}\mathrm{Nd} + ^{4}_{2}\mathrm{He} \]

Understanding Positron Emission

Positron emission occurs when a proton in the nucleus is converted into a neutron, emitting a positron \((\beta^+\)). The atomic number decreases by 1, and the mass number remains unchanged.

Writing the Equation for Positron Emission

For \(^{125} \mathrm{Ba}\), the atomic number 56 decreases to 55 (becomes Cesium). The balanced nuclear equation is:\[ ^{125}_{56}\mathrm{Ba} \rightarrow ^{125}_{55}\mathrm{Cs} + \beta^+ \]

Key concepts

Beta Emission

Beta emission is a fascinating nuclear process that involves the transformation of a neutron into a proton within an unstable nucleus. During this change, the nucleus emits a beta particle, which is either an electron (\( \beta^-\), or \( e^- \)). This emission results in an increase in the atomic number by one, while the mass number remains constant. This change significantly alters the identity of the element, turning it into a different element in the periodic table with one higher atomic number.
Understanding this process is crucial as it helps explain natural radioactivity and its applications in various fields, such as medical imaging and radioactive dating.
For example, in the beta emission of \(^{157}_{63}\mathrm{Eu}\), the europium atom (atomic number 63) becomes gadolinium (atomic number 64), with the equation:

• \[ ^{157}_{63}\mathrm{Eu} \rightarrow ^{157}_{64}\mathrm{Gd} + \beta^- \]

Electron Capture

Electron capture is an intriguing process where an inner orbital electron is drawn into the nucleus and captured by a proton, transforming it into a neutron. This capture decreases the atomic number by one while the mass number stays unchanged. Electron capture is the opposite of beta emission, involving a proton being transformed back into a neutron.
This process occurs in some isotopes when it is energetically favorable for an electron to be captured rather than being emitted. It happens naturally and plays a role in certain types of radioactive decay.
For \(^{126}\mathrm{Ba}\), the capture of an electron changes its atomic number from 56 to 55, resulting in cesium through the equation:

• \[ ^{126}_{56}\mathrm{Ba} + e^- \rightarrow ^{126}_{55}\mathrm{Cs} \]

Alpha Emission

Alpha emission involves the release of an alpha particle from the nucleus of an atom. An alpha particle is essentially a helium nucleus (\( \alpha = ^{4}_{2}\mathrm{He} \)), consisting of two protons and two neutrons. This emission causes a decrease in the atomic number by two and the mass number by four.
Alpha emission is a common decay mode in heavy, radioactive nuclei and is part of their natural instinct to reduce size and become more stable.
In nuclear equations, this process is straightforward and results in a change of element. For instance, \(^{146}_{62}\mathrm{Sm}\) (samarium) becomes \(^{142}_{60}\mathrm{Nd}\) (neodymium), following this balanced equation:

• \[ ^{146}_{62}\mathrm{Sm} \rightarrow ^{142}_{60}\mathrm{Nd} + ^{4}_{2}\mathrm{He} \]

Positron Emission

Positron emission is a nuclear process that involves the transformation of a proton into a neutron with the ejection of a positron, which is the antimatter counterpart of an electron, denoted as \( \beta^+ \).This process decreases the atomic number by one while leaving the mass number unchanged.
The emission of a positron is one way radioactive decay can occur, often seen in proton-rich nuclides. It serves important functions in medical applications, most notably in positron emission tomography (PET) scans.
For example, when \(^{125}_{56}\mathrm{Ba}\) undergoes positron emission, it changes into cesium \(^{125}_{55}\mathrm{Cs}\), as shown in the equation:

• \[ ^{125}_{56}\mathrm{Ba} \rightarrow ^{125}_{55}\mathrm{Cs} + \beta^+ \]