- The reaction follows the law of conservation of mass number and atomic number.
- Solve for the mass number: \(230 = 4 + A \implies A = 226\).
- Solve for the atomic number: \(90 = 2 + Z \implies Z = 88\).
- Identify the element with atomic number 88 as Radium, resulting in \({ }_{88}^{226} \text{Ra}\), which is option b.
- Final answer: \(\boxed{{ }_{88}^{226} \text{Ra}}\).
Explanation
- Problem Analysis and Conservation Laws
In the given nuclear reaction \({ }_{90}^{230} \text{Th} \rightarrow { }_{2}^{4} \text{He} + X\), we are observing an alpha decay process. To determine the unknown particle \(X\), we must apply the fundamental conservation laws of nuclear reactions, which state that the total mass number and the total atomic number must remain constant before and after the reaction. The mass number is the superscript (top value), and the atomic number is the subscript (bottom value).
- Applying Conservation Laws
Let the unknown component \(X\) be represented as \({ }_{Z}^{A} \text{Y}\).
First, we apply the conservation of mass number:
\[230 = 4 + A\]
Solving for \(A\), we get:
\[A = 230 - 4 = 226\]
Next, we apply the conservation of atomic number:
\[90 = 2 + Z\]
Solving for \(Z\), we get:
\[Z = 90 - 2 = 88\]
- Identifying the Isotope and Conclusion
Now that we have the atomic number \(Z = 88\) and the mass number \(A = 226\), we identify the element. An atomic number of 88 corresponds to the element Radium (Ra) on the periodic table. Combining these, the unknown component \(X\) is \({ }_{88}^{226} \text{Ra}\). Comparing this result to the given options, it clearly matches option b.
Examples
Nuclear decay equations are essential in fields like medicine, archaeology, and energy. For instance, carbon-14 dating uses the principle of radioactive decay to determine the age of ancient organic materials. By calculating missing components in nuclear equations—as we did here with Thorium decay—scientists can understand and predict the transformation of radioactive elements over time, which is fundamental to dating historical artifacts and managing nuclear fuel.