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Uranium- \(238\left({ }_{92}^{238} \mathrm{U}\right)\) is radioactive and decays into a succession of different elements. The following particles are emitted before the nucleus reaches a stable form: \(\alpha, \beta, \beta, \alpha, \alpha, \alpha, \alpha, \alpha, \beta, \beta, \alpha, \beta, \beta\), and \(\alpha\) ( \(\beta\) stands for "beta particle," \(e^{-}\) ). What is the final stable nucleus? The original nucleus emitted 8 alpha particles and 6 beta particles. When an alpha particle is emitted. \(Z\) decreases by 2, since the alpha particle carries away a charge of \(+2 e .\) A beta particle carries away a charge of \(-1 e\), and so as a result the charge on the nucleus must increase to \((Z+1) e\). We then have, for the final nucleus, $$ \begin{array}{l} \text { Final } Z=92+6-(2)(8)=82 \\ \text { Final } A=238-(6)(0)-(8)(4)=206 \end{array} $$ The final stable nucleus is \({ }_{82}^{206} \mathrm{~Pb}\).

Step by step solution

01

Initial Understanding

We begin with Uranium-238, denoted as \(_{92}^{238} \text{U}\), which undergoes a series of radioactive decays. Understanding the names and charges of emitted particles is crucial. An alpha particle consists of 2 protons and 2 neutrons and reduces atomic number \(Z\) by 2 and mass number \(A\) by 4. A beta particle increases \(Z\) by 1 without changing \(A\).

02

Calculate Final Atomic Number

To find the final atomic number, we account for each type of emission. Starting at \(Z = 92\), we lose \(2 \times 8 = 16\) from 8 alpha emissions, and we gain \(1 \times 6 = 6\) from 6 beta emissions. The calculation is performed as: \(92 - 16 + 6 = 82\).

03

Calculate Final Mass Number

Start with \(A = 238\). Each alpha emission decreases the mass number by 4. With 8 alpha particles emitted, we have a decrease of \(8 \times 4 = 32\). Beta emissions don't affect mass number so we keep the counts the same. Calculate \(238 - 32 = 206\).

04

Identify the Final Stable Nucleus

Using the calculated \(Z = 82\) and \(A = 206\), we identify the corresponding element. The periodic table indicates that \(Z = 82\) is the element Lead (Pb). Thus, the final stable nucleus is \(_{82}^{206} \text{Pb}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Alpha Particles

In the world of radioactive decay, alpha particles play a significant role. An alpha particle is essentially the nucleus of a helium atom. It has 2 protons and 2 neutrons, which account for its mass and charge. When an alpha particle is emitted from a nucleus, it results in a decrease of the atomic number, denoted as \( Z \), by 2. This is because the nucleus is losing two protons.
Moreover, the mass number, represented by \( A \), decreases by 4 because the alpha particle also contains two neutrons. This means that for every alpha particle emitted, the original element transforms into another element further down the periodic table.
Alpha Decay Effects:

• Reduction in Atomic Number: \( Z - 2 \)
• Reduction in Mass Number: \( A - 4 \)

Through the process of alpha decay, heavy elements like Uranium-238 change into lighter elements, until a stable nucleus is reached.

Beta Particles

Beta particles are essentially high-energy, high-speed electrons or positrons emitted during the decay of radioactive nuclei. In beta decay, a neutron in unstable nuclei is transformed into a proton and an electron. The electron, known as a beta particle, is then emitted from the nucleus.
This process increases the atomic number, \( Z \), by 1 while the mass number \( A \) remains unchanged. Because beta particles carry away a negative charge of \(-1e\), the nucleus compensates by adding a proton.
Beta Decay Effects:

• Increase in Atomic Number: \( Z + 1 \)
• Mass Number Unchanged: \( A = \text{constant} \)

Thus, through beta decay, a nucleus can increase its atomic number, effectively moving to a higher element on the periodic table, while maintaining its overall mass.

Atomic Number

The atomic number, symbolized by \( Z \), is pivotal in understanding an element's identity. It represents the number of protons found within the nucleus of an atom. Since each element on the periodic table is defined by its unique atomic number, changes in \( Z \) result in element transformation.
During radioactive decay:

• Alpha Decay: The atomic number decreases by 2, as a helium nucleus is lost.
• Beta Decay: The atomic number increases by 1, owing to the conversion of a neutron into a proton.

Understanding these changes is crucial in radioactive decay sequences, as they guide the transition from one element to another.

Mass Number

The mass number, denoted by \( A \), is a fundamental property of an atom that consists of the total number of protons and neutrons within the nucleus. Unlike atomic number, the mass number is not singularly defining but provides insight into the isotope of an element.
In radioactive decay:

• Alpha Decay: The mass number decreases by 4 due to the emission of a helium nucleus, which contains 2 protons and 2 neutrons.
• Beta Decay: The mass number remains unchanged since this process involves the transformation within the nucleus without proton-neutron alteration.

The mass number helps in calculating and identifying isotopes and their transformations during decay processes, such as in the decay chain of Uranium-238 to Lead-206.