Problem 17 - Chemistry The Central Science In Si Units

Question

The naturally occurring radioactive decay series that begins with \({ }_{92}^{235} \mathrm{U}\) stops with formation of the stable \({ }_{82}^{207} \mathrm{~Pb}\) nucleus. The decays proceed through a series of alpha-particle and beta-particle emissions. How many of each type of emission are involved in this series?

Step-by-Step Solution

Answer

There are 7 alpha decays and 4 beta decays in the series.

Step 1: Understand Alpha and Beta Decay

In radioactive decay, an alpha particle \( (_2^4\mathrm{He}) \) consists of 2 protons and 2 neutrons, so the emission of an alpha particle decreases the atomic number by 2 and the mass number by 4. A beta particle is an electron \( (_{-1}^0e) \) emitted during the conversion of a neutron to a proton, effectively increasing the atomic number by 1 without changing the mass number.

Step 2: Determine the Overall Change in Atomic and Mass Numbers

Starting with \({}_{92}^{235} \mathrm{U}\), the series ends with \({}_{82}^{207} \mathrm{~Pb}\). Therefore, the total change in atomic number is \(92 - 82 = 10\), and the total change in mass number is \(235 - 207 = 28\).

Step 3: Calculate the Number of Alpha Decays

Each alpha decay reduces the atomic number by 2 and the mass number by 4. Let \( x \) be the number of alpha decays. Therefore, we have \(4x = 28\) leading to \(x = 7\). Thus, there are 7 alpha decays.

Step 4: Calculate the Number of Beta Decays

Let \( y \) be the number of beta decays. Each beta decay increases the atomic number by 1. Since the total change in atomic number is 10 and the alpha decays accounted for a change of 7\times2=14, the beta decays must offset this by increasing it by 4 (to achieve a net change of 10). Hence, \( y = 4 \). Thus, there are 4 beta decays.

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay where an unstable nucleus emits an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, which can be denoted as \(_2^4\mathrm{He}\). This process results in the reduction of the atomic number of the element by 2 and the mass number by 4.
In practice, when a nucleus undergoes alpha decay, it essentially loses a helium nucleus.
This loss has significant impacts:

• Reduction in atomic number: The original element transforms into a new element that is two steps before it in the periodic table.
• Reduction in mass number: The mass number decreases by four units.

Understanding alpha decay is crucial in predicting the new element formed after decay and is a common step in transforming heavy elements into lighter, more stable forms.

Beta Decay

Beta decay occurs when a neutron in the nucleus converts into a proton, emitting a beta particle, which is essentially an electron, denoted by \(_{-1}^0\mathrm{e}\). This process results in an increase in the atomic number by 1 while the mass number remains unchanged.
There are several key points to note about beta decay:

• Atomic number increase: As the neutron changes to a proton, the nucleus becomes one atomic number higher.
• Mass number constant: The overall count of nucleons (protons and neutrons) stays the same, keeping the mass number unchanged.

Beta decay is important in balancing the changes brought about by alpha decay in decay series. It ensures that the nucleus returns to a stable and less radiating form.

Atomic Number

The atomic number of an element, symbolized as \(Z\), is the number of protons in the nucleus of an atom.
It determines the identity of an element and its position on the periodic table.
The atomic number is crucial because:

• Element identity: It defines what element the atom represents, such as hydrogen having an atomic number of 1.
• Chemical properties: It heavily influences the chemical behavior of the element, dictating its reactions and interactions with other elements.
• Changes in decay: In radioactive decay processes like alpha and beta decay, the atomic number changes, leading to the transformation of the element into a different one.

These transformations are significant for understanding radioactive decay series, where elements change to become stable.

Mass Number

The mass number of an element, denoted as \(A\), is the total number of protons and neutrons in an atom's nucleus.
Unlike the atomic number, the mass number tells us about the isotope of an element.
This number is essential for the following reasons:

• Isotope identification: Atoms of the same element with different mass numbers are isotopes, like \(^3\mathrm{He}\) and \(^4\mathrm{He}\).
• Understanding decay: During radioactive decay, the mass number changes, especially in processes like alpha decay.
• Energy calculations: It's used in calculating nuclear binding energies, essential for understanding nuclear reactions.

Grasping the concept of mass number is vital for evaluating how an atom changes throughout a decay series and for predicting the stability of the resulting element.