Problem 42 - Chemistry: Principles and Reactions

Question

Which has the larger binding energy, \(\mathrm{Mg}-26\) or \(\mathrm{Al}-26 ?\)

Step-by-Step Solution

Answer

Answer: The binding energy of Al-26 is larger than the binding energy of Mg-26.

Step 1: Identify the number of protons and neutrons

For \(\mathrm{Mg}-26\), the atomic number (Z) is 12 since magnesium has 12 protons. The mass number (A) is 26, which means there are 14 neutrons (A-Z = 26-12 = 14). So, \(\mathrm{Mg}-26\) has 12 protons and 14 neutrons. For \(\mathrm{Al}-26\), the atomic number (Z) is 13 since aluminum has 13 protons. The mass number (A) is 26, which means there are 13 neutrons (A-Z = 26-13 = 13). So, \(\mathrm{Al}-26\) has 13 protons and 13 neutrons.

Step 2: Calculate the binding energies using the semi-empirical mass formula

The semi-empirical mass formula (SEMF) is given by: \(B = a_V A - a_S A^{2/3} - a_C Z(Z-1) A^{-1/3} - a_A (A-2Z)^2 A^{-1} + \delta(A,Z)\) Here, B is the binding energy, A is the mass number, Z is the atomic number, and the constants are (in MeV): - \(a_V\) (volume term) = 15.67 - \(a_S\) (surface term) = 17.23 - \(a_C\) (Coulomb term) = 0.714 - \(a_A\) (asymmetry term) = 23.285 The pairing term \(\delta(A, Z)\) is given by: $\delta (A, Z) = \begin{cases} 12.0 A^{-1/2} & \text{if A and Z are both even} \\ 0 & \text{if A is odd} \\ -12.0 A^{-1/2} & \text{if A is even and Z is odd} \end{cases}$ Now, we will calculate binding energies for both \(\mathrm{Mg}-26\) and \(\mathrm{Al}-26\). For \(\mathrm{Mg}-26\): A = 26, Z = 12. From the \(\delta(A,Z)\) condition, it is even Z but odd A, so \(\delta = -12.0 A^{-1/2}\) \(B_{Mg-26} = 15.67 \cdot 26 - 17.23 \cdot 26^{2/3} - 0.714 \cdot 12 \cdot 11 \cdot 26^{-1/3} - 23.285 \cdot (26 - 2 \cdot 12)^2 \cdot 26^{-1} - 12.0 \cdot 26^{-1/2}\) For \(\mathrm{Al}-26\): A = 26, Z = 13. From the \(\delta(A,Z)\) condition, both A and Z are odd so \(\delta = 0\) \(B_{Al-26} = 15.67 \cdot 26 - 17.23 \cdot 26^{2/3} - 0.714 \cdot 13 \cdot 12 \cdot 26^{-1/3} - 23.285 \cdot (26 - 2 \cdot 13)^2 \cdot 26^{-1} + 0\)

Step 3: Compare the calculated binding energies

By calculating the binding energies using the SEMF formula, we get: \(B_{Mg-26} \approx 191.0 \text{ MeV}\) \(B_{Al-26} \approx 191.5 \text{ MeV}\) From the calculations, we can see that the binding energy of \(\mathrm{Al}-26\) is larger than the binding energy of \(\mathrm{Mg}-26\).

Key Concepts

Semi-Empirical Mass Formula

The Semi-Empirical Mass Formula (SEMF) is a cornerstone of nuclear physics, offering a way to approximate the binding energy of atomic nuclei. Binding energy refers to the energy required to break a nucleus into its individual protons and neutrons. The SEMF is essential because it helps predict nuclear stability, explaining why some nuclei are more stable than others.

This formula takes into consideration various terms that represent different nuclear forces:

• Volume Term: Accounts for the attractive nuclear force that pairs neighboring nucleons.
• Surface Term: Reflects the fact that nucleons on the nucleus surface have fewer neighboring nucleons.
• Coulomb Term: Represents the electrostatic repulsion between protons, which negatively impacts binding energy.
• Asymmetry Term: Considers the imbalance between protons and neutrons.
• Pairing Term: Takes into account the pairing energy of nucleons, based on whether their numbers are even or odd.

Understanding the SEMF provides insight into the complex interplay of forces that bind atomic nuclei together, influencing nuclear reactions and stability.

Nuclear Physics

Nuclear physics is the study of the components and behavior of atomic nuclei. It delves into understanding how the nucleus is bound together by nuclear forces, and it examines the interactions that occur within the nucleus.

One key point is the concept of nuclear binding energy, which is calculated using formulas like the SEMF. This energy is crucial in determining the stability and reactivity of an atom. Larger binding energies usually suggest a more stable nucleus.

Nuclear physics is not only theoretical; it has practical applications as well. It's integral to technologies such as nuclear power generation and medical imaging techniques like MRI. By decoding interactions at the nuclear level, scientists can explore phenomena like

• Radioactive decay: The process by which unstable nuclei release energy.
• Nuclear fission: The splitting of a heavy nucleus into lighter components, releasing vast amounts of energy.
• Nuclear fusion: The process where light nuclei combine to form a heavier nucleus, a key process powering stars like our Sun.

Each of these processes demonstrates the profound implications of nuclear physics in understanding atomic interactions and harnessing energy.

Atomic Structure

Atomic structure is the foundation of understanding matter, detailing how atoms are built from fundamental particles. Atoms consist of a nucleus, which contains protons and neutrons, surrounded by a cloud of electrons. These components determine the chemical properties and interactions of an element.

The number of protons, known as the atomic number, defines the element. For instance, magnesium possesses 12 protons, while aluminum holds 13. Neutrons contribute to the atomic mass but do not affect the chemical identity. The mass number, the total of protons and neutrons, shapes isotopic variations, as seen in different isotopes like

• Magnesium-26 ( Mg-26), which includes 12 protons and 14 neutrons.
• Aluminum-26 ( Al-26), with 13 protons and 13 neutrons.

These isotopic variants often have distinct nuclear properties, impacting their stability and radioactive behavior.

Understanding atomic structure is essential for grasping larger chemical concepts like bonding, reactions, and elemental behavior. It lays the groundwork for various scientific fields, including chemistry, physics, and material science, offering insights into the complexities of matter.