Question

For Al-28, calculate (a) the mass defect. (b) the binding energy in \(\mathrm{kJ} / \mathrm{mol}\).

Short answer

The mass defect for the Aluminum-28 isotope is found to be 0.242653 atomic mass units (amu). This mass defect, when converted to kilograms, results in a binding energy of 3.620963×10^{-12} Joules for one nucleus. The binding energy per mole for the Aluminum-28 isotope is then calculated to be 2.17908×10^9 kilojoules per mole (kJ/mol).

Step-by-step solution

Gather necessary information

To determine the mass defect of Al-28, we first need the following information: - The number of protons in the Aluminum nucleus (Z): 13 - The number of neutrons in the Aluminum nucleus (N): 15 (as the mass number is 28, and mass number = no. of protons + no. of neutrons) - The mass of one proton: 1.007276 amu (atomic mass unit) - The mass of one neutron: 1.008665 amu - The mass of the Aluminum-28 nucleus: 27.98191 amu (can be found in a reference table)

Calculate the sum of individual nucleons' masses

Now, we need to calculate the total mass of the individual protons and neutrons in the Aluminum-28 atom as follows: Total mass of protons = (number of protons) × (mass of one proton) Total mass of protons = (13) × (1.007276 amu/proton) = 13.094588 amu Total mass of neutrons = (number of neutrons) × (mass of one neutron) Total mass of neutrons = (15) × (1.008665 amu/neutron) = 15.129975 amu Sum of individual nucleons' masses = Total mass of protons + Total mass of neutrons Sum of individual nucleons' masses = 13.094588 amu + 15.129975 amu = 28.224563 amu

Calculate the mass defect

The mass defect (Δm) can be calculated by subtracting the mass of the Aluminum-28 nucleus from the sum of the individual nucleons' masses: Mass defect (Δm) = Sum of individual nucleons' masses - Mass of the Aluminum-28 nucleus Mass defect (Δm) = 28.224563 amu - 27.98191 amu = 0.242653 amu So, the mass defect (Δm) of Al-28 is 0.242653 amu.

Convert the mass defect to kg

Before calculating the binding energy, we need to convert the mass defect (Δm) from amu to kg. We can do it using the conversion factor, 1 amu = 1.66054×10^{-27} kg: Mass defect (Δm) in kg = 0.242653 amu × 1.66054×10^{-27} kg/amu = 4.02902×10^{-28} kg

Calculate the binding energy

Using Einstein's equation, \(E=mc^2\), we can calculate the binding energy, where m is the mass defect in kg and c is the speed of light in m/s (c = 2.998×10^8 m/s): Binding energy (E) = (4.02902×10^{-28} kg) × (2.998×10^8 m/s)^2 = 3.620963×10^{-12} J

Calculate the binding energy per mole

To determine the binding energy per mole, multiply the binding energy for one nucleus by Avogadro's number, and then convert the energy unit to kJ: Binding energy per mole = (3.620963×10^{-12} J) × (6.022×10^23 nuclei/mol) = 2.17908×10^12 J/mol Binding energy per mole = 2.17908×10^12 J/mol × (1 kJ / 1000 J) = 2.17908×10^9 kJ/mol So, the binding energy of Al-28 is 2.17908×10^9 kJ/mol.

Key concepts

Nuclear Chemistry

Nuclear chemistry is a subfield of chemistry focusing on reactions and changes in the atomic nucleus. It involves studying radioactive elements and isotopes, nuclear transformations, fission and fusion processes, and the way nuclear reactions release or absorb energy.

This field is crucial in understanding the inner workings of atoms, specifically how the nucleus achieves stability. A fascinating aspect of nuclear chemistry is the concept of 'mass defect', which refers to the difference in mass between the total mass of an atom's nucleons (protons and neutrons) when separate, and their mass when bound together in a nucleus.

The mass defect illustrates how mass can be converted into energy, particularly in nuclear reactions, as governed by Einstein's mass-energy equivalence principle. This becomes evident when observing that the binding energy, which is the energy required to break the nucleus apart into its individual protons and neutrons, can be derived from the mass defect. Nuclear chemistry thus provides critical insights into the fundamental principles that govern both chemical and atomic reactions.

Atomic Mass Unit

The atomic mass unit (amu) is a standard unit of mass that quantifies the mass of atoms or molecules. It's defined as one twelfth the mass of a carbon-12 atom, where a carbon-12 atom has exactly 12 amu.

This tiny unit of mass is essential in chemistry, especially when dealing with the infinitesimally small masses of atoms and subatomic particles. By convention, the mass of protons and neutrons is very close to 1 amu, which simplifies the calculations in nuclear chemistry, like determining the mass defect. For example, an Aluminum-28 atom has a specific predicted mass based on its number of protons and neutrons, yet when measured, this mass is slightly less due to mass defect. This discrepancy represents mass that has been converted into binding energy, holding the nucleus together. Understanding amu is fundamental when grasping concepts like mass defect because it is the standard scale used to express atomic masses and thereby calculate the changes in mass involved in nuclear reactions.

Einstein's Mass-Energy Equivalence

Einstein's mass-energy equivalence principle is encapsulated in the famous equation, \(E=mc^2\), where \(E\) stands for energy, \(m\) for mass, and \(c\) for the speed of light. The equation states that mass and energy are interchangeable; mass can be transformed into energy and vice versa, with the speed of light squared acting as the conversion factor.

This principle is pivotal in understanding nuclear reactions. In the case of the mass defect, the 'lost' mass upon formation of a nucleus doesn't just disappear; rather, it's converted into binding energy, which is the energy required to separate the nucleus into its constituent protons and neutrons. When you calculate the binding energy of a nucleus like Aluminum-28, you're essentially determining the energetic equivalent of the mass defect using Einstein's equation. This profound relationship between mass and energy explains not only nuclear stability but also the immense energy released in nuclear fission and fusion reactions, which are harnessed in power generation and weapons development.

Avogadro's Number

Avogadro's number, approximately \(6.022 \times 10^{23}\), is the number of constituent particles, usually atoms or molecules, contained in one mole of a substance. It is named after the Italian scientist Amedeo Avogadro. This huge number is a key component of the mole concept and provides a bridge between the atomic scale and the macroscopic scale of material amounts in chemistry. For instance, Avogadro's number enables chemists to count atoms or molecules by weighing, since one mole of a substance has its mass equal to its molecular or atomic mass expressed in grams.

When applied to nuclear chemistry, Avogadro's number allows us to convert per-nucleus quantities, such as the binding energy determined from mass defects, to a per-mole quantity which is more meaningful in chemical contexts. In calculating the binding energy per mole for Aluminum-28, Avogadro's number is used to scale the nuclear binding energy from a single nucleus up to an amount of substance that can be handled and measured in a laboratory setting, making the abstract atomic-scale concept of binding energy accessible and applicable in practical chemistry.