Problem 57 - Chemistry: The Central Science

Question

How much energy must be supplied to break a single \({ }^{21} \mathrm{Ne}\) nucleus into separated protons and neutrons if the nucleus has a mass of \(20.98846\) amu? What is the nuclear binding energy for \(1 \mathrm{~mol}\) of \({ }^{21} \mathrm{Ne}\) ?

Step-by-Step Solution

Answer

The energy required to break a single Ne-21 nucleus into separated protons and neutrons is \(5.3425 \times 10^{-12} J\), and the nuclear binding energy for 1 mol of Ne-21 is \(3.216 \times 10^{12} J\).

Step 1: Calculate the binding energy of Ne-21 nucleus

First, we need to calculate the mass defect of the nucleus. Mass defect is the difference between the mass of the nucleus and the sum of the masses of its constituent protons and neutrons. Ne-21 has 10 protons and 11 neutrons. Mass of a proton is 1.00728 amu and that of a neutron is 1.00867 amu. Mass of 10 protons = 10 × 1.00728 amu = 10.0728 amu Mass of 11 neutrons = 11 × 1.00867 amu = 11.09537 amu Total mass of protons and neutrons = 10.0728 amu + 11.09537 amu = 21.16817 amu Now, we need to find the mass defect: Mass defect = Total mass of protons and neutrons - Mass of Ne-21 nucleus Mass defect = 21.16817 amu - 20.98846 amu = 0.17971 amu Now, calculate the binding energy for the Ne-21 nucleus: Binding energy (per nucleus) = Mass defect × (1 amu in kg) × (speed of light)^2 The mass of 1 amu in kg is approximately 1.6605 × 10^{-27} kg, and the speed of light is 2.9979 × 10^{8} m/s. Binding energy (per nucleus) = 0.17971 amu × 1.6605 × 10^{-27} kg/amu × (2.9979 × 10^{8} m/s)^{2} Binding energy (per nucleus) = 5.3425 × 10^{-12} J

Step 2: Calculate the nuclear binding energy for 1 mol of Ne-21

Since there are 6.022 × 10^{23} atoms in 1 mol, we now calculate the binding energy for 1 mol of Ne-21: Binding energy (per mol) = Binding energy (per nucleus) × Avogadro's Number Binding energy (per mol) = 5.3425 × 10^{-12} J × 6.022 × 10^{23} Binding energy (per mol) = 3.216 × 10^{12} J So, the nuclear binding energy for 1 mol of Ne-21 is 3.216 × 10^{12} J.

Key Concepts

Mass Defect

In the realm of nuclear physics, 'mass defect' is a fundamental phenomenon that refers to the discrepancy between the calculated mass of an atomic nucleus and the sum of the masses of its individual protons and neutrons. When protons and neutrons come together to form a nucleus, the total mass of the nucleus is found to be less than the sum of its parts. This 'missing' mass is actually converted into energy, which is then used to hold the nucleus together; this energy is known as the binding energy.

To put it simply, during the nuclear formation, some mass is transformed into energy following Einstein’s famous equation, \( E = mc^2 \) where \( E \) stands for energy, \( m \) for mass, and \( c \) the speed of light in a vacuum. In the case of \( {}^{21}Ne \) given in the exercise, students are challenged to calculate the mass defect by subtracting the atomic mass unit (amu) of the nucleus from the combined mass of the individual protons and neutrons. Understanding mass defect is crucial for students to comprehend the concept of nuclear stability and the forces at play within an atom's core.

Nuclear Physics

Nuclear physics is the branch of physics that studies the constituents and interactions of atomic nuclei. It encompasses a variety of concepts, including the ones you are learning like mass defect and binding energy. The heart of nuclear physics lies in understanding the forces that hold the nucleus together, despite the repulsive electromagnetic force between the positively charged protons.

This field explains phenomena such as radioactivity, nuclear fission and fusion, and the behavior of nuclei under extreme conditions. As part of studying nuclear physics, students delve into the intricacies of how energy is generated from the nucleus and how it plays a vital role in both natural processes, like the stars shining, and human-made technologies such as nuclear reactors and medical imaging.

Avogadro's Number

Avogadro's number, denoted commonly as \( N_A \) and named after the scientist Amedeo Avogadro, is a constant that represents the number of particles found in one mole of any substance. \( N_A = 6.022 \times 10^{23} \) entities per mole, which can refer to atoms, ions, or molecules depending on the context.

In the exercise you're working through, Avogadro's number is essential for transitioning from the individual nucleus to a larger, more relatable scale - precisely one mole of \( {}^{21}Ne \) nuclei. It’s not only fundamental to nuclear physics but also to chemistry and physics at large, aiding in the understanding of the microscopic world on a macroscopic scale. Avogadro's number bridges the gap between an individual particle and the amount of substance we can measure in the laboratory.

Energy Conversion Calculations

Energy conversion calculations play a critical role in the study of physical systems, where various forms of energy are transformed from one type to another. In the context of our nuclear binding energy exercise, we're specifically focused on the conversion of mass into energy. \( E = mc^2 \) embodies this concept, where a tiny amount of mass can be converted into a significant amount of energy.

In the provided solution, students utilize this conversion calculation to determine the binding energy released from the mass defect, indicative of the strength of the bonds within the nucleus. Performing such calculations is pivotal in physics and engineering, aiding in the design of power plants, understanding of stellar processes, and even the development of medical treatments involving radioactive isotopes.