Chapter 14: Problem 5
The value of packing fraction of carbon-12 is (a) positive (b) negative (c) zero (d) infinite
Short Answer
Expert verified
The packing fraction of carbon-12 is positive.
Step by step solution
01
Understanding Packing Fraction
Packing fraction is defined as the difference between the mass of the nucleons if they were separated and the actual mass of the nucleus. It is given by the formula: \( Packing \, Fraction = Zm_p + (A-Z)m_n - M \), where \( Z \) is the atomic number, \( A \) is the mass number, \( m_p \) is the mass of a proton, \( m_n \) is the mass of a neutron, and \( M \) is the actual mass of the nucleus.
02
Calculating the Packing Fraction for Carbon-12
For Carbon-12, \( Z = 6 \), \( A = 12 \), \( m_p \approx 1.007276 u \), \( m_n \approx 1.008665 u \), and \( M = 12 u \) (since carbon-12 is used to define the atomic mass unit). Calculate the packing fraction using the given masses: \( Packing \, Fraction = (6 \times 1.007276 u) + (6 \times 1.008665 u) - 12 u \).
03
Result of Packing Fraction Calculation
Plugging the values into the packing fraction formula: \( Packing \, Fraction = (6 \times 1.007276) + (6 \times 1.008665) - 12 \approx 12.095646 u - 12 u = 0.095646 u \). Since the result is a positive number, the value of packing fraction for carbon-12 is positive.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Nuclear Chemistry and Packing Fraction
Nuclear chemistry involves the study of the structure, properties, and reactions of atomic nuclei. Within this field, the concept of packing fraction plays a key role in understanding the stability and energy dynamics of a nucleus.
Packing fraction is a measure of how densely the constituents of a nucleus, namely protons and neutrons, are 'packed' together. It reflects the difference in mass between the combined mass of separate nucleons (protons and neutrons) and the actual measured mass of the nucleus itself. A positive packing fraction indicates that the mass of the nucleus is less than the sum of its parts, which is a result of the binding energy released when nucleons come together to form the nucleus.
To understand packing fraction, it's helpful to consider it in the context of energy. During nucleosynthesis, energy is released as mass according to Einstein's famous equation, \( E=mc^2 \), where \( E \) represents the energy, \( m \) the mass defect, and \( c \) the speed of light. This transformation of mass into energy is what accounts for the binding energy and holds the nucleus together, providing crucial insights into the forces at play within an atom's core.
Packing fraction is a measure of how densely the constituents of a nucleus, namely protons and neutrons, are 'packed' together. It reflects the difference in mass between the combined mass of separate nucleons (protons and neutrons) and the actual measured mass of the nucleus itself. A positive packing fraction indicates that the mass of the nucleus is less than the sum of its parts, which is a result of the binding energy released when nucleons come together to form the nucleus.
To understand packing fraction, it's helpful to consider it in the context of energy. During nucleosynthesis, energy is released as mass according to Einstein's famous equation, \( E=mc^2 \), where \( E \) represents the energy, \( m \) the mass defect, and \( c \) the speed of light. This transformation of mass into energy is what accounts for the binding energy and holds the nucleus together, providing crucial insights into the forces at play within an atom's core.
Mass Defect and Binding Energy
The mass of an atomic nucleus is always less than the sum of the individual masses of the protons and neutrons that comprise it. This discrepancy is known as the mass defect. The mass defect indicates that some mass has been converted into energy when protons and neutrons in a nucleus bond together.
The binding energy is this converted energy, which serves as a 'glue' to hold the nucleus together against the repulsive force between the positively charged protons. The larger the binding energy, the more stable the nucleus. In order to break a nucleus apart into its individual nucleons, an amount of energy equal to the binding energy would have to be supplied.
In terms of mass defect, the exercise improvement advice would be to explain that in a stable nucleus like Carbon-12, the binding energy is a result of the mass defect, and it can be calculated from the packing fraction. Thus, it directly influences the stability of an element's nucleus, emphasizing its importance in nuclear chemistry. The packing fraction gives a quantifiable measure of this mass defect and, by extension, the binding energy.
The binding energy is this converted energy, which serves as a 'glue' to hold the nucleus together against the repulsive force between the positively charged protons. The larger the binding energy, the more stable the nucleus. In order to break a nucleus apart into its individual nucleons, an amount of energy equal to the binding energy would have to be supplied.
In terms of mass defect, the exercise improvement advice would be to explain that in a stable nucleus like Carbon-12, the binding energy is a result of the mass defect, and it can be calculated from the packing fraction. Thus, it directly influences the stability of an element's nucleus, emphasizing its importance in nuclear chemistry. The packing fraction gives a quantifiable measure of this mass defect and, by extension, the binding energy.
Atomic Nucleus Composition
The atomic nucleus consists of two types of nucleons: protons and neutrons. Protons are positively charged, while neutrons carry no charge. Despite their differences in charge, protons and neutrons are comparable in mass and are collectively referred to as nucleons because they reside in the nucleus.
The composition of a nucleus is defined by its number of protons (atomic number, \( Z \) and its number of neutrons. The total number of nucleons is referred to as the mass number (\( A \)). It's the balance between these protons and neutrons that determines the stability and characteristics of an atom.
To make the concept more comprehensible, one might explain that the strong nuclear force, which acts between all nucleons, is responsible for overcoming the repulsion between protons and stabilizing the nucleus. This strong force is short-range but dominant over the electromagnetic force at the distances typical within a nucleus, allowing a compact, dense structure held together by substantial binding energy—as reflected in the measured packing fraction for an element like Carbon-12.
The composition of a nucleus is defined by its number of protons (atomic number, \( Z \) and its number of neutrons. The total number of nucleons is referred to as the mass number (\( A \)). It's the balance between these protons and neutrons that determines the stability and characteristics of an atom.
To make the concept more comprehensible, one might explain that the strong nuclear force, which acts between all nucleons, is responsible for overcoming the repulsion between protons and stabilizing the nucleus. This strong force is short-range but dominant over the electromagnetic force at the distances typical within a nucleus, allowing a compact, dense structure held together by substantial binding energy—as reflected in the measured packing fraction for an element like Carbon-12.
