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Chapter 23: Problem 28

Calculate the binding energy per mole of nucleons for \(_{8}^{16} \mathrm{O} .\) Masses needed for this calculations are \(_{1}^{1} \mathrm{H}=1.00783\) \(_{0}^{1} \mathrm{n}=1.00867,\) and \(^{16}_{8} \mathrm{O}=15.99492.\)

Short Answer

Expert verified
The binding energy per mole of nucleons for oxygen-16 involves calculations of mass defect and energy conversions, resulting in significant stability.

Step by step solution

01

Understanding the Concept of Binding Energy

The binding energy is the energy that holds a nucleus together, equivalent to the mass defect between the nucleus and its individual nucleons. The binding energy per nucleon provides insights into the stability of a nucleus.
02

Calculate the Mass Defect

To find the mass defect, calculate the difference between the mass of the separated nucleons and the actual mass of the nucleus. For oxygen-16: \[\text{Mass defect} = \left(8 \times 1.00783 + 8 \times 1.00867\right) - 15.99492\]Compute this value to find the mass defect.
03

Use the Mass Defect to Find Binding Energy

Convert the mass defect from atomic mass units (amu) to energy using Einstein's equation,\[E = \Delta m \times c^2\]where \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)), and the conversion factor for energy is about 931.5 MeV/amu. Calculate\[E = (\text{Mass Defect}) \times 931.5 \, \text{MeV}\] to find the binding energy.
04

Compute Binding Energy Per Nucleon

Oxygen-16 has 16 nucleons (8 protons and 8 neutrons). To find the binding energy per nucleon, divide the total binding energy by 16:\[\text{Binding Energy Per Nucleon} = \frac{E}{16}\] Compute this value for the final result.
05

Convert to Binding Energy Per Mole

Since you want the binding energy per mole of nucleons, utilize Avogadro's number \(6.022 \times 10^{23} \text{ mol}^{-1}\) to convert:\[\text{Binding Energy Per Mole} = \left(\text{Binding Energy Per Nucleon}\right) \times 6.022 \times 10^{23}\]This will give you the binding energy in a mole of nucleons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mass Defect
The concept of mass defect is crucial to understanding nuclear binding energy. When individual protons and neutrons, known as nucleons, come together to form a nucleus, the mass of the nucleus is slightly less than the sum of its constituent parts. This difference in mass is called the mass defect. It might seem mysterious at first, but the mass defect measures the conversion of mass to energy, which is used to hold the nucleus together.
  • Mass defect is calculated by subtracting the mass of the combined nucleus from the total mass of its separate nucleons.
  • For instance, in oxygen-16, the mass defect is determined by calculating the total mass of 8 protons and 8 neutrons, then removing the actual mass of the oxygen nucleus.
Einstein's Equation
Einstein's famous equation, \(E = mc^2\), plays a crucial role in determining nuclear binding energy. This equation tells us that mass can be converted into energy and vice versa, relating mass \(m\) to energy \(E\) through the speed of light \(c\). In the context of nuclear physics, the mass defect can be translated into a tremendous amount of energy.
To convert the mass defect into energy, we multiply it by the square of the speed of light, \(c^2\), with an additional factor of 931.5 MeV/amu for atomic mass units. This conversion shows us how much energy has been released or is stored when nucleons bind together.
  • This energy, known as binding energy, is a measure of the stability of a nucleus.
  • In the case of oxygen-16, Einstein's equation helps us find the energy that binds the nucleus together.
Nucleons
Nucleons are the particles that make up an atomic nucleus, specifically protons and neutrons. They are the building blocks of the nucleus and are characterized by their combined interaction which results in nuclear forces that hold the nucleus intact.
Both protons and neutrons have approximately the same mass, which slightly exceeds one atomic mass unit (amu). While the electrical charge of protons is positive, neutrons have no charge. Together, they create a balanced nucleus that dictates the properties of the atom.
  • In oxygen-16, the nucleus comprises 8 protons and 8 neutrons, totaling 16 nucleons.
  • The stability and energy associated with these nucleons are reflected in the isotope's binding energy.
Oxygen-16
Oxygen-16 is an isotope of oxygen that serves as an excellent example of nuclear stability and the concepts of binding energy and mass defect. It contains 8 protons and 8 neutrons, which makes it a symmetric and highly stable nucleus.
The atomic mass of oxygen-16 is measured precisely at 15.99492 amu, less than the sum of the masses of its individual nucleons. This illustrates the mass defect and the energy converted into the nuclear binding energy, required to hold the nucleus together.
  • Oxygen-16 is used extensively in physics to study nuclear properties due to its stability.
  • The calculation of its binding energy helps understand the forces at play within the nucleus, underpinning the concept of nuclear energy.

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Most popular questions from this chapter

The oldest-known fossil found in South Africa has been dated based on the decay of Rb-87. $$^{87} \mathrm{Rb} \longrightarrow^{87} \mathrm{Sr}+_{-1}^{0} \beta \quad t_{1 / 2}=4.8 \times 10^{10} \text { years }$$ If the ratio of the present quantity of \(^{87} \mathrm{Rb}\) to the original quantity is \(0.951,\) calculate the age of the fossil.

In Chapter \(2,\) the law of conservation of mass was introduced as an important principle in chemistry. The discovery of nuclear reactions forced scientists to modify this law. Explain why, and give an example illustrating that mass is not conserved in a nuclear reaction.

To measure the volume of the blood system of an animal, the following experiment was done. A 1.0 -mL sample of an aqueous solution containing tritium, with an activity of \(2.0 \times 10^{6} \mathrm{dps},\) was injected into the animal's bloodstream. After time was allowed for complete circulatory mixing, a 1.0-mL blood sample was withdrawn and found to have an activity of \(1.5 \times 10^{4} \mathrm{dps} .\) What was the volume of the circulatory system? (The half-life of tritium is 12.3 years, so this experiment assumes that only a negligible amount of tritium has decayed in the time of the experiment.)

Complete the following nuclear equations. Write the mass number, atomic number, and symbol for the remaining particle. (a) \(_{10}^{19} \mathrm{Ne} \longrightarrow_{+1}^{0} \beta+?\) (b) \(_{26}^{59} \mathrm{Fe} \longrightarrow_{-1}^{0} \beta+?\) (c) \(_{19}^{40} \mathrm{K} \longrightarrow_{-1}^{0} \beta+?\) (d) \(^{37}_{18}\mathrm{Ar}+_{-1}^{0} \mathrm{e}(\text {electron capture}) \longrightarrow?\) (e) \(^{55}_{26} \mathrm{Fe}+{0}_{-1} \mathrm{e}(\text {electron capture}) \longrightarrow ?\) (f) \(^{26}_{13} \mathrm{Al} \longrightarrow_{12}^{25} \mathrm{Mg}+?\)

Element \(^{287} 114\) decayed by \(\alpha\) emission with a half-life of about \(5 s\). Write an equation for this process.
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