Why is mass defect negative

In the case of nuclei, the binding energy is so great that it accounts for a significant amount of mass. The actual mass is always less than the sum of the individual masses of the constituent protons and neutrons because energy is removed when when the nucleus is formed. This energy has mass, which is removed from the total mass of the original particles.

This mass, known as the mass defect, is missing in the resulting nucleus and represents the energy released when the nucleus is formed. Mass defect M d can be calculated as the difference between observed atomic mass m o and that expected from the combined masses of its protons m p , each proton having a mass of 1. Mass must be in units of kg. Once this energy, which is a quantity of joules for one nucleus, is known, it can be scaled into per-nucleon and per-mole quantities.

To convert to joules per nucleon, simply divide by the number of nucleons. Nuclear binding energy can also apply to situations when the nucleus splits into fragments composed of more than one nucleon; in these cases, the binding energies for the fragments, as compared to the whole, may be either positive or negative, depending on where the parent nucleus and the daughter fragments fall on the nuclear binding energy curve.

If new binding energy is available when light nuclei fuse, or when heavy nuclei split, either of these processes result in the release of the binding energy. This energy—available as nuclear energy—can be used to produce nuclear power or build nuclear weapons. When a large nucleus splits into pieces, excess energy is emitted as photons, or gamma rays, and as kinetic energy, as a number of different particles are ejected. For example, the nuclear mass defect for carbon 12 C is 0.

On the other hand, 12 C with the atomic mass of One would only be able to differentiate the two usages of the term mass defect based on the context in which it is discussed. The goal of this discussion is not to review different applications of mass defect in mass spectrometry, which can be found elsewhere [ 12 ], but to discuss mass defect in general.

The importance of mass defect in nuclear physics is reviewed and its significance in mass spectral analysis is discussed. It is argued that nuclear and chemical mass defects are not the same and their difference is highlighted at the end, by looking at the mass defect plots of the elements in the periodic table.

While nuclear mass defect reflects a physical property, chemical mass defect does not, and it is based on a convention that carbon has a mass defect of zero. Mass of the nucleus is slightly less than the added masses of its constituent protons and neutrons and this mass difference is called nuclear mass defect [ 16 ]:.

The energy equivalent of the nuclear mass defect is known as the nuclear binding energy. In other words, binding energy is the energy released with the formation of a nucleus from its nucleons, or is the energy required to break a nucleus into its individual components. The nuclear mass defect is a fundamental property of a nucleus and is a fixed value corresponding to a certain amount of binding energy for that nucleus.

Mass defect and binding energy are important factors in the energy involved in nuclear reactions. Looking at how mass defect and binding energy change from one element to another will make the relationship between mass defect, binding energy, and nuclear energy more apparent. A plot of nuclear mass defect versus mass number for different elements is shown in Figure 1. Nuclear mass defect versus mass number for the most abundant isotopes. Masses are based on the 12 C mass scale.

The nuclear mass defect changes with mass number from zero for hydrogen to a value close to —2 for uranium. The nuclear mass defect per nucleon, which is mass defect divided by mass number, is a more useful value. It provides a more meaningful way of comparison between different elements and is plotted against mass number in Figure 2a. The corresponding energy, the binding energy per nucleon, is the amount of energy that is released per nucleon upon the formation of a nucleus and is an indication of its stability Figure 2b.

If we start from hydrogen and move to heavier atoms on the curve in Figure 2b , we find that binding energy per nucleon increases and reaches a maximum around 56, the mass number for iron. If we continue past iron, the binding energy per nucleon decreases gradually. Therefore, the medium mass nuclei are the most stable. Iron has the highest binding energy per nucleon and is the most stable nucleus 62 Ni is more stable than 56 Fe but it is not the most abundant isotope of nickel.

The difference in the binding energy between elements provides an opportunity for producing energy if elements with lower binding energy per nucleon are converted into more stable elements with higher binding energy per nucleon. Figure 2b shows that heavy nuclei gain stability and therefore give off energy if they are fragmented into two mid-sized nuclei. The process of splitting a nucleus into smaller nuclei is known as fission.

The breakdown of the uranium nucleus into more stable nuclei as a result of collisions with neutrons releases energy and is an example of a fission reaction [ 17 ]:. This is the process by which nuclear energy is produced in nuclear power plants. Energy is also released if light nuclei are combined or fused together to form more massive nuclei with greater binding energy per nucleon than that of reacting species.

This too is a change towards a greater stability. The process that is called fusion is exothermic only for the nuclei of mass number below The reaction of deuterium and tritium to form helium is an example of a fusion reaction [ 17 ]:. Nuclear fusion is the source of energy of the sun and other stars. Combination of hydrogen nuclei to form more complex nuclei was first proposed as the mechanism of production of stellar energy in after the publication of masses of isotopes by Aston [ 18 ].

The difference in the binding energy per nucleon between hydrogen and helium is much more than between uranium and a mid-mass element such as iron and as a result hydrogen fusion can produce more energy, kilogram for kilogram, than the nuclear fission of uranium. The released energy as a result of the formation of nuclei can be compared to the heat of formation of molecules.

The heat of formation of a molecule is the energy released with the formation of a molecule from its elements and it is a measure of the stability of the molecule. A large heat of formation is an indication of a stable molecule since a large amount of energy is required to decompose the molecule into its constituent atoms. The energy released with the formation of a nucleus from its constituent protons and neutrons is a measure of the stability of the nucleus in a similar way.

The heat of formation of molecules is considerably less than the energy released with the formation of nuclei. For comparison, the energy released as a result of the formation of a carbon nucleus from protons and neutrons is 8. This corresponds to a mass loss of 4. Unlike nuclear mass defect, the mass loss is too minute to be measured and is largely ignored.

The energy change in chemical reactions is calculated from the heat of formation of the reactants and products. Similarly, we can calculate the amount of energy released or consumed in a nuclear reaction. The Q -value is the energy involved in a nuclear reaction and is defined as:. A positive Q -value is an indication of an energetically favored reaction.

The Q -value for the reaction 3 , for example, is calculated using atomic masses of reactants and products:. The neutral isotopic mass is not always given in isotope tables, but the mass excess is listed instead in units of mass or energy MeV. It is defined as the difference between the measured atomic mass m and the mass number A :.

Since the sum of the mass numbers on either side of a nuclear reaction is the same, A is conserved. We are left with an equation for the Q -value that depends only on the mass excess. For the above example, one can use the mass excess in mass unit u to find the Q -value:.

However, it is easier to use the energy equivalent of the mass excess, which is commonly listed in tables [ 19 ]:. The value that is reported in the tables of nuclear properties is the mass excess in units of energy rather than the mass. The above example shows that the use of mass excess makes prediction of the Q -values straightforward and simplifies the calculation of energy involved in nuclear reactions.

Definition of mass defect in general chemistry and physics textbooks is consistent with Equation 1 and what has been discussed so far [ 17 , 20 ]. The definition used in mass spectrometry is discussed next. Application of mass spectrometry was extended from mainly atomic to molecular analysis as commercial instruments became available to meet the demands of the chemical and petroleum industry.

Resolution of molecular ions with the same nominal mass but arising from different combination of elements became possible with the increasing resolution of the instruments.

Petroleum samples contain many compounds with the same nominal mass but with different elemental compositions that could be identified by high resolution instruments. A goal of petroleum analysis was to identify compounds based on their class, type, and the degree of alkylation. Compound class is collectively defined as all of elemental compositions with the same heteroatom content. Compounds with the same heteroatom but with various numbers of hydrogen in their empirical formula belong to different compound types.

Compound type arises from different number of double bonds or rings in the molecule. Within the same class and type, there are many compounds with varying degrees of alkylation, which differ only in the numbers of methylene CH 2 groups in their formula. They are commonly known as homologous series. Presence of a large number of different molecules made the interpretation of the mass spectra of petroleum samples a difficult task.

The high-resolution spectra of such samples with many resolved peaks at the same nominal mass demanded a new approach for data interpretation. A data interpretation strategy was developed based on the chemical mass defect. Mass defect was defined as the difference between the accurate mass of the ion in question and a reference hydrocarbon ion with the same nominal mass [ 10 ]. This approach was used to identify several new compound classes and types not reported before.

Chemical mass defect was later defined as the difference between the nominal mass and the measured monoisotopic mass by Kendrick and was utilized to facilitate analysis of petroleum samples [ 11 ]:.

Nominal mass is the mass calculated using the integer mass of the most abundant isotopes of each element. It is equivalent to mass number expressed in mass unit [ 21 ] and they are both represented by the same symbol, A, here.

Therefore, chemical mass defect is the difference between the monoisotopic mass and a whole number mass, which may not be the closest integer mass. Nominal mass is the closest integer mass to the monoisotopic mass for low molecular weight compounds but this is not necessarily true at higher masses where the difference between monoisotopic and nominal masses can be quite large [ 22 ].

For example, polystyrene, C 4 H 9 C 8 H 8 H, has a nominal mass of u and monoisotopic mass of Chemical mass defect as defined in Equation 6 is what is currently used in mass spectral analysis. On this mass scale, the repeating mass of methylene does not change the mass defect and all of the compounds in a homologous series that belong to the same class and type will have the same chemical mass defect.

Plotting chemical mass defect versus nominal mass will help visualize all of the compounds present in the spectrum in a way that would not be possible by just viewing the spectrum. Figure 3 shows an example of such a plot for peaks from the high-resolution mass spectrum of a crude oil sample [ 23 ]. Compounds belonging to the same class and type but with different number of CH 2 groups will fall on a horizontal line on this plot.

Similarly, compounds of the same class but different type differ by two hydrogens and will fall on horizontal lines separated by the mass defect of H 2. Compounds belonging to different classes are now readily identified because their chemical mass defect will be displaced vertically from each other. Visualization of a complex mass spectrum is simplified by using a simple two-dimensional graphical display of the data based on chemical mass defect.

Patterns are recognizable on the plot, and the outlier data are easily identified. Identification of a few compounds on the plot, at least one from each class, is the key to identifying the majority of the compounds.

Such a plot has been used for analyzing data from a single high resolution mass spectrum of a crude oil sample containing several thousand ion peaks [ 24 ]. Class and type assignment for so many compounds in the sample was accomplished by taking advantage of their chemical mass defect, a task that would be difficult to achieve in the absence of such a powerful data interpretation strategy.

A CH 2 based mass scale is historically the first one used for the analysis of crude oil samples by mass defect. Other mass scales 16 O- and H 2 -based for example have also been used to plot data on two-dimensional plots similar to the one shown in Figure 3 , and are useful for environmental samples [ 25 ].

The use of more than two mass scales for graphical visualization of data on higher-order plots makes data interpretation easier and increases the number of assigned chemical formulas [ 26 ]. Six different classes of compounds are identified. Different compound types are listed under each class.

Points on horizontal lines differ in the number of CH 2 groups and belong to homologous series. Reprinted from reference Study Materials.

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