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Name: Hope
Status: educator
Age: 9-12
Location: CA
Country: N/A
Date: 12/29/2004


Question:
I teach high school chemistry. I am having problems explaining to a student (and myself!) why the mass number for a particular isotope is not a whole number. I understand that the atomic mass is a weighted average of isotopes and therefore should be a decimal but why aren't isotopes whole number values for mass if we are using atomic mass units with protons-1 and neutrons-1? Does it have to do with nuclear binding forces? I am pretty darn confused about this. I have another question but I will put it in another box. Thanks for any/all your help and happy holidays!


Replies:
The "mass defect" -- the fact that the mass of an isotope is less that the sum of its nucleons is due to the fact that when nucleons combine (either by fusion, for elements with atomic number less than iron[A=26]; or fission for elements with atomic number greater than iron) there is a release of energy due to the nuclear binding force, which source of the "mass defect" you correctly identified. The energy released as a result of the binding of the nucleons is equivalent to a small amount of mass of the matter -- Einstein's famous equation: E=mc^2, so the energy release E < 0 causes the mass of the isotope to have a mass deficiency m < 0. The site below gives a nice slide presentation of the effect.

http://www.eas.asu.edu/~holbert/eee460/massdefect.html

Vince Calder


Hi Hope! I am sorry but you are making some confusion with atomic weight and mass number. You mentioned correctly that every element has these 3 characteristic "numbers": atomic number, mass number and atomic weight. The atomic number Z corresponds to its place at the Periodic Table and is its number of protons. The mass number A is the number of protons + the number of neutrons, so an integer. The atomic weight is related to the atomic mass (or mass number)by the following way: the atomic weight is the average(weighted, as you say) mass of the atoms in a representative sample of an element.

Each atom in the sample has a "particular" atomic mass. For example, about 3/4 of the element chlorine have an atomic mass of 35 amu. and about 1/4 have an atomic mass of 37 amu. So the average mass in the sample will be ( 5 x 35) + ( 3 x 37 ) divided by 8 = = 35.5 amu and that will be the atomic weight

Now we said above that the mass number is the number of neutrons added to the number of protons (so not a decimal number!) The mass number of the most common isotope (same place in the table) can be obtained from the periodic table. Many tables have the atomic weight and not the atomic mass.(so the confusion!) If you take the decimal from the number at the periodic table and round it to the nearest whole number, you have the mass number. For example the atomic weight of Iron (Fe) is 55.847. When rounded it gives a mass number of 56. The atomic number of Fe is 26 so most Fe atoms have 30 x (56-26) neutrons. In addition, all neutral Fe atoms have 26 protons and 26 electrons. Atoms of the same element with a different number of neutrons are called isotopes. The most common isotope of an element is the one represented at the periodic table. Another way to denote atomic mass is like that: the most common isotope of uranium is uranium-238 which has 92 protons, 92 electrons, and 146 neutrons. (92 + 146) = 238

Another isotope is uranium-235 with 92 protons, 92 electrons, and 143 neutrons. (92+ 143) = 235 So atomic weight is a decimal number because it corresponds to an average relation. Atomic mass is always an integer.

Thanks for asking NEWTON! And for the greetings!

Mabel
(Dr. Mabel Rodrigues)


Hi Hope,

An excellent question!

Each isotope does not weigh an integer number of atomic mass units. Protons and neutrons do not weigh exactly 1 u. That is an approximation. In addition, do not forget the mass of an electron, which is small but not zero. Finally, there is a certain amount of nuclear mass which is manifested in the binding energy of the nucleus (the energy which holds the nucleus together). This tends to make the mass of the atom smaller than the sum of the masses of isolated electrons, protons and neutrons.

facts:
-a proton weighs 1.007276 u
-a neutron weighs 1.008665 u
-an electron weighs 0.000549 u (Modern Physics, K. Krane, 1983)

If you added these together to estimate the weight of an atom of 3He, which has 2 protons, 1 neutron and 2 electrons, you'd get a "theoretical atomic mass" of 2*1.00728 + 1.00867 + 2*0.00055 = 3.02433 u .

The actual atomic mass of 3He is 3.01603 u. The difference in masses is attributed to the mass which is tied up in the energy needed to bind the two neutrons to the proton. I am, a little rusty on these calculations as I have not done them since the mid 1980's, but I will give it a shot using my dogeared copy of Krane's Modern Physics textbook.

We can obtain the nuclear binding energy of any isotope using the general formula

B = [N*m_(neutron) + Z*m_(H atom) - m_(isotope atomic mass)]*c^2

where N = number of neutrons and Z=number of protons. Converting from Joules to MeV gives us a conversion factor from mass to energy and lets' us drop the c^2. Since there are 931.5 MeV per u, for 3He we have

B(MeV) = (1*1.00867 + 2*1.007825 - 3.016029)*931.5 = 7.72 MeV, or 2.57 MeV per nucleon (2 protons + 1 neutron).

This turns out to be a much smaller energy per nucleon then 4He (about 7 MeV/nucleon), which is the most stable isotope of He.

In terms of mass, this is

B(u) = 1*1.00867 + 2*1.007825 - 3.016029 = 0.008291 u

If you add this to the actual atomic mass, you get 3.01603 + 0.00829 = 3.02432 (compare to the theoretical mass of 3.02433 above).

Hope this helps.

Dr. Topper


Does it have to do with nuclear binding forces?

Yes, exactly so!

That and "E=mc2" will directly give you the "mass-defects" that are bothering you.

A few things:

- 1 amu is similar in size to 1 GeV of mass/energy. About 7% difference. This really helps me intuitively. It is also 1.00 gram per mole, almost by definition.

- electrons weigh about 1/2000 or 0.0005 of a proton or neutron, 1 amu = mass[proton] + mass[electron] - mass[energy released when fusing free particles to make this nucleus] 1 amu depends on the isotope.

- the proton and neutron are different enough that they can never be assigned the same exact weight.

A neutron = [ proton + electron + neutrino + energy ( < 1MeV) ]. So a neutron must be noticeably heavier than a proton.

mass[proton + electron] =~ mass [neutron] =~ 1amu

- it might help thinking and discussion to define terms:

ami - "atomic mass index", the integer count of nucleons in the isotope alias- "nucleon count"?

amu - a unit of mass, some very small floating-point number of kilograms, chosen so the weight of each ami or nucleon, at some strategic point or average among all the isotopes, is 1.0000 amu.

- in which isotope of which element should we consider the proton or average nucleon to weigh exactly 1 amu?

For a proton (un-influenced by nuclear nest-mates) to be exactly 1, we should choose H to be 1.0005 gm/mol. Then every other isotope and element will have protons lighter than 1.0

To have the lightest nucleons be 1.0, so all others are heavier, we should choose the main isotope of Iron, Fe56, to be 56.000 or perhaps Cobalt (only one isotope, 59 ) to be 59.000

We may have muffed that one for the time being. We chose one in the middle, carbon-12. For carbon 12, 1 amu = 1.0000 grams per mole.

This has the advantages:
- the element we use most in chemistry (carbon) has a round number, and

- that changes in nucleon weight over the whole spectrum from H (heavier nucleons) to Fe (lightest nucleons) to U238 (heavier again) will extend both above and below 1.0 This way the maximum deviation from 1.0 may not seem as large, and use of rounded integers is hopefully more accurate on average. I do not think it really helps much. Fortunately, it is all a wash if you have good calculating ability. A computer spreadsheet program can make it all easy. "Basic"-language programming can do it too.

My start on a spreadsheet shows that (excepting H and He), an "ami" goes up to maybe 1.002 and down to 0.999 gm/mol:

                 amu       per ami               decay     decay
  Atom    ami   (gm/mol)   (gm/mol)    GeV        (MeV)     time
    e      e     0.0005   0.00055    0.00051        -
    p      1               1.00728   0.93828        -
    n      1     1.00867   1.00867   0.93957    0.7825     12 min
    H      1     1.00797   1.00797   0.93893        -
    D      2     2.0140    1.00700                  -
    T      3     3.01605   1.00535             0.01861     12.2yr
  He3      3     3.01603   1.00534                  -
  He4      4     4.00260   1.00065                  -
  He5      5     5.0123    1.00246
  ...
  Be9      9     9.01218   1.00135
  ...
  C12     12    12.00000   1.00000
  C13     13    13.00335   1.00026
  C14     14
  ...
   Sc     45    44.95592   0.99902
 Ti46     46    45.95263   0.99897
 Ti47     47    46.95180   0.99897
 Ti49     49    48.94787   0.99894
 Ti50     50     49.9448   0.99890
  V50     50     49.9472   0.99894
  V51     51     50.9440   0.99890
 Cr52     52     51.9405   0.99886
   Mn     55     54.9381   0.99887
 Fe56     56     55.9349   0.99884
 Fe57     57     56.9354   0.99887
 Fe58     58     57.9333   0.99885
   Co     59     58.9332   0.99887
 Ni58     58     57.9353   0.99888
 Ni60     60     59.9332   0.99889
 Cu63     63     62.9298   0.99889
 Cu65     65     64.9278   0.99889
  ...
 U233    233    233.0395   1.00017
 U235    235    235.0439   1.00019
 U238    238    238.0508   1.00021
  ...


Jim Swenson



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