Mass Number for Isotopes
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!
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.
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
Each atom in the sample has a "particular" atomic
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!
(Dr. Mabel Rodrigues)
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
-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.
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
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
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
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Update: June 2012