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4.5.2: How to calculate the mass defect?

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    The atomic masses of all stable and long-lived elements have been determined with a very high precision, and they can be readily found on the Web -usually, given in a.m.u. (often shortened to single "u"), which means "atomic mass units". The most useful units for the discussion in this section are not a.m.u., but the e.m.u. energy equivalent expressed in MeV. A highly precise value of the a.m.u. \(\mathrm{MeV} / \mathrm{c}^{2}\) conversion factor can be found in the NIST Web page: 1 a.m.u. \(=9 \cdot 314940954 * 0^{5} \mathrm{MeV} / \mathrm{c}^{2}\).

    Let's, for example, consider the U-235. It's atomic mass (taken from the Web page of the Nuclear Data Center, Japan Atomic Energy Agency) is \(235.043931368\) a.m.u., and after performing the multiplication one gets the value of \(2.189420329 * 10^{5} \mathrm{MeV} / \mathrm{c}^{2}\). There is one "trick" that has to be done here: namely, the tables almost always list the values of atomic masses, whereas we need the mass of the nucleus - so we have to subtract the masses of all electrons in the atom. The energy equivalent of an electron is \(0.511\) \(\mathrm{MeV} / \mathrm{c}^{2}\) and there are 92 of then in an Uranium atom, so after subtracting the total mass of the electrons, we get the value of \(2.188950209^{*} 10^{5} \mathrm{MeV} / c^{2}\) for the mass of the nucleus.

    Now, we need the energy-equivalents for the masses of a single proton and a singleneutron - again, from the NIST Web page we find: \(m_{p}=938.2720813\) \(\mathrm{MeV} / \mathrm{c}^{2}\) and \(m_{n}=939.5654133 \mathrm{MeV} / c^{2}\). In the U-235 nucleus there are 92 protons and 143 neutrons, and the total mass of such combination is:

    \[ 92 \times m_{p}+143 \times m_{n}=2.206788856 \times 10^{5} \mathrm{MeV} / c^{2} \]

    Subtraction the \(m\) ass of the nucleus from the above result yields the mass defect value:

    \[ \Delta M=(2.206788856-2.188950209) \times 10^{5} \mathrm{MeV} / c^{2}=1783.864682 \mathrm{MeV} / c^{2} \]

    This method is straightforward, but tedious - but trustworthy values can be obtained more conveniently from the "Nuclear Binding Energy Finder" provided on the Web by Wolfram Alpha. For U-235 this service returns the value of \(7.590907 \mathrm{MeV} / c^{2}\) for a single nucleon. Multiplication by 235 yields \(1783.863145 \mathrm{MeV} / c^{2}\) for the entire nucleus, in excellent agreement with the result obtained by the more laborious method.

    A "technical" remark: In the above calculations, all decimal numbers of the values provided by the Web sources were taken - which was an unnecessary "overkill", the calculations in the next subsection demonstrate that taking just six decimals is a sufficient precision. But if the final result is to be a difference between two similar very large numbers, it never hurts to be cautious.

    4.5.2: How to calculate the mass defect? is shared under a CC BY 1.3 license and was authored, remixed, and/or curated by Tom Giebultowicz.

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