### Convert 1 amu to MeV

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**Solution:**

1) Determine mass of 1 amu (in kg):

By definition, the mass of 1 atom of C-12 is 12 amu.
Therefore, the mass of one mole of C-12 atoms is, by definition, 12 g

Avogadro's Number is 6.0221409 x 10^{23} mol¯^{1}

12 g/mol / 6.0221409 x 10^{23} mol¯^{1} = 1.9926468 x 10¯^{23} g (mass of one atom of C-12 in g)

(1.9926468 x 10¯^{23} g) (1 kg / 1000 g) = 1.9926468 x 10¯^{26} kg (mass of one atom of C-12 in kg)

1.9926468 x 10¯^{26} kg/atom / 12 amu/atom = 1.660539 x 10¯^{27} kg/amu (this is the mass of 1 amu)

2) Use Einstein's mass-energy equation to determine Joules in one amu:

E = mc^{2}
E = (1.660539 x 10¯^{27} kg) (2.99792458 x 10^{8} m/s)^{2}

E = 1.49242 x 10¯^{10} kg-m^{2}/s^{2} (this is Joules)

3) Convert J to eV, then MeV

1 eV = 1.60217733 x 10¯^{19} J
1.49242 x 10¯^{10} J / 1.60217733 x 10¯^{19} J/eV = 931494893 eV

(931494893 eV) (1 MeV / 10^{6} eV) = 931.494893 MeV

Often, 931.5 MeV is the value used in problem solving.

4) 931.5 MeV is the __energy__ in one amu. Is it possible to express the mass of one amu using MeV in some way? I'm glad you asked:

E = mc^{2}
931.5 MeV = mc^{2}

m = 931.5 MeV/c^{2}

The use of 931.5 MeV/c^{2} for the mass of 1 u is wide-spread. So much so that the c^{2} is often not written and you must infer its presence by context.

5) To sum up:

1 u = 931.5 MeV (expressed as energy)
1 u = 931.5 MeV/c^{2} (expressed as mass)

In solving problems, you may see this unit:

931.5 MeV/u-c^{2}
read it as 931.5 MeV/c^{2} per 1 u

By the way, if you look around the Internet, you will find values different than the 931.494893 value I calculated. This is because different authors will use values that they have rounded off differently than the ones I used.

For example, I used 1.660539 x 10¯^{27} kg for the mass of 1 amu. Often, you will see it as 1.66 x 10¯^{27} kg.

The cumulative effect of these rounding off decisions will be seen in slightly different answers from different authors.

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