Determine the difference in the rest masses of a proton and a neutron from their rest energies. Use a value of 938.3 MeV for proton rest energy and 939.6 MeV for the neutron rest energy.
We can record the proton rest energy as 𝑒 sub zero 𝑝 and the neutron rest energy as 𝑒 sub zero 𝑛. If we call the rest mass of the proton 𝑚 sub zero 𝑝 and the rest mass of the neutron 𝑚 sub zero 𝑛, then the difference between these rest masses, which we can call Δ𝑚 is equal to 𝑚 sub zero 𝑛 minus 𝑚 sub zero 𝑝.
To solve for Δ𝑚, we can recall Einstein’s relationship between energy and mass, which says that an object’s rest energy 𝑒 sub zero equals its rest mass 𝑚 sub zero times the speed of light squared. If we apply this relationship to solve for the differences in the rest masses of a proton and a neutron, then we can write that 𝑒 sub zero 𝑛 minus 𝑒 sub zero 𝑝 equals 𝑚 sub zero 𝑛 minus 𝑚 sub zero 𝑝 times 𝑐 sqtuared, where we treat the speed of light 𝑐 as exactly 3.00 zero times 10 to the eighth metres per second.
Looking at our equation, we recognize that this difference in masses can be written Δ𝑚, the quantity we’re solving for. When we rearrange this equation to solve for Δ𝑚, we see it’s equal to the difference in rest energies of the proton and neutron over 𝑐 squared. When we plug in for these three values, before calculating our solution, we multiply our numerator by a conversion factor between joules and electron volts, which will allow us to give our final answer in terms of units of kilograms.
When we calculate Δ𝑚 including our conversion factor, we find that to two significant figures it equals 2.3 times 10 to the negative 30th kilograms. That’s the difference in rest masses between a neutron and a proton.