Video: Finding the Rest Masses of the Proton and the Neutron from Their Rest-Mass Energy

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.

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Video Transcript

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 π‘š sub zero 𝑝 and the rest mass of the neutron π‘š sub zero 𝑛, then the difference between these rest masses, which we can call Ξ”π‘š is equal to π‘š sub zero 𝑛 minus π‘š sub zero 𝑝.

To solve for Ξ”π‘š, 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 π‘š 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 π‘š sub zero 𝑛 minus π‘š 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 Ξ”π‘š, the quantity we’re solving for. When we rearrange this equation to solve for Ξ”π‘š, 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 Ξ”π‘š 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.

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