The natural abundance of \({ }^{3} \mathrm{He}\) is \(0.000137 \% .\) (a) How many protons, neutrons, and electrons are in an atom of \({ }^{3}\) He? (b) Based on the sum of the masses of their subatomic particles, which is expected to be more massive, an atom of \({ }^{3}\) He or an atom of \({ }^{3} \mathrm{H}\) (which is also called tritium)? (c) Based on your answer to part (b), what would need to be the precision of a mass spectrometer that is able to differentiate between peaks that are due to \({ }^{3} \mathrm{He}^{+}\) and \({ }^{3} \mathrm{H}^{+}\) ?

To determine the number of protons, neutrons, and electrons in a helium-3 atom, we need to analyze its atomic notation: \({ }^{3}\mathrm{He}\). The superscript 3 indicates the atomic mass number (A) which is the sum of protons and neutrons in the nucleus. The element symbol (He) represents helium, which has an atomic number (Z) of 2, indicating the number of protons in its nucleus. Number of protons (Z) = 2 Since the atom is electrically neutral, the number of electrons should also be equal to the number of protons. Number of electrons = 2 To find the number of neutrons, we need to subtract the number of protons from the atomic mass number (A). Number of neutrons (N) = A - Z = 3 - 2 = 1 So, a helium-3 atom has 2 protons, 1 neutron, and 2 electrons.

To compare the sum of the masses of subatomic particles in helium-3 and tritium atoms, we need to consider the masses of protons, neutrons, and electrons in each atom. For helium-3: Number of protons = 2, mass of 1 proton ≈ \(1.007\, u\) Number of neutrons = 1, mass of 1 neutron ≈ \(1.009\, u\) Number of electrons = 2, mass of 1 electron ≈ \(0.0005\, u\) Total mass of helium-3 atom ≈ 2\((1.007\, u)\) + \(1.009\, u\) + 2\((0.0005\, u)\) ≈ \(2.015\, u\) For tritium: Number of protons = 1, mass of 1 proton ≈ \(1.007\, u\) Number of neutrons = 2, mass of 1 neutron ≈ \(1.009\, u\) Number of electrons = 1, mass of 1 electron ≈ \(0.0005\, u\) Total mass of tritium atom ≈ \(1.007\, u\) + 2\((1.009\, u)\) + \(0.0005\, u\) ≈ \(3.025\, u\) Comparing the masses of helium-3 and tritium, we can conclude that tritium atom is expected to be more massive.

Based on the answer to part (b), we need to determine the precision of a mass spectrometer that can differentiate between helium-3 and tritium. To do this, we need to find the mass difference between the two ions of interest: Mass difference = Mass of \({ }^{3}\mathrm{T}^{+}\) - Mass of \({ }^{3}\mathrm{He}^{+}\) ≈ \(3.025\, u\) - \(2.015\, u\) ≈ \(1.010\, u\) The precision of the mass spectrometer needs to be smaller than the mass difference to differentiate between the two peaks: Precision of mass spectrometer < \(1.010\, u\) So, the mass spectrometer should have a precision better than \(1.010\, u\) to differentiate between peaks due to helium-3 and tritium ions.