Question

The natural abundance of ${}^3\mathrm{He}$ is $0.000137\mathrm{\%}$. (a) How many protons, neutrons, and electrons are in an atom of ${}^3\mathrm{He}$ ? (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 for 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 ${}^3{\mathrm{H}}^{+}?$

Short answer

(a) A helium-3 atom has 2 protons, 1 neutron, and 2 electrons. (b) Both helium-3 and tritium have a mass of about 3 amu based on the sum of their subatomic particles. (c) The required precision of a mass spectrometer to differentiate between helium-3 and tritium ions is approximately $0.0014$ amu.

Step-by-step solution

Determine the number of protons, neutrons, and electrons in helium-3

To find the number of protons, neutrons, and electrons in helium-3, we look at the given symbol ${}^3\mathrm{He}$. The superscript 3 indicates that the mass number of helium (total number of protons and neutrons) is 3. Since helium has an atomic number of 2, it means that it has 2 protons and 1 neutron (3 - 2 = 1). An electrically neutral atom has equal numbers of protons and electrons, so helium-3 also has 2 electrons. The composition of helium-3 will be 2 protons, 1 neutron, and 2 electrons.

Compare the masses of helium-3 and tritium atoms

To determine which atom is more massive between ${}^3\mathrm{He}$ and ${}^3\mathrm{H}$, we need to compare the masses of their subatomic particles. We already know the composition of helium-3 to be 2 protons, 1 neutron, and 2 electrons. Tritium has the symbol ${}^3\mathrm{H}$, indicating that it has a mass number of 3. Since hydrogen has an atomic number of 1, it means tritium has 1 proton, 2 neutrons, and 1 electron. The masses of each subatomic particle are approximately: - Proton: 1 atomic mass unit (amu) - Neutron: 1 amu - Electron: negligible compared to protons and neutrons The mass of a helium-3 atom is the sum of the masses of 2 protons, 1 neutron, and 2 (negligible) electrons, which is approximately 3 amu. Similarly, the mass of a tritium atom is the sum of the masses of 1 proton, 2 neutrons, and 1 (negligible) electron, which is also approximately 3 amu. Based on the sum of the masses of their subatomic particles, both helium-3 and tritium have a mass of about 3 amu.

Calculate the precision required for a mass spectrometer to differentiate between helium-3 and tritium ions

In this step, we will calculate the precision required for a mass spectrometer to distinguish between helium-3 and tritium ions. When both these atoms are ionized, they lose one electron each, thus forming ${}^3{\mathrm{He}}^{+}$ and ${}^3{\mathrm{H}}^{+}$. The mass difference between the two ions can be calculated by considering that a neutron has slightly more mass than a proton. Neutron mass: $1.008664915$ amu Proton mass: $1.007276466812$ amu Mass difference: $1.008664915-1.007276466812=0.001388448188$ amu The mass difference is very small, and to differentiate between the peaks of ${}^3{\mathrm{He}}^{+}$ and ${}^3{\mathrm{H}}^{+}$ in a mass spectrometer, the instrument must be capable of distinguishing masses with a precision greater than this difference. Therefore, the required precision of a mass spectrometer to differentiate between these ions is approximately $0.0014$ amu.

Key concepts

He-3 Composition

Helium-3, denoted as ${}^3\text{He}$, is a unique isotope of helium that is less commonly found in nature. The isotopic symbol tells us much about its composition. The superscript "3" indicates the mass number, which is the sum of protons and neutrons in the nucleus. ${}^3\text{He}$ has a mass number of 3 and an atomic number of 2, signifying that it contains 2 protons.
For ${}^3\text{He}$, to find the neutrons, subtract the atomic number from the mass number: $$3-2=1$$ neutron. Electrically neutral, this atom also contains 2 electrons, matching the number of protons. Thus, helium-3 comprises:

• 2 protons
• 1 neutron
• 2 electrons

The rarity of ${}^3\text{He}$ is highlighted by its low natural abundance, approximately 0.000137%. This makes it a significant isotope with specialized applications, such as in cryogenics and nuclear studies.

Subatomic Particles

Subatomic particles are the fundamental components of all atoms, essential in determining atomic structure. There are three primary subatomic particles:

• Protons: Positively charged particles located in the nucleus. They have a relative mass of approximately 1 atomic mass unit (amu).
• Neutrons: Neutral particles also found in the nucleus. Similar to protons, they have a mass near 1 amu, slightly heavier than a proton.
• Electrons: Negatively charged particles orbiting the nucleus. Their mass is negligible when compared to protons and neutrons, often ignored in mass calculations.

When comparing ${}^3\text{He}$ and tritium (${}^3\text{H}$), we note that both isotopes have a mass number of 3. However, their compositions differ in the number of protons and neutrons:

• ${}^3\text{He}$: 2 protons, 1 neutron
• ${}^3\text{H}$: 1 proton, 2 neutrons

Despite their differences, both isotopes have an approximate mass of 3 amu, emphasizing the need for precise measurement tools to distinguish between them.

Mass Spectrometry

Mass spectrometry is a powerful analytical technique used to measure the masses and relative concentrations of atoms and molecules. It works by ionizing chemical compounds to generate charged particles and measuring their mass-to-charge ratios.
Distinguishing ${}^3{\text{He}}^{+}$ ions from ${}^3{\text{H}}^{+}$ in mass spectrometry requires high precision, as both ions have almost identical masses. The crucial difference stems from the slightly greater mass of a neutron compared to a proton:

• Neutron mass: $1.008664915$ amu
• Proton mass: $1.007276466812$ amu
• Mass difference: $0.001388448188$ amu

For a mass spectrometer to differentiate these isotopes, it must boast enough precision to resolve this minute mass difference, roughly $0.0014$ amu. Advances in mass spectrometry have allowed scientists to make such fine distinctions, playing a critical role in research and industrial applications.