Question

The natural abundance of \(^{3} \mathrm{He}\) is 0.000137\(\%\) . (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 \(^{3} \mathrm{He}\) 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 \(^{3} \mathrm{H}^{+}\) ?

Short answer

In an atom of \(^{3}\mathrm{He}\), there are 2 protons, 1 neutron, and 2 electrons. Based on the sum of their subatomic particles' masses, an atom of \(^{3}\mathrm{H}\) (tritium) is expected to be more massive than an atom of \(^{3}\mathrm{He}\). To differentiate between peaks due to \(^{3}\mathrm{He}^{+}\) and \(^{3}\mathrm{H}^{+}\), a mass spectrometer should have a precision of approximately 0.000433 or better.

Step-by-step solution

Finding the number of protons, neutrons, and electrons in \(^{3}\mathrm{He}\)

We can use the isotopic notation to determine the number of protons, neutrons, and electrons in an atom of \(^{3}\mathrm{He}\). The isotope notation for He-3 is written as \(_{2}^{3}\mathrm{He}\), where 2 is the atomic number (Z) representing the number of protons, and 3 is the mass number (A) representing the total number of protons and neutrons. From the notation: Number of protons (Z) = 2 Number of neutrons (N) = A - Z = 3 - 2 = 1 Since the atom is neutral, the number of electrons equals the number of protons: Number of electrons = 2 Thus, there are 2 protons, 1 neutron, and 2 electrons in an atom of \(^{3}\mathrm{He}\).

Comparing masses of \(^{3}\mathrm{He}\) and \(^{3}\mathrm{H}\)

Now, let's compare the masses of \(^{3}\mathrm{He}\) and \(^{3}\mathrm{H}\) based on the sum of their subatomic particles. Given the masses of proton, neutron, and electron as: Mass of a proton (m_p) = \(1.0073 \ \mathrm{amu}\), Mass of a neutron (m_n) = \(1.0087 \ \mathrm{amu}\), and Mass of an electron (m_e) = \(5.486 \times 10^{-4} \ \mathrm{amu}\). We can then calculate the approximate masses for \(^{3}\mathrm{He}\) and \(^{3}\mathrm{H}\). For \(^{3}\mathrm{He}\): mass = (number of protons)×(mass of proton) + (number of neutrons)×(mass of neutron) + (number of electrons)×(mass of electron) mass = (2×1.0073 + 1×1.0087 + 2×5.486 \times 10^{-4})\ \mathrm{amu} mass ≈ 3.0157\ \mathrm{amu} For \(^{3}\mathrm{H}\), let's denote the number of neutrons by N': Number of neutrons (N') = mass number of \(^{3}\mathrm{H}\) - Z = 3 - 1 = 2 Mass of \(^{3}\mathrm{H}\): mass = (number of protons)×(mass of proton) + (number of neutrons)×(mass of neutron) + (number of electrons)×(mass of electron) mass = (1×1.0073 + 2×1.0087 + 1×5.486 \times 10^{-4})\ \mathrm{amu} mass ≈ 3.0160\ \mathrm{amu} Thus, the atom of \(^{3}\mathrm{H}\) is expected to be more massive than the atom of \(^{3}\mathrm{He}\).

Determining mass spectrometer precision

To differentiate between the peaks due to \(^{3}\mathrm{He}^{+}\) and \(^{3}\mathrm{H}^{+}\), the mass spectrometer should be able to resolve masses with a precision equal to or better than the mass difference between these ions. Considering that their difference comes from only one additional neutron in the case of \(^{3}\mathrm{H}^{+}\), we can calculate the required precision as Required precision = \(\frac{\text{mass difference}}{\text{average mass}}\) Since we are concerned only with the ions, we can ignore the mass of the electrons: mass difference = \(|\text{mass}(^{3}\mathrm{H}^{+}) - \text{mass}(^{3}\mathrm{He}^{+})|\) where mass(\(^{3}\mathrm{H}^{+}\)) includes the mass of 1 proton and 2 neutrons, and mass(\(^{3}\mathrm{He}^{+}\)) includes the mass of 2 protons and 1 neutron. mass difference = \(|(1×1.0073 + 2×1.0087) - (2×1.0073 + 1×1.0087)|\ \mathrm{amu}\) mass difference = \(0.0013\ \mathrm{amu}\) To find the average mass, we can average the mass of \(^{3}\mathrm{H}^{+}\) and \(^{3}\mathrm{He}^{+}\): average mass = \(\frac{\text{mass}(^{3}\mathrm{H}^{+}) + \text{mass}(^{3}\mathrm{He}^{+})}{2}\) average mass = \(\frac{(1×1.0073 + 2×1.0087) + (2×1.0073 + 1×1.0087)}{2}\ \mathrm{amu}\) average mass = 3.0035\ \mathrm{amu} Now, we can calculate the required precision: required precision = \(\frac{0.0013\ \mathrm{amu}}{3.0035\ \mathrm{amu}}\) required precision ≈ 0.000433 Hence, to differentiate between peaks due to \(^{3}\mathrm{He}^{+}\) and \(^{3}\mathrm{H}^{+}\), a mass spectrometer should have a precision of approximately 0.000433 or better.

Key concepts

Helium-3

Helium-3, denoted as \(^{3} \mathrm{He}\), is a light, stable isotope of helium with two protons and one neutron. This isotope is unique due to its low natural abundance, making up only about 0.000137% of naturally occurring helium.
Helium-3 is non-radioactive, which means it does not naturally decay into other elements or isotopes over time. This makes it particularly interesting for scientific applications.
In Helium-3, the atomic number (Z) is 2, indicating that there are two protons in its nucleus, balancing out the two electrons and making it electrically neutral.
The mass number (A) of 3 arises from the total number of protons and neutrons combined, meaning \(^{3} \mathrm{He}\) has one neutron. The precise mass of \(^{3} \mathrm{He}\) atoms can be explored using its subatomic components, accounting for its defined behavior and interactions in a laboratory setting.

Tritium

Tritium, or \(^{3}\mathrm{H}\), is an isotope of hydrogen that consists of one proton and two neutrons. The presence of two neutrons makes tritium radioactive, with a half-life of about 12.3 years.
Tritium decays by beta emission into helium-3, which is a stable form of helium. This characteristic makes it significant in nuclear fusion and other scientific applications.
The atomic number for tritium is 1, representing its single proton and distinguishing it from other isotopes of hydrogen, such as protium and deuterium.
The additional neutron in tritium gives it an increased mass compared to helium-3, contributing to its higher energy and reactive nature. The notable difference in mass between \(^{3} \mathrm{H}\) and \(^{3} \mathrm{He}\) allows scientists to differentiate them using precise measurement techniques such as mass spectrometry.

Subatomic Particles

Subatomic particles form the fundamental building blocks of an atom, principally including protons, neutrons, and electrons. Each particle carries distinct properties and charges:

• **Protons**: They possess a positive charge and reside in the nucleus, defining the atomic number of an element.


• **Neutrons**: Neutrally charged, neutrons also reside in the nucleus and contribute to the atomic mass.


• **Electrons**: These negatively charged particles orbit the nucleus and are crucial for atomic bonding and reactions.

In isotopes like helium-3 and tritium, the combination and count of these subatomic particles define the characteristics and behavior of the atom.
The specific number of protons determines the identity of the element, whereas the number of neutrons can vary, leading to different isotopes. The electrons, matching the number of protons in a neutral atom, dictate the element's reactivity and bonding potential.

Mass Spectrometry Precision

Mass spectrometry is utilized to measure the mass-to-charge ratio of ions, providing insights into an atom's composition. This technology is pivotal in distinguishing isotopes like helium-3 and tritium due to their slight but crucial mass differences.
The precision of a mass spectrometer refers to its ability to resolve two different mass peaks closely situated. For a mass spectrometer to effectively differentiate between \(^{3}\mathrm{He}^{+}\) and \(^{3}\mathrm{H}^{+}\), it must possess a precision of approximately 0.000433.
Such meticulous precision ensures reliable identification and comparison of these isotopes and plays a vital role in fields like nuclear physics and chemistry, where understanding atomic differences is essential.
Advances in mass spectrometry over time have enhanced the capability to analyze elements with unprecedented accuracy, thereby refining our understanding of atomic and molecular compositions.