Problem 88 - Chemistry: The Central Science

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 \({ }^{3} \mathrm{H}^{+}\)?

Step-by-Step Solution

Answer

(a) A \({ }^{3} \mathrm{He}\) atom has 2 protons, 1 neutron, and 2 electrons. (b) The mass of \({ }^{3} \mathrm{He}\) is slightly less than the mass of \({ }^{3} \mathrm{H}\) (tritium). (c) To differentiate between \({ }^{3} \mathrm{He}^{+}\) and \({ }^{3} \mathrm{H}^{+}\), a mass spectrometer would need a precision better than \(5.45 \times 10^{-4}\) atomic mass units.

Step 1: (a) Protons, neutrons, and electrons in \({ }^{3} \mathrm{He}\)

We can determine the number of protons, neutrons, and electrons in an atom of \({ }^{3} \mathrm{He}\). \({ }^{3} \mathrm{He}\) represents helium-3, which has an atomic mass of 3. The atomic number of helium is 2, which means there are 2 protons and 2 electrons in a helium atom. The number of neutrons can be determined by subtracting the atomic number from the atomic mass. Number of neutrons = Atomic mass - Atomic number = \(3 - 2 = 1\) So, \({ }^{3} \mathrm{He}\) has 2 protons, 1 neutron, and 2 electrons.

Step 2: (b) Comparing \({ }^{3} \mathrm{He}\) and \({ }^{3} \mathrm{H}\) masses

Now we'll compare the mass of an atom of \({ }^{3} \mathrm{He}\) with an atom of tritium (\({ }^{3} \mathrm{H}\)) based on the sum of the masses of their subatomic particles. Protons and neutrons have nearly equal masses (approximately 1 atomic mass unit each), while electrons have a much smaller mass (approximately 1/1836 atomic mass units). The mass of \({ }^{3} \mathrm{He}\) can be calculated as follows: Mass of \({ }^{3} \mathrm{He}\) = 2 protons + 1 neutron + 2 electrons = \(2 + 1 + 2 \times (1/1836) = 3 + 2/1836\) The mass of \({ }^{3} \mathrm{H}\) can be calculated as follows: Mass of \({ }^{3} \mathrm{H}\) = 1 proton + 2 neutrons + 1 electron = \(1 + 2 + 1 \times (1/1836) = 3 + 1/1836\) Comparing both masses, we can see that the mass of \({ }^{3} \mathrm{He}\) is slightly less than the mass of \({ }^{3} \mathrm{H}\).

Step 3: (c) Precision of mass spectrometer

We now need to find the precision of a mass spectrometer that can differentiate between \({ }^{3} \mathrm{He}^{+}\) and \( { }^{3} \mathrm{H}^{+}\) ions. The small mass difference between \({ }^{3} \mathrm{He}^{+}\) and \({ }^{3} \mathrm{H}^{+}\) can be calculated as: Mass difference = \(3 + 2/1836 - (3 + 1/1836) = 1/1836\) To differentiate between the peaks that are due to \({ }^{3} \mathrm{He}^{+}\)and \({ }^{3} \mathrm{H}^{+}\), the mass spectrometer needs a precision better than 1/1836 atomic mass units, which is \(5.45 \times 10^{-4}\) units.

Key Concepts

Helium-3

Helium-3, often denoted as extsuperscript{3}He, is a light, stable isotope of helium with an atomic mass of 3. Unlike its more common form, helium-4, which contains two neutrons, helium-3 contains only one neutron:

• It has 2 protons, as indicated by the atomic number of helium.
• It also has 2 electrons, corresponding to the number of protons, to ensure the atom is neutral.
• With only 1 neutron, it makes helium-3 unique and less abundant compared to helium-4.

Helium-3 is noteworthy due to its rarity and unique nuclear properties. This makes it useful in certain nuclear research and applications. Its relative scarcity occurs because it is less commonly formed in stellar processes or found naturally on Earth. Understanding extsuperscript{3}He is essential in isotope analysis for both academic research and practical applications.

Mass Spectrometer Precision

Mass spectrometers are instruments used to measure the mass-to-charge ratio of ions. When it comes to isotope analysis, such as differentiating between helium-3 and tritium, precision is key.

• High precision means the instrument can distinguish between very similar mass values, as found with extsuperscript{3}He and tritium extsuperscript{3}H.
• The necessary precision to separate these isotopes was calculated to be better than rac{1}{1836} atomic mass units.
• This translates to detecting mass differences as small as 0.000545 atomic mass units.

Precision in mass spectrometry is essential for accurate isotope distinction. This can have significant implications in scientific research and industries where isotopic composition needs to be measured precisely. Therefore, enhancements in mass spectrometer technology continually aim to increase this precision, supporting a wide range of analytical applications.

Subatomic Particles

Atoms are composed of subatomic particles: protons, neutrons, and electrons. Understanding the behavior and properties of these particles is crucial when analyzing isotopes.

• Protons are positively charged and contribute to the atomic number of an element.
• Neutrons have no charge and, along with protons, make up the atomic mass.
• Electrons are negatively charged and determine the chemical behavior of the atom, but they contribute minimally to its mass.

In the context of isotopes, changing the number of neutrons leads to different isotope forms of the same element. Subatomic particle composition affects both mass and stability of atomic nuclei. Therefore, analyzing the subatomic structure of isotopes helps in applications like nuclear physics, chemistry, and environmental science.

Protons and Neutrons

Protons and neutrons, collectively known as nucleons, reside in the nucleus of an atom and play key roles in determining the identity and mass of the element.

• Protons define the element itself; for example, helium always has two protons.
• Neutrons contribute to the isotope variation of an element without changing its chemical identity.

For example, helium-3 has 2 protons and 1 neutron, while tritium ( extsuperscript{3}H) has 1 proton and 2 neutrons. The difference in neutron counts leads to the slight mass difference between isotopes, which can be detected using precise instrumentation like a mass spectrometer. Understanding the distribution of protons and neutrons is essential in fields like nuclear energy and astrophysics, where altering the nuclear configuration can have significant implications.