Problem 88 - Chemistry: The Central Science in SI Units, Global Edition

Question

The natural abundance of \({ }^{3}\) 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 \({ }^{3}\) He or an atom of \({ }^{3} \mathrm{H}\) (which is also called tritium) \(?(\mathbf{c})\) 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) \({ }^{3}\mathrm{He}\) is expected to be slightly less massive than \({ }^{3}\mathrm{H}\) (tritium) due to having one more proton and one less neutron. (c) The required precision of a mass spectrometer that is able to differentiate between \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\) is approximately 0.000467 or 0.0467%.

Step 1: Part (a): Number of protons, neutrons, and electrons in a \({ }^{3}He\) atom

An atom of \({ }^{3}\mathrm{He}\) (helium-3) is represented by the notation \({ }^{3}\mathrm{He}\). The number 3 (the superscript) is the mass number, which represents the sum of the protons and neutrons in the atom's nucleus. The element symbol, He, tells us it's a helium atom, and we know helium has an atomic number (number of protons) of 2. To find the number of protons, neutrons, and electrons in the \({ }^{3}\mathrm{He}\) atom, we can use the mass number and atomic number: - Protons: The atomic number of helium is 2, meaning there are 2 protons. - Neutrons: The mass number is 3 (protons plus neutrons). Since there are 2 protons, then there must be 1 neutron to make the mass number 3. - Electrons: In a neutral atom, the number of electrons equals the number of protons. So, there are 2 electrons in a \({ }^{3}\mathrm{He}\) atom.

Step 2: Part (b): Comparison of \({ }^{3}He\) and \({ }^{3}H\) atomic masses

To compare the atomic masses of \({ }^{3}\mathrm{He}\) and \({ }^{3}\mathrm{H}\) (also known as tritium), we can look at the composition of their nuclei (protons and neutrons). \({ }^{3}\mathrm{He}\) has 2 protons and 1 neutron, while \({ }^{3}\mathrm{H}\) (tritium) has 1 proton and 2 neutrons. The mass of a proton is approximately 1 atomic mass unit (amu), and the mass of a neutron is also approximately 1 amu. Therefore, the sum of the masses of the subatomic particles in \({ }^{3}\mathrm{He}\) and \({ }^{3}\mathrm{H}\) will both be approximately 3 amu. However, the mass of a proton is slightly less than the mass of a neutron. Since \({ }^{3}\mathrm{He}\) has 1 more proton and 1 less neutron than \({ }^{3}\mathrm{H}\), we can expect \({ }^{3}\mathrm{He}\) to be slightly less massive than \({ }^{3}\mathrm{H}\).

Step 3: Part (c): Precision of a mass spectrometer

In order to differentiate between \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\) in a mass spectrometer, the instrument must be precise enough to distinguish the slight difference in mass between these two ions. Since we found out that \({ }^{3}\mathrm{He}\) is slightly less massive than \({ }^{3}\mathrm{H}\) due to having one more proton, the required precision can be calculated as: Precision = \(\frac{Mass\ difference\ between\ }^{3}\mathrm{He}^{+}\mathrm{\ and\ }^{3}\mathrm{H}^{+}}{Average\ mass\ of\ }^{3}\mathrm{He}^{+}\mathrm{\ and\ }^{3}\mathrm{H}^{+}\) The mass difference between a proton and a neutron is approximately 0.0014 amu. So, the mass difference between \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\) is approximately 0.0014 amu. The average mass of \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\) is approximately 3 amu. Precision = \(\frac{0.0014\ amu}{3\ amu} \approx 0.000467\) Hence, the required precision of a mass spectrometer that is able to differentiate between peaks that are due to \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\) is approximately 0.000467 or 0.0467%.

Key Concepts

Atomic Structure

Understanding atomic structure is fundamental in chemistry. Atoms are the basic units of matter and consist of three main subatomic particles:

• Protons: positively charged particles found in the nucleus.
• Neutrons: neutral particles, also located in the nucleus.
• Electrons: negatively charged particles orbiting the nucleus.

The atomic number of an element is defined by the number of protons present in the nucleus. For helium, this number is 2. Helium-3 ( 3 He) is an isotope characterized by having a mass number of 3. This mass number indicates the combined total of protons and neutrons. In the case of helium-3, with 2 protons, it has 1 neutron. Electrons, in a neutral atom, will match the number of protons, so helium-3 also has 2 electrons. Knowing the structure helps predict chemical behavior and interactions.

Mass Spectrometry

Mass spectrometry is a powerful analytical technique used to measure the mass-to-charge ratio of ions. It involves ionizing chemical compounds to generate charged molecules and measuring their characteristics:

• The sample is vaporized, and its components are ionized.
• The ions are accelerated in an electric field to different velocities based on their mass-to-charge ratios.
• A detector records the ions and their relative abundance.

This technique can differentiate isotopes such as helium-3 and tritium ( 3 H) by their slight mass differences. While their mass numbers are similar, accurate instruments can detect small discrepancies due to different proton and neutron numbers. This requires highly sensitive mass spectrometers that can distinguish tiny variations, enhancing our understanding of isotopic compositions.

Helium-3

Helium-3 is a fascinating isotope of helium, different from its more common counterpart helium-4. It's scientifically significant for several reasons:

• Low natural abundance, less than 0.0002%, makes it rare and valuable.
• Its unique nuclear properties are useful in research, cryogenics, and nuclear fusion experiments.
• Unequal distribution of protons and neutrons, having an extra proton compared to tritium, gives helium-3 distinct characteristics.

Due to its rarity and unique properties, helium-3 is a subject of interest in fields such as nuclear physics and energy. Its applications extend beyond basic science to potential future technologies, making it an essential element to study.