Problem 82 - 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 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 \({ }^{3} \mathrm{H}^{+} ?\)

Step-by-Step Solution

Answer

(a) In a \({ }^{3}\mathrm{He}\) atom, there are 2 protons, 1 neutron, and 2 electrons. (b) \({ }^{3}\mathrm{H}\) (tritium) is expected to be more massive than \({ }^{3}\mathrm{He}\) due to the extra neutron. (c) The mass spectrometer should have a precision of at least 0.001388448188 amu to distinguish between \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\).

Step 1: Determine the number of protons, neutrons, and electrons in \({ }^{3}\mathrm{He}\) atom

In a \({ }^{3}\mathrm{He}\) atom, the atomic number (Z) is 2 (for helium) and the mass number (A) is 3. The atomic number represents the number of protons in the nucleus, which is equal to the number of electrons in a neutral atom. The mass number (A) is the sum of protons and neutrons. So, using this information, we can find the number of neutrons by subtracting the atomic number from the mass number. Number of protons = Z = 2 Number of electrons = 2 (since the atom is neutral) Number of neutrons = A - Z = 3 - 2 = 1

Step 2: Compare the mass of \({ }^{3}\mathrm{He}\) and \({ }^{3}\mathrm{H}\) atoms

First, we need to find the mass of each atom based on the sum of their subatomic particle masses. Mass of \({ }^{3}\mathrm{He}\) = 2(proton) + 1(neutron) + 2(electrons) Mass of \({ }^{3}\mathrm{H}\) = 1(proton) + 2(neutrons) + 1(electron) Since both atoms have the same number of nucleons (A = 3), the mass difference between \({ }^{3}\mathrm{He}\) and \({ }^{3}\mathrm{H}\) mainly comes from the extra neutron in \({ }^{3}\mathrm{H}\) and the extra proton-electron pair in \({ }^{3}\mathrm{He}\). Neutrons are slightly more massive than protons; hence, \({ }^{3}\mathrm{H}\) (tritium) is expected to be more massive.

Step 3: Determine the required precision of a mass spectrometer

To distinguish between \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\), the mass spectrometer should be able to resolve the peaks corresponding to the mass difference between them, which is mainly due to the difference in neutron and proton masses. Mass of Neutron = 1.008664915 amu Mass of Proton = 1.007276466812 amu Mass Difference = Mass of Neutron - Mass of Proton = 1.008664915 - 1.007276466812 = 0.001388448188 amu The mass spectrometer should have a precision of at least 0.001388448188 amu to differentiate between \({ }^{3}\mathrm{He}^{+}\) and \({ }^{3}\mathrm{H}^{+}\).

Key Concepts

Atomic Number

Every element on the periodic table is defined by its atomic number (Z), which indicates the number of protons found in the nucleus of an atom. Protons carry a positive charge, and since atoms are electrically neutral, the atomic number also tells us the number of electrons in a neutral atom. Understanding the atomic number is crucial in identifying the element itself because each element has a unique number of protons. For helium, which is represented by He in chemical notation, the atomic number is 2, which means it contains two protons and two electrons in its neutral state.

The atomic number serves as a fundamental concept in chemistry for balancing chemical equations, understanding chemical bonding, and analyzing isotopic composition. By knowing the atomic number, one can infer properties of an element, such as its position on the periodic table, and predict how it will react with other elements.

Mass Number

The mass number (A) of an atom reveals the total number of protons and neutrons in the atom's nucleus. While the atomic number allows us to identify the element, the mass number helps us differentiate between the isotopes of that element. For instance, helium-3 (³He) has a mass number of 3, indicating a sum of 3 nucleons (subatomic particles in the atom's nucleus) - 2 protons and 1 neutron.

The mass number is significant because isotopes have the same atomic number (same number of protons) but different mass numbers (different number of neutrons), which leads to unique physical and chemical properties. An atom's weight is primarily determined by its mass number, as protons and neutrons have roughly the same mass and contribute most to the overall mass of the atom, while electrons contribute negligibly.

Subatomic Particles

Atoms are composed of three main subatomic particles: protons, neutrons, and electrons. Protons, located in the nucleus, carry a positive charge and largely determine the atom's identity and properties. Neutrons, also in the nucleus, have no electrical charge but add mass to the atom and can impact its stability. Electrons orbit the nucleus in a 'cloud' and, although they have much less mass than protons or neutrons, they are responsible for chemical bonding and electricity because they carry a negative charge.

The subatomic particles' arrangement distinguishes between different elements and isotopes. In the given example of helium-3, we see the atom consists of 2 protons, 1 neutron, and 2 electrons in a neutral state. Helium-3 is an isotope because it varies in neutron count from the most common helium isotope, helium-4, which has 2 neutrons. This difference in subatomic particle count within isotopes leads to diverse physical characteristics and varying behaviors in chemical reactions.

Mass Spectrometry

Mass spectrometry is an analytical technique that measures the mass-to-charge ratio of ions to identify and quantify molecules in a sample. By converting particles to ions, mass spectrometry can separate them based on their mass-to-charge ratio using magnetic and electric fields. The precision of a mass spectrometer is paramount, as it determines the machine's ability to distinguish between ions with similar mass-to-charge ratios.

In context with isotopes like ³He and ³H, a mass spectrometer must be able to resolve the tiny mass differences between their ions (like ³He⁺ and ³H⁺), which are due to the differences in the mass of neutrons and protons. The precision required to differentiate between these ions is on the order of 0.001388448188 atomic mass units (amu), a testament to the instrument's sensitivity. Such precision allows scientists to study subtle nuances of chemical compounds, biological molecules, and materials, making mass spectrometry a powerful tool in research and industry.