Problem 20 - Chemistry The Central Science

Question

Each of the following nuclei undergoes either beta decay or positron emission. Predict the type of emission for each: \((\mathbf{a})\) tritium, \(_{1}^{3} \mathrm{H},(\mathbf{b})_{38}^{89} \mathrm{Sr},(\mathbf{c})\) iodine-120, \((\mathbf{d})\) (d) silver-102.

Step-by-Step Solution

Answer

All four nuclei will undergo beta decay: Tritium (_{1}^{3}H), Strontium (_{38}^{89}Sr), Iodine-120 (_{53}^{120}I), and Silver-102 (_{47}^{102}Ag). This is because their neutron-to-proton ratios indicate that they are neutron-rich.

Step 1: Tritium (_{1}^{3}H)

To find the neutron-to-proton ratio, we calculate: Neutrons = mass number - atomic number Neutrons = 3 - 1 = 2 Ratio = Neutrons/Protons Ratio = 2/1 = 2 Tritium (_{1}^{3}H) is neutron-rich, as the ratio is higher than 1, so it will undergo beta decay.

Step 2: Strontium (_{38}^{89}Sr)

Calculate the neutron-to-proton ratio: Neutrons = mass number - atomic number Neutrons = 89 - 38 = 51 Ratio = Neutrons/Protons Ratio = 51/38 ≈ 1.342 Strontium (_{38}^{89}Sr) is also neutron-rich, so it will undergo beta decay.

Step 3: Iodine-120 (_{53}^{120}I)

Calculate the neutron-to-proton ratio: Neutrons = mass number - atomic number Neutrons = 120 - 53 = 67 Ratio = Neutrons/Protons Ratio = 67/53 ≈ 1.264 Iodine-120 (_{53}^{120}I) is neutron-rich, so it will undergo beta decay.

Step 4: Silver-102 (_{47}^{102}Ag)

Calculate the neutron-to-proton ratio: Neutrons = mass number - atomic number Neutrons = 102 - 47 = 55 Ratio = Neutrons/Protons Ratio = 55/47 ≈ 1.170 Silver-102 (_{47}^{102}Ag) has a neutron-to-proton ratio close to the optimal ratio for stable nuclei (found mostly in lighter elements), but it is still considered neutron-rich. Thus, silver-102 will undergo beta decay. In conclusion, all four nuclei will undergo beta decay, as their neutron-to-proton ratios indicate that they are neutron-rich.

Key Concepts

Beta Decay

In nuclear chemistry, beta decay is a fundamental process where a neutron transforms into a proton, emitting an electron (beta particle) in the process. This happens in neutron-rich nuclei, which have an excess of neutrons compared to protons. During beta decay, the atomic number of the nucleus increases by one, resulting in the transformation into a different element. For example, when tritium undergoes beta decay, it transforms into helium-3. The mass number remains the same during this process. Beta decay helps nuclei achieve a more stable state by reducing the neutron-to-proton ratio, which is crucial for maintaining nuclear stability.

Neutron-to-Proton Ratio

The neutron-to-proton ratio is a helpful concept to predict the stability of a nucleus and the type of decay it might undergo. Calculate it by dividing the number of neutrons by the number of protons.

• A ratio greater than 1 often indicates the nucleus is neutron-rich.
• When this ratio is much higher than the typical stable range, it suggests that beta decay is more likely to occur.

Understanding this ratio is essential because it helps scientists and students determine whether a particular nucleus might undergo beta decay or another type of radioactive decay. The ratio also provides a basis for predicting how a nucleus can reach a more stable state.

Nuclear Stability

Nuclear stability refers to the ability of a nucleus to remain unchanged and not spontaneously decay. Nuclei are considered stable when they have an optimal balance between protons and neutrons, often indicated by an ideal neutron-to-proton ratio. For lighter elements, the ratio is closer to 1. As elements become heavier, a slightly higher ratio is necessary to balance the increasing electrostatic forces between protons.

• Stable nuclei do not undergo transformations like beta decay.
• Instability in a nucleus often leads to radioactive decay processes like beta decay, which work to rectify the imbalance.

Nuclear stability is a key aspect of nuclear chemistry that explains why certain isotopes are radioactive while others remain stable over time.

Nuclear Chemistry

Nuclear chemistry is the branch of chemistry that deals with the study of the nucleus, nuclear reactions, and their applications. It includes the examination of radioactive decay, nuclear fission, nuclear fusion, and more. Understanding nuclear chemistry is not only crucial in academic settings, but it also has numerous practical applications such as in medicine, energy production, and archaeological dating.

• In medicine, nuclear chemistry is used in diagnostic imaging and cancer treatments.
• Nuclear chemistry helps in generating energy through processes such as nuclear fission in power plants.
• Carbon dating, a technique to determine the age of ancient artifacts, relies on principles of nuclear chemistry.

This field covers many aspects of how nuclei interact, transform, and apply to real-world problems, providing a deeper comprehension of how the universe functions at a fundamental level.