Problem 12 - Chemistry: The Central Science

Question

Write balanced nuclear equations for the following transformations: (a) gold-191 undergoes electron capture; (b) gold-201 decays to a mercury isotope; (c) gold198 undergoes beta decay; (d) gold-188 decays by positron emission.

Step-by-Step Solution

Answer

The balanced nuclear equations for the given transformations of gold isotopes are as follows: (a) \(_{79}^{191}\textrm{Au} + _{-1}^{0}e \rightarrow _{78}^{191}\textrm{X}\) (b) \(_{79}^{201}\textrm{Au} \rightarrow _{80}^{201}\textrm{Hg} + _{-1}^{0}\beta\) (c) \(_{79}^{198}\textrm{Au} \rightarrow _{80}^{198}\textrm{Hg} + _{-1}^{0}\beta\) (d) \(_{79}^{188}\textrm{Au}\rightarrow_{78}^{188}\textrm{X} + _{1}^{0}\beta^{+}\)

Step 1: (a) Gold-191 undergoes electron capture

Electron capture involves a proton in the nucleus of an atom capturing an electron from an inner shell, combining to form a neutron. Let's write the balanced nuclear equation for gold-191: 1. Au (Z=79, A=191) 2. Capture of an electron (e⁻) 3. Decrease the atomic number (Z) by 1 4. The mass number (A) remains the same Resulting equation: \(_{79}^{191}\textrm{Au} + _{-1}^{0}e \rightarrow _{78}^{191}\textrm{X}\)

Step 2: (b) Gold-201 decays to a mercury isotope

In this case, we're directly given the decay product (a mercury isotope) and the transformation undergone. Decay to a mercury isotope typically means undergoing beta decay (emission of an electron). 1. Au (Z=79, A=201) 2. Emission of a beta particle ( β⁻ ) 3. Increase the atomic number (Z) by 1 4. The mass number (A) remains the same Resulting equation: \(_{79}^{201}\textrm{Au} \rightarrow _{80}^{201}\textrm{Hg} + _{-1}^{0}\beta\)

Step 3: (c) Gold198 undergoes beta decay

For gold-198 undergoing beta decay, we will write the balanced nuclear equation by following similar steps as in part (b): 1. Au (Z=79, A=198) 2. Emission of a beta particle ( β⁻ ) 3. Increase the atomic number (Z) by 1 4. The mass number (A) remains the same Resulting equation: \(_{79}^{198}\textrm{Au} \rightarrow _{80}^{198}\textrm{Hg} + _{-1}^{0}\beta\)

Step 4: (d) Gold-188 decays by positron emission

Positron emission is a decay process where a proton in the nucleus is converted into a neutron, while releasing a positron (a positively charged electron). Let's write the balanced nuclear equation for gold-188: 1. Au (Z=79, A=188) 2. Emission of a positron (β⁺) 3. Decrease the atomic number (Z) by 1 4. The mass number (A) remains the same Resulting equation: \(_{79}^{188}\textrm{Au}\rightarrow_{78}^{188}\textrm{X} + _{1}^{0}\beta^{+}\)

Key Concepts

Electron Capture

Electron capture is a process that occurs when an inner orbital electron is drawn into the nucleus and combines with a proton to produce a neutron and a neutrino. This process decreases the atomic number by one while the mass number remains unchanged, as the proton becomes a neutron. It is a type of radioactive decay that can lead to changes in the atomic number of an atom, thus transforming it into a different element.

For instance, when gold-191 undergoes electron capture, the resulting equation is: \[\begin{equation}_{79}^{191}\text{Au} + _{-1}^{0}e \rightarrow _{78}^{191}\text{X}\end{equation}\]In this equation, Au represents the gold isotope, and 'e' denotes the captured electron. The atomic number drops from 79 to 78, indicating the transformation to a different element (notated as X), which in this case would be platinum.

Beta Decay

Beta decay represents a common form of radioactive decay where a beta particle (which can be an electron or a positron) is emitted. There are two types of beta decay: beta-minus (\( \beta^- \)), where an electron is emitted, and beta-plus (\( \beta^+ \)), involving the emission of a positron.

In beta-minus decay, a neutron turns into a proton plus an electron and an antineutrino. This increases the atomic number by one but leaves the mass number constant. For gold-198, the balanced nuclear equation showing beta-minus decay is: \[\begin{equation}_{79}^{198}\text{Au} \rightarrow _{80}^{198}\text{Hg} + _{-1}^{0}\beta\end{equation}\]Here, the emission of the beta particle results in an increase in the atomic number, changing the element from gold to mercury (Hg).

Positron Emission

Positron emission, or beta-plus (\( \beta^+ \)) decay, is when a proton in the nucleus is transformed into a neutron, a positron, and a neutrino. This process decreases the atomic number by one and keeps the mass number the same because a positively charged proton has become a neutrally charged neutron. Importantly, a positron is the antimatter counterpart to an electron, having the same mass as an electron but a positive charge.

When gold-188 decays by positron emission, the balanced nuclear equation is: \[\begin{equation}_{79}^{188}\text{Au}\rightarrow _{78}^{188}\text{X} + _{1}^{0}\beta^{+}\end{equation}\]After emitting a positron, gold-188 transforms into the element with the atomic number 78 (platinum), and a new positron (\( \beta^+ \)) is created.

Nuclear Chemistry

Nuclear chemistry delves into the reactions and changes that occur within the nuclei of atoms. It focuses on different types of decay and transmutation of elements, as well as the energy changes that accompany these transformations. Understanding nuclear chemistry is essential for explaining phenomena such as radioactivity, fission, and fusion.

Techniques in nuclear chemistry are used not just in theoretical applications but also for practical purposes like medical imaging, cancer treatment (through radiation therapy), nuclear power production, and radioisotope dating in archaeology and geology. The balanced nuclear equations for transformations, like the ones involving various isotopes of gold, provide insights into the stability of nuclei and the mechanisms underlying radioactive decay.