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Cis-trans isomerization is a fascinating type of stereoisomerism where two isomers, referred to as "cis" and "trans," differ based on the position of substituent groups around a double bond or a ring structure. In chemical structures with a double bond, rotation around the bond is restricted, leading to this type of isomerism.
The cis isomer has substituent groups on the same side of the double bond, whereas in the trans isomer, they are on opposite sides. This distinct difference in spatial arrangement confers unique physical and chemical properties to each isomer.

• **Cis Isomer**: Substituents on the same side.
• **Trans Isomer**: Substituents on opposite sides.

This conversion between cis and trans isomers can occur under certain conditions such as light exposure, heat, or through catalysts. In the exercise given, a radioactive carbon-labeled cis-dichloroethene isomer undergoes isomerization, affecting the equilibrium positions as part of a dynamic process.

The equilibrium constant, denoted as \(K_c\), represents the ratio of the concentrations of products to reactants at equilibrium for a reversible chemical reaction. It's a key concept in understanding chemical equilibrium, indicating the extent to which reactants are converted into products. High \(K_c\) values typically suggest a reaction favoring product formation, while lower values indicate reactant preference.
In the problem, the equilibrium between cis and trans isomers was analyzed. When 40% of radioactive cis molecules converted to trans form, the \(K_c\) was calculated using:\[K_c = \frac{\text{[trans]}}{\text{[cis]}}\]Given that 60% of the original cis molecules remain, and 40% converted to trans, the equilibrium constant became:

• \(K_c = \frac{0.4}{0.6} = 0.667\)

Therefore, this indicates the trans form is slightly less favored under the given conditions. Understanding \(K_c\) helps predict and interpret the direction and completeness of reactions in equilibrium.

Radioactive isotopes, also known as radioisotopes, have unstable atomic nuclei which decay over time, emitting radiation. These isotopes are crucial in a variety of scientific and medical applications.

• They are used as tracers in biochemical experiments to track chemical processes and paths.
• In medicine, they help in diagnostic imaging and cancer treatments.

In the given problem, a radioactive carbon isotope \(^{14}C\) is used to study the dynamic equilibrium between cis and trans isomers of dichloroethene. The labeled (radioactive) molecules allow us to trace how isotopic atoms distribute between isomers at equilibrium.
This radioisotope approach helps in understanding not just chemical equilibria but also reaction mechanisms and molecular interactions, providing insights that stable isotopes could not easily offer. The emission of radiation during decay is detectable, thus making it possible to follow reactions and transformations at a molecular level.