For which isotope is the half-life the shortest among the following: Ir-192, Au-198, and Tc-99?

The isotope with the shortest half-life among the options given is Technetium-99 (Tc-99). Tc-99 has a half-life of approximately 6 hours, which is relatively short compared to the other isotopes listed.

This short half-life makes Tc-99 particularly useful in medical applications, especially for diagnostic imaging in nuclear medicine. Its rapid decay means that it can be used to provide timely information while minimizing the patient's overall radiation exposure, as the radioactivity decreases quickly after administration.

In contrast, Iridium-192 (Ir-192) has a half-life of about 73.8 days, and Gold-198 (Au-198) has a half-life of approximately 2.7 days, both of which are significantly longer than that of Tc-99. Uranium-238 (U-238) possesses an extremely long half-life of around 4.5 billion years, making it unsuitable for applications that require short-lived isotopes. Thus, Technetium-99 is the correct choice due to its rapid decay and practical applications in medical imaging.