The off gas from a boiling water nuclear power reactor contains a whole variety of radioactive trash, one of the most troublesome being Xe-133 (half life \(=5.2\) days). This off gas flows continuously through a large holdup tank in which its mean residence time is 30 days, and where we can assume that the contents are well mixed. Find the fraction of activity removed in the tank.

Understanding the half-life of a radioactive substance is vital when dealing with nuclear reactions or processes. The half-life, denoted as \( t_{1/2} \), is the time required for half of the radioactive nuclei in a sample to undergo radioactive decay. In our exercise involving Xe-133 with a half-life of \( 5.2 \) days, this implies that every \( 5.2 \) days, half of the Xe-133 present will have decayed into another element or isotope.

To calculate the decay constant \( \lambda \) from the half-life, we use the formula:\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]. The constant \( \lambda \) gives us the rate at which the atoms are decaying. This information is crucial for figuring out how quickly the Xe-133 activity reduces over time and ultimately helps us to determine the efficiency of radioactive removal in the holdup tank.

Mean residence time \( \tau \) is another critical concept in the handling of radioactive materials within reactors. It represents the average time a particle (or in our case, a radioactive atom) spends in a system before leaving it. For a holdup tank in a nuclear reactor, mean residence time is the time a given unit of off gas spends in the tank.

In the scenario presented, the mean residence time is 30 days. This period is the window available for the decay process to take place before the Xe-133 is removed from the tank. The longer the residence time, the more opportunity for decay and consequently, a higher fraction of radioactive material can be removed.

The Bateman equation is a mathematical model used to calculate the quantity of a radioactive element remaining in a continuously stirred tank reactor (CSTR) over time. In a CSTR, the material within the system is well mixed, and the input and output flows are continuous and steady. This setup allows for a homogeneous concentration of particles throughout the tank.

The Bateman equation for a CSTR in its steady-state form is given by:\[ fraction\;removed = \frac{\lambda}{\lambda + \frac{1}{\tau}} \]. Here, \( \lambda \) is the decay constant, and \( \tau \) is the mean residence time. This equation is particularly apt for determining the fraction of radioactivity removed as it encompasses both the inherent decay rate of the substance and the operational parameters of the reactor system.

Radioactivity removal is the process of reducing the radioactive contamination from a substance or environment. In nuclear reactors, efficient removal of radioactive isotopes is critical for safety and environmental protection. Various methods are implemented depending on the type of radioactive material and the stage of the nuclear process.

In our exercise's context, the removal efficiency is found using the ratio of the decay constant to the sum of the decay constant and the inverse of the mean residence time, as outlined by the Bateman equation for a CSTR. By calculating this, we identify the fraction of Xe-133 activity that is removed during its transit through the holdup tank. Proper maintenance of this removal process is essential to keep the reactor within safe operating conditions and to minimize the impact on the surrounding environment.