During the manufacture of a steel engine component,...

During the manufacture of a steel engine component, radioactive iron $\left(^{59} \mathrm{Fe}\right)$ is included in the total mass of 0.200 $\mathrm{kg}$ . The component is placed in a test engine when the activity due to this isotope is 20.0$\mu \mathrm{Ci}$ . After a $1000-\mathrm{h}$ test period, some of the lubricating oil is removed from the engine and found to contain enough 59 $\mathrm{Fe}$ to produce 800 disintegrations/min/L of oil. The total volume of oil in the engine is 6.50 L. Calculate the total mass worn from the engine component per hour of operation. The half-life of 59 $\mathrm{Fe}$ is $45.1 \mathrm{d} .$

During the manufacture of a steel engine component, radioactive iron $\left(^{59} \mathrm{Fe}\right)$ is included in the total mass of 0.200 $\mathrm{kg}$ . The component is placed in a test engine when the activity due to this isotope is 20.0$\mu \mathrm{Ci}$ . After a $1000-\mathrm{h}$ test period, some of the lubricating oil is removed from the engine and found to contain enough 59 $\mathrm{Fe}$ to produce
800 disintegrations/min/L of oil. The total volume of oil in the engine is 6.50 L. Calculate the total mass worn from the engine component per hour of operation. The half-life of 59 $\mathrm{Fe}$ is $45.1 \mathrm{d} .$

Key Concepts

Tracer Techniques

Tracer techniques involve introducing a detectable radioactive isotope into a system to study its behavior without significantly altering the system's overall properties. By tracking the activity associated with the tracer, one can infer rates of processes such as mass loss or material transfer. This method is widely applicable in both industrial quality control and environmental studies.

Activity Measurement

Activity represents the rate at which a radioactive sample disintegrates, typically measured in disintegrations per unit time such as disintegrations per second, minute, or in curies. It is directly proportional to the number of radioactive atoms present and is a key parameter for linking observed measurements to the underlying mass and decay properties of the substance.

Exponential Decay Equation

The exponential decay equation describes the decrease of a quantity over time according to an exponential law. In the context of radioactive decay, it relates the remaining amount of a substance to its initial amount, the decay constant, and the elapsed time. This equation is fundamental in calculating the reduction in activity or mass of material in a system undergoing decay.

Half-Life

Half-life is the time required for half the quantity of a radioactive substance to decay. This parameter is crucial for calculating how much of a substance remains over a certain period and forms a direct link between the decay process and time. It is used to determine decay constants and to model the behavior of radioactive materials in both laboratory and industrial settings.

Radioactive Decay

This concept refers to the process by which unstable atomic nuclei lose energy by emitting radiation. In problems involving radioactive materials, the decay follows predictable statistical laws, allowing us to relate the activity of a radioactive substance to the number of nuclei present and the time elapsed. It is essential in understanding how the properties of the material change over time in a variety of scientific and engineering applications.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a nucleus will decay. It is mathematically related to the half-life via the natural logarithm and appears in the exponential decay formula. Understanding the decay constant allows for precise predictions of the remaining quantity of a radioactive substance after any given time period.

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