Suppose 50.0 mg of potassium- 45, a beta emitter was...

Suppose $50.0 \mathrm{mg}$ of potassium- $45,$ a beta emitter was isolated in pure form. After one hour, only 3.1 mg of the radioactive material was left. What is the half-life of potassium- $45 ?$

Key Concepts

Exponential Decay Law

The exponential decay law describes how the quantity of a radioactive substance decreases over time. It states that the remaining amount of a substance is proportional to the exponential of the negative product of the decay constant and time. This law is mathematically expressed as N(t) = N? e^(??t) or alternatively using half-life as N(t) = N? 2^(?t/T?/?), where N? is the initial quantity, ? is the decay constant, t is time, and T?/? is the half-life.

Half-Life

The half-life of a radioactive isotope is the time required for half of the initial amount of the substance to decay. It is a characteristic property of each radioactive element and is related to the decay constant by the equation T?/? = ln(2)/?. Understanding half-life is crucial for determining how quickly a radioactive material will decay, which is essential in fields such as nuclear medicine, radiometric dating, and nuclear power management.

Logarithmic Equation Solving

In dealing with exponential decay problems, logarithms are used to solve for unknown variables such as the decay constant or half-life. By applying logarithmic operations to both sides of the exponential decay equation, one can linearize the relationship and isolate the variable of interest. This method is fundamental in mathematics and physics when manipulating exponential relationships, enabling the determination of decay rates from empirical data.

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