A freshly prepared sample of a certain radioactive isotop...

A freshly prepared sample of a certain radioactive isotop has an activity of 10.0 $\mathrm{mCi}$ . After 4.00 $\mathrm{h}$ , its activity 8.00 $\mathrm{mCi}$ . (a) Find the decay constant and half-life (b) How many atoms of the isotope were contained in th freshly prepared sample? (c) What is the sample's activity 30.0 $\mathrm{h}$ after it is prepared?

Key Concepts

Radioactivity and Activity

Activity refers to the number of nuclear decays per unit time and is often measured in units such as curies or becquerels. It is directly proportional to both the decay constant and the number of radioactive atoms present (A = ?N). This concept is crucial for understanding how the observed activity of a sample changes over time and for relating macroscopic measurements to microscopic properties.

Exponential Decay Equation

The exponential decay equation, typically written as N(t) = N0 exp(-?t), describes how the number of undecayed atoms decreases over time. This law also applies to the activity of the sample, since activity is proportional to the number of radioactive atoms. The equation is central to solving problems related to decay constants, half-lives, and predicting the behavior of radioactive materials over time.

Half-Life

Half-life is the time required for half of the radioactive nuclide in a sample to decay. It is a characteristic property of each radioactive isotope and is inversely related to the decay constant. The half-life provides an intuitive measure of the persistence of the radioactivity over time and is commonly used to describe how quickly a radioactive substance loses its activity.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time that a single nucleus will decay. It is directly related to the rate of decay and appears in the exponential decay law. A larger decay constant indicates a more rapid decay of the sample. It serves as a key connection between the measurable activity of a radioactive sample and the number of unstable nuclei present.

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This process occurs spontaneously and is statistically described as an exponential decay, where the number of undecayed nuclei decreases over time. The behavior follows a predictable pattern that can be described mathematically, making it fundamental in nuclear physics and applications such as dating and medical diagnostics.

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