If there is 10 μmol of the radioactive isotope ^32 P...

If there is $10 \mu$ mol of the radioactive isotope $^{32}$ P (half-life 14 days) at $t=0,$ how much $^{32} \mathrm{P}$ will remain at $(\mathrm{a}) 7$ days
(b) 14 days,
(c) 21 days, and
(d) 70 days?

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Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose energy by emitting radiation. This fundamental phenomenon in nuclear chemistry and physics follows specific probabilistic laws, meaning that while it is unpredictable exactly when a particular nucleus will decay, the decay rate of a large population of nuclei can be described mathematically.

Half-Life

The half-life is the period of time required for half of the atoms in a radioactive sample to decay. It is a characteristic property of each radioactive isotope and is crucial for determining how quickly a radioactive substance diminishes, independent of the initial quantity present.

Exponential Decay

Exponential decay is the mathematical description of the decrease in the quantity of a substance over time when the rate of decay is proportional to the current amount. This concept is commonly expressed with the formula N(t) = N? × (1/2)^(t / T?/?), which is used to calculate the remaining amount at any given time.

Decay Constant

The decay constant is a parameter that quantifies the likelihood of decay for a single nucleus per unit time. It is directly related to the half-life via the formula ? = ln(2)/T?/? and provides an alternative way to model and understand the exponential decay process.

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