The decay constant of a radioactive material represents the fraction of nuclei of a particular isotope that

Options

A. will eventually decay if you wait long enough
B. Will decay per unit time
C. Are present in an element containing a mixture of isotopes
D. Are present in a chemical compounds containing elements of different atomic numbers

Answer

To solve the question regarding the decay constant of a radioactive material, we need to understand what the decay constant represents in the context of radioactive decay.

### Step-by-Step Solution:

1. **Understanding Radioactive Decay**:
- Radioactive decay is a random process by which unstable nuclei lose energy by emitting radiation. The rate of decay is characterized by the decay constant (λ).

2. **Decay Constant (λ)**:
- The decay constant is defined as the probability per unit time that a nucleus will decay. It is a measure of the likelihood that a given nucleus will decay in a specific time interval.

3. **Radioactive Decay Law**:
- The mathematical expression for radioactive decay is given by:
\[ N(t) = N_0 e^{-\lambda t} \]
where:
- \(N(t)\) = number of undecayed nuclei at time \(t\),
- \(N_0\) = initial number of nuclei,
- \(λ\) = decay constant,
- \(e\) = base of the natural logarithm.

4. **Fraction of Nuclei Decaying**:
- The fraction of nuclei that decay in a small time interval (Δt) can be expressed as:
\[ \text{Fraction decayed} = \frac{\Delta N}{N_0} = \frac{N_0 - N(t)}{N_0} = 1 - e^{-\lambda \Delta t} \]
- For small Δt, this fraction approaches λΔt as Δt approaches zero.

5. **Conclusion**:
- Therefore, the decay constant (λ) represents the fraction of the nuclei of a particular isotope that will decay per unit time. This means that if you have a large number of nuclei, the decay constant gives you the expected number of nuclei that will decay in a unit time interval.

### Final Answer:
The decay constant of a radioactive material represents the fraction of the nuclei of a particular isotope that will decay per unit time.

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