Each statement that follows refers to a comparison between two radioisotopes, A and X. Indicate whether each of the following statements is true or false. (a) If the half-life for A is shorter than the half-life for X, A has a larger decay rate constant.

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Understand the relationship between half-life and decay rate constant. The half-life \( t_{1/2} \) of a radioactive isotope is related to its decay rate constant \( k \) by the equation: \( t_{1/2} = \frac{0.693}{k} \).

Recognize that a shorter half-life means the isotope decays more quickly.

From the equation \( t_{1/2} = \frac{0.693}{k} \), observe that if the half-life \( t_{1/2} \) is shorter, the decay rate constant \( k \) must be larger to maintain the equality.

Apply this understanding to the problem: If isotope A has a shorter half-life than isotope X, then isotope A must have a larger decay rate constant than isotope X.

Conclude that the statement is true based on the mathematical relationship between half-life and decay rate constant.

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Half-Life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. It is a crucial concept in nuclear chemistry, as it helps determine the stability and longevity of a radioisotope. A shorter half-life indicates that a substance decays more quickly, which is essential for understanding the behavior of radioactive materials.

Decay Rate Constant

The decay rate constant (λ) is a proportionality constant that quantifies the rate at which a radioactive substance decays. It is related to the half-life by the equation λ = ln(2) / t½. A larger decay rate constant signifies a faster decay process, which is directly linked to the half-life of the isotope.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process can occur in various forms, including alpha, beta, and gamma decay. Understanding radioactive decay is essential for comparing isotopes, as it influences their half-lives and decay rate constants, thereby affecting their stability and applications in fields like medicine and energy.