In an experiment on the transport of nutrients in the root...

In an experiment on the transport of nutrients in the root structure of a plant, two radioactive nuclides $\mathrm{X}$ and $\mathrm{Y}$ are used. Initially, 2.50 times more nuclei of type $\mathrm{X}$ and $\mathrm{Y}$ present than of type Y Precisely three days later, there are 4.20 times more nuclei of type $\mathrm{X}$ than of type $\mathrm{Y}$ . Isotope $\mathrm{Y}$ has a half-life of $1.60 \mathrm{d} .$ What is the half-life of isotope $\mathrm{X}$ ?

Key Concepts

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation, transforming into a different element or a lower-energy state. This process occurs at random at the level of individual atoms and is statistically described by decay laws that quantify the rate of decay over time.

Half-Life

Half-life is the time required for half of the radioactive nuclei in a given sample to decay. It is a fundamental characteristic of any radioactive isotope, remaining constant regardless of the initial amount of the substance, and is used to measure the stability of the nucleus.

Exponential Decay

Exponential decay describes the mathematical behavior of radioactive decay. It states that the number of undecayed nuclei decreases exponentially with time, and the decay rate can be modeled using exponential functions based on the decay constant and half-life.

Decay Constant

The decay constant is a parameter that quantifies the probability of a radioactive decay event occurring per unit time. It provides a link between the half-life and the rate of decay, where a higher decay constant corresponds to a shorter half-life, and is central to the formulation of the exponential decay law.

Ratio Comparison in Radioactive Kinetics

Ratio comparison in radioactive kinetics involves analyzing the relative quantities of different isotopes or materials over time. By comparing ratios at different times, one can deduce unknown decay parameters such as half-lives or decay constants, making it a useful approach in multi-isotope experiments.

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