The compound Xe(CF3)2 decomposes in a firstorder reaction to...

The compound $\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}$ decomposes in a firstorder reaction to elemental Xe with a half-life of 30. minutes. If you place $7.50 \mathrm{mg}$ of $\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}$ in a flask, how long must you wait until only $0.25 \mathrm{mg}$ of $\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}$ remains?

The compound $\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}$ decomposes in a firstorder reaction to elemental Xe with a half-life of
30. minutes. If you place $7.50 \mathrm{mg}$ of $\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}$ in a flask, how long must you wait until only $0.25 \mathrm{mg}$ of $\mathrm{Xe}\left(\mathrm{CF}_{3}\right)_{2}$ remains?

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Key Concepts

Exponential Decay

Exponential decay refers to the process where the decrease in the amount of a substance follows an exponential relationship with time. For first-order reactions, this is mathematically represented as [A] = [A]? exp(-kt), linking the remaining amount of reactant ([A]) to time, the initial amount ([A]?), and the rate constant (k).

Logarithmic Calculations

Logarithmic calculations are used to solve for variables such as time in the exponential decay equation. Taking the natural logarithm allows one to linearize the exponential relationship, enabling the determination of reaction time from the ratio of the remaining concentration to the initial concentration.

First-Order Reaction Kinetics

First-order reactions are defined by a rate that is directly proportional to the concentration of a single reactant. This leads to an exponential decay in the quantity of the reactant over time, and the reaction rate can be described by the differential equation d[A]/dt = -k[A], where k is the rate constant.

Half-Life

The half-life of a reaction is the amount of time required for the concentration of a reactant to decrease to half of its initial value. In first-order reactions, the half-life is constant and independent of the initial concentration, and it is quantitatively related to the rate constant by the formula t?/? = ln(2)/k.

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