(a) Given that ^238 U has a half-life of 4.47 ×10^9 years, find its decay constant r. (b) How m

step 1 So in the first part of this question, we are given that uranium 238 has a half life of 4 .47 ti

(a) Given that ^238 U has a half-life of 4.47 ×10^9 years,...

(a) Given that ${ }^{238} \mathrm{U}$ has a half-life of $4.47 \times 10^{9}$ years, find its decay constant $r$. (b) How many atoms are there in $1 \mathrm{~g}$ of pure ${ }^{238} \mathrm{U} ?(\mathrm{c})$ How many decays are there per second in $1 \mathrm{~g}$ of ${ }^{238} \mathrm{U} ?$

Key Concepts

Activity

Activity is a measure of the rate at which a sample of radioactive material decays and is expressed in terms of disintegrations per unit time (typically per second). It is calculated by multiplying the decay constant by the number of radioactive nuclei present. This concept is fundamental to determining how many decay events occur in a sample, which is important in both theoretical studies and practical applications of radioactivity.

Avogadro's Number and Molar Mass

Avogadro's number is the number of constituent particles, usually atoms or molecules, in one mole of a substance. When combined with the concept of molar mass—the mass of one mole of a substance—this concept allows for the conversion between mass and the number of atoms. It is essential for calculations involving the number of atoms present in a given mass of a radioactive material.

Half-Life

The half-life of a radioactive substance is the time required for half of the nuclei in a sample to decay. It is an intrinsic property of the substance, independent of the initial amount present. The half-life provides a convenient way to quantify the rate of decay and is commonly used to compare the stabilities of different isotopes.

Decay Constant

The decay constant is a parameter that quantifies the probability per unit time of a single nucleus decaying. It is mathematically related to the half-life by the formula r = ln(2) / (half-life). This constant plays a crucial role in the exponential decay law and allows for the calculation of the remaining quantity of a substance after a given time.

Exponential Decay

Exponential decay describes the process by which the number of radioactive nuclei decreases over time. It is characterized by a constant probability per unit time that a nucleus will decay, leading to a decay rate that is proportional to the number of undecayed nuclei. This concept forms the basis for understanding how quantities like half-life and decay constant relate to the decrease in the number of atoms over time.

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