The half-life for the αdecay of uranium 23892 U is 4.47 ×10^9 yr Determine the age (in years) of

step 1 So the formula we're going to need here is n is equal to n not times 2 to the negative t divided

The half-life for the αdecay of uranium 23892 U is 4.47...

The half-life for the $\alpha$ decay of uranium $\stackrel{238}{92} \mathrm{U}$ is $4.47 \times 10^{9} \mathrm{yr}$ Determine the age (in years) of a rock specimen that contains 60.0$\%$ of its original number of $\stackrel{238}{92} \mathrm{U}$ atoms.

Key Concepts

Relationship Between Decay Constant and Half-Life

The decay constant (?) is directly related to the half-life (T1/2) of a radioactive material through the equation T1/2 = ln(2)/?. This relationship is essential for converting between the decay rate and the time scale of decay, enabling the computation of the age of materials from measured proportions of radioactive elements.

Radioactive Decay

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable nucleus accompanied by the emission of particles or radiation. This process is random at the level of individual atoms but predictable in large numbers, forming the basis for analysis in many scientific disciplines including physics, chemistry, and geology.

Half-Life

Half-life is the time required for half of the radioactive atoms in a sample to decay. It is a characteristic property of each radioactive isotope and is crucial for understanding the rate at which the substance decays. The concept of half-life allows scientists to determine the age of samples based on the remaining proportion of radioactive material.

Exponential Decay Law

The exponential decay law mathematically describes the decrease in the number of radioactive atoms over time. It is expressed by the formula N = N0 * e^(-?t), where N is the number of atoms at time t, N0 is the initial number of atoms, and ? is the decay constant. This relationship is fundamental in modeling decay processes and calculating the elapsed time since the formation of a sample.

Logarithmic Functions in Solving Exponential Equations

Logarithmic functions are used to solve equations involving exponentials, such as those arising in radioactive decay problems. By applying the natural logarithm, one can linearize the exponential decay law and isolate the time variable, making it possible to calculate the elapsed time given the remaining fraction of the radioactive substance.

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