The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. If a sample of U-238 initially contained 1.5⨉10¹⁸ atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms does it contain today?

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Identify the given values: the half-life of U-238 is 4.5 billion years, the initial number of atoms is 1.5 \(\times\) 10^{18}, and the time elapsed is 13.8 billion years.

Use the formula for radioactive decay: N(t) = N_0 \(\times\) (1/2)^{t/t_{1/2}}, where N(t) is the number of atoms remaining, N_0 is the initial number of atoms, t is the time elapsed, and t_{1/2} is the half-life.

Substitute the given values into the formula: N(t) = 1.5 \(\times\) 10^{18} \(\times\) (1/2)^{13.8/4.5}.

Calculate the exponent: 13.8/4.5 to determine how many half-lives have passed.

Evaluate the expression to find the remaining number of U-238 atoms, N(t).

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주요 개념

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Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate characterized by the half-life, which is the time required for half of the radioactive atoms in a sample to decay. For U-238, this half-life is 4.5 billion years, meaning that after this period, only half of the original amount of U-238 remains.

Half-Life

Half-life is a fundamental concept in nuclear chemistry that quantifies the time it takes for half of a radioactive substance to decay. It is a constant for each isotope and does not depend on the initial amount of the substance or environmental conditions. Understanding half-life is crucial for calculating the remaining quantity of a radioactive isotope after a given period, as seen in the decay of U-238 over billions of years.

Exponential Decay

Exponential decay describes the process where a quantity decreases at a rate proportional to its current value. In the context of radioactive decay, the number of remaining atoms decreases exponentially over time, following the formula N(t) = N0 * (1/2)^(t/T), where N0 is the initial quantity, t is the elapsed time, and T is the half-life. This concept is essential for calculating the remaining U-238 atoms after 13.8 billion years.

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. If a sample of U-238 initially contained 1.5⨉1018 atoms when the universe was formed 13.8 billion years ago, how many U-238 atoms does it contain today?