A proposed nuclear theory suggests that the relative abun-dances of the uranium isotopes 235U and 238U were approximately equal at the time they were formed. Today, the observed ratio of these isotopes is 7.25 x 10^-3. Give that the half-lives for radioactive decay are 7.04 x 19^8 are y for and 4.47 x 10^9 y for , calculate the age of the elements.

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Step 1: Understand the problem. We are given the current ratio of the isotopes 235U and 238U, and their respective half-lives. We are asked to calculate the age of the elements, which is the time elapsed since the isotopes were formed.

Step 2: Use the formula for radioactive decay, which is N = N0 * (1/2)^(t/T), where N is the final amount, N0 is the initial amount, t is the time elapsed, and T is the half-life. In this case, we know that the initial amounts of 235U and 238U were equal, so we can set up a ratio of the final amounts: (235U/238U) = (1/2)^(t/T_235U) / (1/2)^(t/T_238U).

Step 3: Substitute the given values into the equation. We know that the current ratio of 235U to 238U is 7.25 x 10^-3, the half-life of 235U is 7.04 x 10^8 years, and the half-life of 238U is 4.47 x 10^9 years.

Step 4: Simplify the equation. The terms with base (1/2) can be combined by subtracting the exponents, giving us (235U/238U) = (1/2)^(t/T_235U - t/T_238U).

Step 5: Solve the equation for t. This will involve taking the logarithm of both sides and rearranging the equation to isolate t. The final equation will be t = (log(235U/238U) / log(1/2)) * (T_235U*T_238U / (T_238U - T_235U)).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 nuclei in a sample to decay. Understanding this concept is crucial for calculating the age of materials using isotopic ratios.

Half-Life

The half-life of a radioactive isotope is the time it takes for half of a sample of that isotope to decay into its daughter products. Each isotope has a unique half-life, which is constant and independent of the initial amount of the substance. This concept is essential for determining the age of geological or archaeological samples through radiometric dating.

Isotope Ratios

Isotope ratios refer to the relative abundances of different isotopes of an element in a sample. In the context of radiometric dating, comparing the ratio of parent isotopes (like 235U) to daughter isotopes (like 207Pb) allows scientists to estimate the time elapsed since the formation of the sample. This ratio is critical for calculating the age of elements based on their decay rates.

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