How old is a rock that contains 83.2% of the amount of uranium-238 it contained when it was formed?

Verified step by step guidance

insert step 1> Determine the decay constant (\( \lambda \)) for uranium-238 using its half-life. The half-life of uranium-238 is approximately 4.5 billion years. Use the formula \( \lambda = \frac{0.693}{\text{half-life}} \).

insert step 2> Use the decay formula \( N = N_0 e^{-\lambda t} \), where \( N \) is the remaining quantity of uranium-238, \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

insert step 3> Since the rock contains 83.2% of the original uranium-238, set \( N = 0.832 N_0 \). Substitute this into the decay formula to get \( 0.832 N_0 = N_0 e^{-\lambda t} \).

insert step 4> Simplify the equation to \( 0.832 = e^{-\lambda t} \).

insert step 5> Solve for \( t \) by taking the natural logarithm of both sides: \( \ln(0.832) = -\lambda t \). Rearrange to find \( t = \frac{\ln(0.832)}{-\lambda} \).

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 unstable atomic nuclei lose 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. Understanding this concept is crucial for determining the age of a rock based on the remaining quantity of a radioactive isotope.

Half-Life

Half-life is a specific time period in which half of the original amount of a radioactive substance decays into a different element or isotope. For uranium-238, the half-life is approximately 4.5 billion years. By knowing the percentage of uranium-238 remaining in a rock, one can calculate how many half-lives have passed and thus estimate the rock's age.

Radiometric Dating

Radiometric dating is a technique used to date materials by comparing the abundance of a radioactive isotope to its decay products. In the case of uranium-238, scientists measure the ratio of uranium-238 to lead-206, its stable decay product, to determine the age of the rock. This method provides a reliable means of estimating geological time scales and the age of ancient materials.

Related Practice

Textbook Question

Consider the reaction:

2 HBr (g) → H₂ (g) + Br₂ (g)

a. Express the rate of the reaction in terms of the change in concentration of each of the reactants and products.

Textbook Question

Consider the reaction:

2 HBr (g) → H₂ (g) + Br₂ (g)

b. In the first 20.0 s of this reaction, the concentration of HBr dropped from 0.600 M to 0.512 M. Calculate the average rate of the reaction during this time interval.

Textbook Question

Consider the reaction:

2 HBr (g) → H₂ (g) + Br₂ (g)

c. If the volume of the reaction vessel in part b was 1.25 L, what amount of Br₂ (in moles) was formed during the first 15.0 s of the reaction?