Question

Uranium dating Uranium- 238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of \(\mathrm{U}-238\) and lead, and they determine that \(85 \%\) of the original \(\mathrm{U}-238\) remains: the other \(15 \%\) has decayed into lead. How old is the rock?

Short answer

Answer: The rock is approximately 1.295 billion years old.

Step-by-step solution

Write the radioactive decay formula

The radioactive decay formula is: $$ N(t) = N_0 \cdot (0.5)^{\frac{t}{t_{1/2}}} $$ Where: - \(N(t)\) is the amount of radioactive substance remaining after time \(t\), - \(N_0\) is the initial amount of radioactive substance, - \(t_{1/2}\) is the half-life of the radioactive substance, - \(t\) is the elapsed time.

Express the remaining percentage of Uranium-238

We are given that 85% of Uranium-238 remains in the rock. We need to express this as a fraction: $$ \frac{N(t)}{N_0} = \frac{85}{100} = 0.85 $$

Plug in the values into the radioactive decay formula

We know the half-life of U-238 (\(t_{1/2} = 4.5\) billion years) and the remaining fraction of U-238 (\( \frac{N(t)}{N_0} = 0.85\)). Plug in these values into the radioactive decay formula and solve for \(t\): $$ 0.85 = (0.5)^{\frac{t}{4.5}} $$

Solve for the age of the rock (t)

We can use logarithms to solve for \(t\): $$ \ln(0.85) = \frac{t}{4.5} \ln(0.5) $$ $$ t = \frac{4.5\cdot \ln(0.85)}{\ln(0.5)} $$ Now, calculate the value of \(t\): $$ t \approx 1.295 \text{ billion years} $$

Conclusion

The rock is approximately 1.295 billion years old.

Key concepts

Uranium-238

Uranium-238, often referred to as U-238, is a common isotope of uranium that plays a crucial role in the process of radioactive dating. This isotope is composed of 92 protons and 146 neutrons, making it one of the heaviest and longest-living isotopes.
One of the fascinating aspects of U-238 is its half-life, which is about 4.5 billion years. This lengthy half-life makes U-238 a valuable tool in geology for dating the age of the Earth and other ancient formations, such as rocks.
Its radioactive decay process involves transforming into various other elements until it eventually becomes lead-206, a stable isotope. This transformation happens very slowly due to its large half-life, allowing scientists to use the concentration of Uranium-238 versus lead to determine how many billions of years have passed since a rock was formed. It's like a natural clock ticking away over immense geological timescales, giving us a glimpse into the distant past.
The use of Uranium-238 in geological dating has provided significant insights into the history and formation of the Earth, helping to validate theories related to continental drift and the age of our solar system.

Half-life

The concept of half-life is essential in understanding radioactive decay processes. Half-life is the period it takes for half of a particular radioactive substance to decay into another form. For U-238, this duration is 4.5 billion years!
This means if we start with a given quantity of Uranium-238, after 4.5 billion years, only half of it will remain. The rest would have decayed into its byproducts, primarily lead.
This property is useful for dating ancient objects because it occurs at a predictable rate, allowing scientists to backtrack and estimate the time it took to reach a certain decay stage. In the case of the given exercise, the 15% decay from 100% provides the clue to determining the age of the rock.

• Predictability: This regularity in decay allows for accurate time measurements over extensive periods.
• Indicator of Age: Helps estimate the time elapsed since the rock's formation by comparing the ratio of remaining Uranium-238 to its decay products.

Thus, half-life is a fundamental concept that aids in deciphering the timeline of historical and geological events.

Logarithms

Logarithms are a powerful mathematical concept employed to solve equations involving exponential functions, which are common in scientific fields dealing with growth and decay, such as physics and biology.
In radioactive decay, the formula describing how a substance diminishes over time is exponential. To isolate the variable representing time (in this case, the age of the rock), logarithms become incredibly useful.
The exercise involves solving the equation\[0.85 = (0.5)^{\frac{t}{4.5}}\]Taking the natural logarithm (ln) of both sides allows us to transform the exponential equation into a linear one, making it simpler to solve for the unknown variable, which is time (\(t\)).

• Transformation Power: Converts multiplicative relationships into additive ones, simplifying the solving process.
• Provides Clarity: Makes exponential decay problems manageable and solvable with basic algebraic techniques.

In this way, logarithms are not just mathematical tools, they are translators that transform complex exponential relationships into a language we can easily understand and solve. So in essence, without logarithms, unraveling the age of ancient rocks through radioactive decay would be much more challenging.