Question

The most common isotope of uranium (uranium-238) has a half-life of \(4.5 \times 10^{9} \mathrm{y}\). If the universe is estimated to have a lifetime of \(1.38 \times 10^{10} \mathrm{y}\), what percentage of uranium- 238 has decayed over the lifetime of the universe?

Short answer

88.6% of uranium-238 has decayed over the universe's lifetime.

Step-by-step solution

Understanding the Problem

We are asked to find what percentage of uranium-238 has decayed over the lifetime of the universe. We are given the half-life of uranium-238 and the estimated age of the universe. The problem involves calculating the decay over a specific period using the concept of half-life.

Formula for Exponential Decay

The formula for radioactive decay is given by \( N(t) = N_0 \times 2^{-t/T} \), where \(N(t)\) is the remaining quantity, \(N_0\) is the initial quantity, \(t\) is the elapsed time, and \(T\) is the half-life.

Substitute Given Values

Using the formula \( N(t) = N_0 \times 2^{-t/T} \), we substitutue \(t = 1.38 \times 10^{10} \text{years}\) and \(T = 4.5 \times 10^{9} \text{years}\).

Calculate Remaining Uranium-238

Calculate \(2^{-1.38 \times 10^{10} / 4.5 \times 10^{9}}\) to find the fraction of uranium-238 remaining. This simplifies to \(2^{-3.067}\).

Evaluate the Exponential Expression

Computing the expression gives \(2^{-3.067} \approx 0.114\). This means approximately 11.4% of the initial uranium-238 remains.

Find Percentage Decayed

To find the percentage of uranium-238 that has decayed, subtract the remaining percentage from 100%. This gives \(100 ext{%} - 11.4 ext{%} = 88.6 ext{%}\).

Key concepts

Uranium-238

Uranium-238 is a naturally occurring isotope of uranium, which is a heavy metal found in Earth's crust. This isotope is of particular interest due to its use in geological dating and nuclear science. It comprises about 99.3% of natural uranium, making it the most prevalent uranium isotope.
One of the unique aspects of uranium-238 is that it undergoes radioactive decay, a process in which an unstable atomic nucleus loses energy by emitting radiation. During this decay process, uranium-238 transforms into thorium-234. This transformation continues through a 14-step decay chain until it ultimately becomes lead-206, a stable isotope.
The presence of uranium-238 is not only crucial in understanding geological time scales but also in providing insights into nuclear fission and energy production.

Half-life

The concept of half-life is fundamental in the study of radioactive elements. A half-life is the time required for half of the radioactive nuclei in a sample to decay. This means that after one half-life period, only half of the original radioactive atoms remain.
In the case of uranium-238, its half-life is incredibly long, at approximately 4.5 billion years. This extended half-life is significant because it allows uranium-238 to serve as a chronometer for dating rocks and other geological features over millions and billions of years.
Moreover, the half-life helps scientists predict how much of a radioactive isotopic material will remain after a given period. It plays a crucial role in understanding not only the age of the Earth but also the age of the universe itself.

Exponential Decay

Exponential decay is a mathematical concept used to describe the process by which a quantity decreases at a rate proportional to its current value. In radioactive processes, this concept is crucial as it dictates how fast nuclei decay over time.
The mathematical representation for radioactive decay is given by the equation: \[ N(t) = N_0 \times 2^{-t/T} \] Here, \( N(t) \) is the quantity of the substance that remains after time \( t \), \( N_0 \) is the initial quantity, and \( T \) is the half-life. The exponent \(-t/T\) shows how the remaining quantity decreases exponentially.
Understanding exponential decay allows us to calculate how long it takes for significant portions of a radioactive sample, like uranium-238, to decay. For instance, using this formula, one can determine that approximately 88.6% of the uranium-238 has decayed over the universe's estimated lifetime.