Question

An ore of uranium is found to contain \({ }_{92}^{238} \mathrm{U}\) and \({ }_{82}^{206} \mathrm{~Pb}\) in the mass ratio of \(1: 0.1 .\) The half-life period of \({ }_{92}^{238} \mathrm{U}\) is \(4.5 \times 10^{9}\) years. Age of the ore is \((\log 2=0.3, \log\) \(\left.\frac{114.9}{103}=0.048\right)\) (a) \(7.2 \times 10^{8}\) years (b) \(7.2 \times 10^{7}\) years (c) \(7.2 \times 10^{9}\) years (d) \(2.16 \times 10^{9}\) years

Short answer

Option (a) \(7.2 \times 10^8\) years.

Step-by-step solution

Equation for the Decay Process

Identify the decay process equation, which is given by the first-order kinetics equation for radioactive decay: \( N = N_0 \times e^{-\frac{t}{T}} \), where \( N \) is the remaining quantity of the substance, \( N_0 \) is the initial quantity, \( t \) is the time elapsed, and \( T \) is the half-life of the substance.

Relate Mass Ratio to Nuclei Ratio

Since the mass ratio of Uranium to Lead is 1:0.1, and knowing that the amount of lead increases as uranium decays, we can assume that the original amount of uranium equals the current amount of uranium plus the current amount of lead. We can express this as \( N_0 = N + Pb \). Given that the atomic masses of Uranium and Lead are approximately the same, for simplicity, we can consider the ratio of their numbers instead of mass and have \( \frac{N_0}{N} = 1 + \frac{Pb}{N} = 1 + \frac{0.1}{1} = 1.1 \).

Determine the Decay Constant

The decay constant \( \frac{1}{T} \) relates to the half-life in the equation \( T = \frac{\text{ln}(2)}{\frac{1}{T}} \). For Uranium-238, the half-life is given as \( T = 4.5 \times 10^9 \) years. We can then find the decay constant \( k \) by using \( k = \frac{\text{ln}(2)}{T} \).

Calculate the Age of the Ore

Using the decay process equation and the decay constant, we rearrange to find the time: \( t = T \times \text{ln}(\frac{N_0}{N}) = T \times \text{ln}(1.1) \). We can use the given logarithmic equivalences to solve for \( t \).

Apply Logarithmic Properties

Substitute the known values into the equation incorporating the given logarithmic information: \( t = 4.5 \times 10^9 \times \text{ln}(1.1) \). With \( \text{ln}(1.1) \approx \frac{\text{ln}(2) + \text{ln}(114.9/103)}{2} \), this becomes \( t = 4.5 \times 10^9 \times \left(0.3 + 0.048 \right)/2 \).Simplify the equation to find the time \( t \).

Perform the Calculations

Plugging in the values and solving for time, we get \( t = 4.5 \times 10^9 \times \left(0.3 + 0.048 \right)/2 = 4.5 \times 10^9 \times 0.174 \). The result is \( t \approx 7.83 \times 10^8 \) years.

Find the Closest Answer Option

Review the answer options and choose the one that is closest to the calculated age: \(7.2 \times 10^8\) years, which is option (a).

Key concepts

Uranium-Lead Dating

Uranium-lead dating is a scientific method for determining the age of geological samples by measuring the radioactive decay of uranium isotopes into stable lead isotopes. This method is based on principles of radioactive decay and isotope ratios. Uranium has two common isotopes used for dating rocks: Uranium-238 and Uranium-235, both of which decay to different isotopes of lead (Lead-206 and Lead-207, respectively).

Over time, Uranium atoms in a sample will decay into Lead at a predictable rate, known as the half-life period. By measuring the proportion of Uranium to Lead, scientists can calculate the time that has passed since the rock was formed. This technique is commonly used in dating the age of the Earth and ancient rocks, thereby providing insights into the early history of our planet.

Half-Life Period

The half-life period is the time required for half the amount of a radioactive substance to undergo decay. It is a constant, characteristic for each radioactive isotope. A half-life is independent of the starting amount of the substance and is not affected by environmental factors, such as temperature or pressure.

Understanding half-life is crucial in the field of radiometric dating, as it sets a clock for the rate at which parent isotopes decay into daughter isotopes within a sample. When analyzing a substance with a known half-life, such as Uranium-238 with a half-life of 4.5 billion years, it provides a means to measure the absolute age of geological materials. Knowing the half-life also assists in making sense of the changes in the ratio between the parent and daughter nuclide over time, which directly correlates with the age of the sample.

First-Order Kinetics Equation

The first-order kinetics equation is used to describe the rate at which radioactive decay occurs. Decay processes that are first-order imply that the rate at which the reaction occurs is directly proportional to the number of parent nuclide atoms present at a given time. This equation is expressed as:
\( N = N_0 \times e^{-kt} \),
where \( N \) is the number of undecayed atoms remaining at time \( t \), \( N_0 \) is the initial number of atoms at time \( t=0 \), \( k \) is the decay constant, and \( e \) is the base of the natural logarithm.

The simplicity of this equation allows us to track radioactive decay over time and calculate how much of the original substance remains after a certain period. This property is fundamental in radiometric dating techniques, as it links the measurable present-day composition of isotopes to the elapsed time since the beginning of the decay process.

Decay Constant

The decay constant, often symbolized as \( k \), quantifies the probability of a radioactive atom decaying per unit time. It is specific to each isotope and plays a critical role in the mathematical description of radioactive decay. The larger the decay constant, the more rapidly a substance decays.

In the context of the half-life equation, the decay constant is inversely related to the half-life period by the equation: \( k = \frac{\text{ln}(2)}{T} \), where \( T \) represents the half-life and \( \text{ln}(2) \) is the natural logarithm of 2. The decay constant is essential for calculating the age of a sample through radiometric dating, and its precision is instrumental in accurately determining half-lives, and accordingly, the ages of geological samples.