Problem 66 - Chemistry and Chemical Reactivity

Question

Radioactive decay series begin with a very long-lived isotope. For example, the half-life of \(^{238} \mathrm{U}\) is \(4.5 \times 10^{9}\) years. Each series is identified by the name of the long-lived parent isotope of highest mass. (a) The uranium-238 radioactive decay series is sometimes referred to as the \(4 n+2\) series because the masses of all 13 members of this series can be expressed by the equation \(m=4 n+2,\) where \(m\) is the mass number and \(n\) is an integer. Explain why the masses are correlated in this way. (b) Two other radioactive decay series identified in minerals in the earth's crust are the thorium-232 series and the uranium-235 series. Do the masses of the isotopes in these series conform to a simple mathematical equation? If so, identify the equation. (c) Identify the radioactive decay series to which each of the following isotopes belongs: \(_{88}^{226} \mathrm{R a}, \quad _{86}^{215} \mathrm{A t}, \quad _{90}^{228} \mathrm{T h}, \quad _{83}^{210} \mathrm{B i}.\) (d) Evaluation reveals that one series of elements, the \(4 n\) \(+1\) series, is not present in the earth's crust. Speculate why.

Step-by-Step Solution

Answer

(a) Mass numbers fit the 4n+2 form due to alpha decay steps. (b) Yes, thorium-232 fits 4n and uranium-235 fits 4n+3 equations. (c) 4n: \(^{226}\text{Ra}\), 4n+2: \(^{210}\text{Bi}\), \(^{228}\text{Th}\), 4n+3: \(^{215}\text{At}\). (d) 4n+1 series may lack stable isotopes or visible remnants.

Step 1: Understanding the Correlation in the 4n+2 Series

The uranium-238 decay series consists of isotopes whose mass numbers are given by the equation \(m = 4n + 2\). In each decay, a parent nucleus transmutes by losing particles, mainly \(\alpha\) particles (which reduces the mass by 4 because an \(\alpha\) particle is a helium nucleus containing 2 protons and 2 neutrons) and sometimes \(\beta\) particles (negligibly affecting mass). As the decay series involves alternating numbers of \(\alpha\) and \(\beta\) decays, the reduction pattern conforms to the expression \(4n+2\).

Step 2: Identifying Equations for Other Series

For the thorium-232 and uranium-235 decay series, a similar pattern can be observed. Thorium-232 is the start of the \(4n\) series, where each member's mass number can be represented by \(m = 4n\). Uranium-235 begins the \(4n+3\) series, and thus the progression for the isotopes follows \(m = 4n + 3\). These equations reflect the preservation of the '4n' pattern, which characterize decay involving predominantly \(\alpha\) loss.

Step 3: Assigning Isotopes to Their Series

For each given isotope:1. \(^\ 226\text{Ra}\) has a mass number \(226\), which fits the \(4n\) series since \(226 = 4(56)\).2. \(^\ 215\text{At}\) fits the \(4n+3\) series as \(215 = 4(53) + 3\).3. \(^\ 228\text{Th}\) matches the \(4n+2\) series with \(228 = 4(56) + 2\).4. \(^\ 210\text{Bi}\) aligns with the \(4n+2\) series, seen as \(210 = 4(52) + 2\).

Step 4: Understanding the Absence of the 4n+1 Series

The \(4n+1\) series is absent on Earth largely due to the nature of isotope stability and half-lives. Isotopes within the \(4n+1\) series may have undergone decay to stability long ago, or they may possess half-lives that have led to their natural depletion. As such, either fewer occurrences or non-beneficial half-lives mean observable elements in this sequence are missing or rare.

Key Concepts

Uranium-238 Decay Series

The Uranium-238 decay series is a fascinating journey through the transformation of one isotope into another, all governed by a predictable pattern in atomic masses. Known as the "4n+2 series," this decay path involves a sequence of isotopes whose mass numbers fit the equation: \(m = 4n + 2\), where \(m\) is the mass number and \(n\) is an integer. This pattern arises because each decay event tends to involve the loss of an alpha particle, which is essentially a helium nucleus that contains two protons and two neutrons - reducing the mass number by 4 each time. Occasionally, beta decay occurs, which changes the atomic number but does not significantly affect the mass number. As this decay progresses over many millions of years, the consistency of mass reduction by 4 accounts for all isotopes in this series conforming to the "4n+2" configuration. This reliable mass correlation makes understanding and predicting the subsequent isotopes in the series much simpler.

Thorium-232 Series

The Thorium-232 series, often identified as the "4n series," is another decay path significant in nuclear physics and geology. It starts with Thorium-232 and includes isotopes whose mass numbers align with the equation: \(m = 4n\). This alignment arises because, similar to the Uranium-238 series, alpha decay is the dominant process, which reduces the mass number by exactly 4 units each time it occurs. As such, every member of the Thorium-232 series fits the quadruple sequence described by the "4n" equation.
The Thorium-232 series has a half-life of 14 billion years, making it incredibly long-lived, which accounts for its presence in detectable amounts within the earth's crust. This intrinsic stability and longevity allow for a useful presence of Thorium-232, making it a fundamental subject of study in understanding radioactive processes and their implications on Earth's geological history.

Uranium-235 Series

The Uranium-235 decay series begins with one of the most famous isotopes used in both nuclear power and weaponry. This series is referred to as the "4n+3 series," with its members' mass numbers fitting the equation \(m = 4n + 3\). This is a result of the specific sequence in which alpha and beta decays occur starting from Uranium-235.
During an alpha decay, the mass number decreases by 4, but a beta decay does not affect the mass number significantly. As a result, the distinct alternating pattern of alpha and beta decays contribute to the predictable changes in isotopes' mass numbers within this decay path. With a half-life of 700 million years, Uranium-235 undergoes transformation over relatively long geological periods, leading to its notable role in Earth's radioactive decay series. This series gives insight into radiometric dating, crucial for determining the age of rocks and understanding geologic time scales.

Alpha and Beta Decay

In radioactive decay series such as Uranium-238, Thorium-232, and Uranium-235, alpha and beta decays are the fundamental processes that transform one isotope into another.

• Alpha decay: This type of decay involves the nucleus emitting an alpha particle (consisting of two protons and two neutrons), effectively decreasing the atomic number by 2 and the mass number by 4. It's the primary decay mode in the series, crucial for shifting isotopes down the decay chain.
• Beta decay: The beta decay is a process where a neutron in the nucleus is transformed into a proton (or vice versa), accompanied by the emission of a beta particle (an electron or positron). This changes the atomic number by 1 but does not significantly alter the mass number.

These processes drive the gradual transformation through different isotopes in a decay series, providing insight into nuclear physics. Understanding them helps explain the sequence of transformations that result in the stable end products of these decay series, specifically in how mass numbers and elemental identity shift over time.