Question

A reported synthesis of the transuranium element bohrium (Bh) involved the bombardment of berkelium-249 with neon-22 to produce bohrium-267. Write a nuclear reaction for this synthesis. The half-life of bohrium-267 is \(15.0\) seconds. If 199 atoms of bohrium-267 could be synthesized, how much time would

Short answer

The nuclear reaction for the synthesis of bohrium-267 is: \[^{249}_{97}Bk + ^{22}_{10}Ne \rightarrow ^{267}_{107}Bh\] It would take approximately 4.85 seconds for half of the 199 synthesized bohrium-267 atoms to decay.

Step-by-step solution

Write the nuclear reaction for the synthesis of bohrium-267

To write the nuclear reaction, we will represent the berkelium-249 and neon-22 particles with their respective atomic and mass numbers. Then we will add these particles together to form bohrium-267 and possibly other particles. The general form of a nuclear reaction can be expressed as: \[{}^{Mass \, number}_{Atomic \, number}Element\] Writing the nuclear reaction for the synthesis of bohrium-267, we have: \[{}^{249}_{97}Bk + {}^{22}_{10}Ne \rightarrow {}^{267}_{107}Bh\]

Determine any other particles produced in the reaction

In this reaction, since the sum of the mass numbers and atomic numbers of both sides of the reaction are equal, no additional particles are produced: \(249 + 22 = 267\) \(97 + 10 = 107\) So the complete nuclear reaction is: \[{}^{249}_{97}Bk + {}^{22}_{10}Ne \rightarrow {}^{267}_{107}Bh\]

Use the half-life formula to calculate the time required for half of the synthesized bohrium-267 to decay

The half-life formula is given by: \[N_t = N_0 \times \frac{1}{2}^{\frac{t}{T}}\] Where - \(N_t\) represents the number of remaining atoms at time \(t\), - \(N_0\) is the initial number of atoms, - \(t\) is the time elapsed, - \(T\) is the half-life of the substance. Since we are looking for the time when half of the synthesized bohrium-267 has decayed, we set: \(N_t = \frac{199}{2} = 99.5 \approx 100\) We know that the half-life (\(T\)) of bohrium-267 is 15.0 seconds, and the initial number of atoms (\(N_0\)) is 199. Now we can find the time elapsed (\(t\)) using the formula: \[100 = 199 \times \frac{1}{2}^{\frac{t}{15.0}}\]

Solve for the time elapsed

Solving for \(t\), we first divide both sides of the equation by 199: \[\frac{100}{199} = \frac{1}{2}^{\frac{t}{15.0}}\] Now take the natural logarithm of both sides: \[\ln\left(\frac{100}{199}\right) = \ln\left(\frac{1}{2}^{\frac{t}{15.0}}\right)\] Using the logarithmic property \(ln(a^b) = b\ln(a)\), we get: \[\ln\left(\frac{100}{199}\right) = \frac{t}{15.0}\ln\left(\frac{1}{2}\right)\] Finally, solve for \(t\): \[t = \frac{15.0 \times \ln\left(\frac{100}{199}\right)}{\ln\left(\frac{1}{2}\right)}\] Calculating the result: \[t \approx 4.85 \, seconds\] So it would take approximately 4.85 seconds for half of the 199 synthesized bohrium-267 atoms to decay.

Key concepts

Nuclear Reaction

Nuclear reactions are processes where the nucleus of an atom, which contains protons and neutrons, transforms into a different element or isotope. These transformations can occur through the collision of two nuclei, as seen in the synthesis of transuranium elements such as bohrium.

In a typical synthesis reaction, a target nucleus is bombarded with particles or other nuclei. This collision can result in the combination of the two, forming a new nucleus with a different number of protons or neutrons. The reaction used to synthesize bohrium involves the bombardment of berkelium-249 with neon-22, resulting in bohrium-267 without emitting additional particles. Critical to the success of such experiments is the careful selection of target nuclei and projectiles, as well as the conditions under which they collide.

Transuranium Elements

Transuranium elements are those beyond uranium on the periodic table, with atomic numbers greater than 92. They do not occur naturally and can only be produced artificially in nuclear reactors or particle accelerators.

Bohrium, element 107, is one such transuranium element created through nuclear reactions that involve combining lighter elements. These elements have high atomic numbers and are usually synthesized in extremely small amounts. Since they have no stable isotopes, their chemical and physical properties are challenging to study. Understanding the nature of these elements helps in expanding our knowledge of chemical bonding, nuclear structure, and the limits of the periodic table.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the initial amount of the substance to decay. This concept is crucial in determining the stability and longevity of isotopes, especially in the context of transuranium elements.

To calculate the half-life, we use the formula:
\(N_t = N_0 \times \frac{1}{2}^{\frac{t}{T}}\)
This formula involves the initial quantity of the substance \(N_0\), the amount left after time \(t\), and the half-life \(T\). In our exercise, we started with 199 atoms of bohrium-267 and wanted to find the time it would take for half of them to decay. Applying the half-life formula, we're able to deduce that it takes approximately 4.85 seconds for this to occur. This calculation is not only essential for scientists working with radioactive materials but also for applications such as medicine, archaeology, and environmental studies.