Question

Many transuranium elements, such as plutonium-232 , have very short half- lives. (For \(^{232} \mathrm{Pu}\) , the half-life is 36 minutes.) However, some, like protactinium- 231 (half-life \(=3.34 \times 10^{4}\) years), have relatively long half-lives. Use the masses given in the following table to calculate the change in energy when 1 mole of \(^{232} \mathrm{Pu}\) nuclei and 1 mole of \(^{231} \mathrm{Pa}\) nuclei are each formed from their respective number of protons and neutrons. (Since the masses of \(^{232} \mathrm{Pu}\) and \(^{231} \mathrm{Pa}\) are atomic masses, they each include the mass of the electrons present. The mass of the nucleus will be the atomic mass minus the mass of the electrons.)

Short answer

The change in energy when 1 mole of each plutonium-232 \( (Pu) \) nucleus and protactinium-231 \( (Pa) \) nucleus is formed can be calculated using Einstein's equation \( E=mc^2 \). First, find the mass of the nuclei, the number of protons and neutrons, and the mass of the protons and neutrons combined for each element. Then, calculate the mass difference and convert it from atomic mass units (u) to kilograms (kg). Lastly, multiply the mass difference in kg by the speed of light squared to find the change in energy for each nucleus formation.

Step-by-step solution

Find the mass of 1 mole of plutonium-232 nuclei and 1 mole of protactinium-231 nuclei

For this step, use the atomic masses given in the table for plutonium-232 (232.0381 u) and protactinium-231 (231.0359 u). Remember that the table gives the mass of the entire atom, including the electrons. Since we want to find the mass of the nucleus, we will need to subtract the mass of the electrons present.

Calculate the mass of the electrons in each atom

To find the mass of the electrons, we will use the following information: - Atomic number of plutonium (Pu) = 94 - Atomic number of protactinium (Pa) = 91 - Mass of an electron = 9.10938356 × 10⁻³¹ kg (which is approximately 5.4858 x 10^-4 u) Multiply the atomic number of each element by the mass of an electron to find the mass of the electrons in the element. Mass of electrons in Pu = (94)*(5.4858 x 10^-4 u) Mass of electrons in Pa = (91)*(5.4858 x 10^-4 u)

Calculate the mass of the plutonium and protactinium nuclei

Now, subtract the mass of the electrons from the atomic masses of the elements to obtain the mass of each nucleus: Mass of Pu nucleus = Atomic mass of Pu - Mass of electrons in Pu Mass of Pa nucleus = Atomic mass of Pa - Mass of electrons in Pa

Determine the number of protons and neutrons in each nucleus

To find the number of protons and neutrons in each nucleus, use the atomic numbers and mass numbers of the elements: - Number of protons in Pu = atomic number of Pu = 94 - Number of neutrons in Pu = mass number of Pu - atomic number of Pu = 232 - 94 = 138 - Number of protons in Pa = atomic number of Pa = 91 - Number of neutrons in Pa = mass number of Pa - atomic number of Pa = 231 - 91 = 140

Calculate the mass of the protons and neutrons combined

Use the following information to calculate the mass of the protons and neutrons combined in each nucleus: - Mass of a proton = 1.007276466812 u - Mass of a neutron = 1.008664915 u Mass of protons and neutrons in Pu: 94 * (Mass of a proton) + 138 * (Mass of a neutron) Mass of protons and neutrons in Pa: 91 * (Mass of a proton) + 140 * (Mass of a neutron)

Calculate the change in energy when 1 mole of each nucleus is formed

To find the change in energy, we can use Einstein's famous equation: \[E=mc^2\] Where E is the energy, m is the mass difference (mass of the combined protons and neutrons minus the mass of the nucleus), and c is the speed of light (2.998 x 10^8 m/s). First, calculate the difference in mass for Pu and Pa formations: Delta_mass(Pu) = Mass of protons and neutrons in Pu - Mass of Pu nucleus Delta_mass(Pa) = Mass of protons and neutrons in Pa - Mass of Pa nucleus Now, convert the mass difference from atomic mass units (u) to kilograms (kg) by multiplying by 1.6605 x 10^-27: Delta_mass(Pu) in kg = Delta_mass(Pu) in u * 1.6605 x 10^-27 Delta_mass(Pa) in kg = Delta_mass(Pa) in u * 1.6605 x 10^-27 Finally, calculate the change in energy for each nucleus formation: Change in energy for Pu formation = Delta_mass(Pu) in kg * c² Change in energy for Pa formation = Delta_mass(Pa) in kg * c²

Key concepts

Atomic Mass

Atomic mass is a fundamental concept in nuclear chemistry and refers to the mass of an atom expressed in atomic mass units (u). It accounts for the sum of the masses of protons, neutrons, and electrons within an atom. However, in precise calculations involving nuclear reactions, especially when computing the energy change, it is vital to focus on the nucleus alone.
The atomic mass provided in tables includes the mass of electrons because it is measured as the entire atom's mass. When isolating the nucleus's mass, one must subtract the mass of the electrons, which is a very small fraction of the total atomic mass. This adjustment is crucial for calculating nuclear properties, such as energy changes and stability.

Half-Life

Half-life is a key term in nuclear chemistry and defines the time required for half the population of a radioactive substance to decay. It is an intrinsic property of unstable atomic nuclei and dictates how quickly they transform into more stable forms.
In the exercise, we are looking at two types of nuclei with vastly different half-lives: plutonium-232 with a short half-life of 36 minutes, and protactinium-231 with a much longer half-life of 34,000 years. This difference means that plutonium-232 decays much faster than protactinium-231, affecting how they are used and studied in nuclear chemistry.

Protons and Neutrons

Protons and neutrons are subatomic particles located in an atom's nucleus, playing a crucial role in atomic structure and nuclear reactions. Protons carry a positive charge and define the atomic number, while neutrons are neutral and contribute to the mass number. Together, these particles define the isotope of an element.
Understanding the correct number of protons and neutrons is essential in nuclear chemistry, as it affects an element's properties, including its atomic mass, stability, and behavior during nuclear reactions. The exercise involves counting these particles to determine the nuclear mass and calculate energy changes, emphasizing their fundamental role in nuclear structure and reactions.

Mass-Energy Equivalence

Mass-energy equivalence is a groundbreaking concept from Einstein's theory of relativity, encapsulated in the equation \(E=mc^2\). This principle shows that mass can be converted into energy and vice versa, highlighting that the mass differences in nuclear reactions correspond to large energy changes.
In the context of the exercise, this relationship is used to calculate the change in energy when forming nuclei from their constituent protons and neutrons. By determining the mass defect (i.e., the difference between the mass of separated nucleons and the nucleus), one can compute the energy released or absorbed during nuclear processes. This calculation underscores the fact that even tiny changes in mass can result in significant energy outputs due to the large value of the speed of light squared \((c^2)\).