Question

The atomic masses of hydrogen-2 (deuterium), helium-4, and lithium-6 are \(2.014102 \mathrm{amu}_{2} 4.002602 \mathrm{amu}\), and \(6.0151228\) amu, respectively. For cach isotope, calculate (a) the nuclear mass, (b) the nuclear binding energy, (c) the nuclear binding energy per nucleon. (d) Which of these three isotopes has the largest nuclear binding energy per nucleon? Does this agrec with the trends plotted in Figure 21.12?

Short answer

In summary, we convert the atomic masses for deuterium, helium-4, and lithium-6 into kg. Then we calculate the mass defect and nuclear binding energy for each isotope using Einstein's mass-energy equivalence formula. Next, we find the nuclear binding energy per nucleon by dividing the nuclear binding energy by the total number of nucleons for each isotope. The isotope with the highest binding energy per nucleon is identified, and we compare the results with the trends plotted in Figure 21.12 to verify if they agree.

Step-by-step solution

Calculate mass in kg for all isotopes

The conversion factor between amu and kg is 1 amu = 1.66054 x 10^{-27} kg. 1. Hydrogen-2 (deuterium): \( m_1 = 2.014102 \times 1.66054 \times 10^{-27} \mathrm{kg} \) 2. Helium-4: \( m_2 = 4.002602 \times 1.66054 \times 10^{-27} \mathrm{kg} \) 3. Lithium-6: \( m_3 = 6.0151228 \times 1.66054 \times 10^{-27} \mathrm{kg} \) b) Nuclear Binding Energy

Calculate mass defect and binding energy for all isotopes

Firstly, we need to calculate the mass defect for each isotope, which is the difference between the total mass of protons and neutrons and the actual mass of the isotope. After that, we will use Einstein's mass-energy equivalence formula to find the nuclear binding energy. For each isotope, the total mass of protons and neutrons in kg must be found. The mass of proton is 1.007276 amu and of neutron 1.008665 amu. Convert these masses to kg: 1. The mass of a proton: \( m_p = 1.007276 \times 1.66054 \times 10^{-27} \mathrm{kg} \) 2. The mass of a neutron: \( m_n = 1.008665 \times 1.66054 \times 10^{-27} \mathrm{kg} \) Now, find the mass defect and nuclear binding energy for each isotope: 1. Hydrogen-2 (deuterium): \( \Delta m_1 = 1 \cdot m_p + 1 \cdot m_n - m_1 \). Then, \( E_1 = \Delta m_1 \cdot c^2 \) 2. Helium-4: \( \Delta m_2 = 2 \cdot m_p + 2 \cdot m_n - m_2 \). Then, \( E_2 = \Delta m_2 \cdot c^2 \) 3. Lithium-6: \( \Delta m_3 = 3 \cdot m_p + 3 \cdot m_n - m_3 \). Then, \( E_3 = \Delta m_3 \cdot c^2 \) c) Nuclear Binding Energy per Nucleon

Calculate binding energy per nucleon for all isotopes

For each isotope, divide the nuclear binding energy by the total number of nucleons (protons and neutrons). 1. Hydrogen-2 (deuterium): \( E_{1n} = \frac{E_1}{2} \) 2. Helium-4: \( E_{2n} = \frac{E_2}{4} \) 3. Lithium-6: \( E_{3n} = \frac{E_3}{6} \) d) Largest Nuclear Binding Energy per Nucleon

Compare binding energy per nucleon and identify isotope

Compare the values of \( E_{1n}, E_{2n}, \) and \( E_{3n} \). The isotope with the largest nuclear binding energy per nucleon is the one with the highest value. e) Check with Trends Plotted

Compare calculated values with Figure 21.12

Look at Figure 21.12 and check if the results from the calculations agree with the trends plotted. If the isotope with the largest nuclear binding energy per nucleon is also the one with the highest value in the graph, then the calculated values agree with the trends plotted.

Key concepts

Mass Defect

The term 'mass defect' is foundational in nuclear physics and pertains to the discrepancy between the predicted mass of a nucleus (assuming it’s just the sum of separate protons and neutrons) and its actual mass. If you could 'weigh' a nucleus on an incredibly precise scale, it would be lighter than the combined mass of the protons and neutrons that constitute it. The lost mass is what we call the mass defect.

This mass hasn't just disappeared—it's been converted into energy that binds the nucleus together, known as nuclear binding energy. This process illustrates the principle of mass-energy equivalence, where mass can be converted into energy and vice versa. The mass defect is crucial for understanding the stability of a nucleus and the energy it holds within its atomic bonds.

Understanding how the mass defect is calculated can enhance a student's grasp of nuclear binding energy and the forces at play within an atom's nucleus.

Binding Energy per Nucleon

Binding energy per nucleon gives us insight into the stability of a nucleus. It’s the average energy needed to remove a nucleon (either a proton or a neutron) from a nucleus. A higher value indicates a more stable and tightly bound nucleus. To find the binding energy per nucleon, we divide the total nuclear binding energy by the number of nucleons in the nucleus.

For example, through careful calculations as provided in the exercise, we can identify which isotopes are more stable. Helium-4 often has one of the highest known binding energy per nucleon values, reflecting its remarkable stability as a nucleus. By comparing the binding energy per nucleon among different isotopes, scientists can predict the relative stability and potential for nuclear reactions such as fission or fusion.

Einstein's Mass-Energy Equivalence

Einstein’s famous mass-energy equivalence formula, represented as E=mc^2, is at the heart of nuclear physics. This revolutionary concept tells us that mass (m) can be converted into an equivalent amount of energy (E), and the 'exchange rate' is provided by the square of the speed of light (c^2). Here, 'c' is a constant representing the speed of light in a vacuum, approximately equal to 299,792,458 meters per second.

When we talk about nuclear binding energy, we're indeed discussing the energy equivalent of the mass defect. The conversion of mass into energy within the atom’s nucleus, as per this equivalence, is what powers the sun and various technologies on Earth, including nuclear power plants. It also explains why, despite their microscopic scale, nuclear reactions release such a significant amount of energy.

Atomic Mass Unit (amu)

In nuclear physics, it's essential to have a standard unit to measure the masses of atoms and subatomic particles. This unit is the atomic mass unit (amu), which is defined as one twelfth of the mass of a carbon-12 atom. An amu is a tiny quantity equivalent to approximately 1.66054 x 10^{-27} kilograms. For practicality, subatomic particles such as protons and neutrons have their masses expressed in amu, which simplifies the comparison of different atoms and isotopes.

Understanding the concept of amu makes it easier to grasp topics like mass defect and binding energy, as it allows for a convenient way to discuss the minuscule masses of particles that would otherwise require unwieldy scientific notation in kilograms.

Isotopes

Isotopes are variations of a chemical element that have the same number of protons but different numbers of neutrons. This difference in neutron count leads to varying atomic masses within the same element. For instance, hydrogen has three common isotopes, each displaying different properties due to their different masses: the most common hydrogen atom with just one proton, deuterium with one proton and one neutron, and tritium with one proton and two neutrons.

Isotopes provide a rich area of study because their differing mass defects and binding energies lead to a wide array of nuclear characteristics and stability. This diversity is vital for many scientific and medical applications, ranging from dating archaeological findings with Carbon-14 to treating cancer with certain radioactive isotopes.