Question

Calculate the mass defect for the formation of an oxygen- 16 nucleus in both grams and \(\mathrm{g} / \mathrm{mol}\), and calculate the binding energy in both \(\mathrm{MeV} /\) nucleon and \(\mathrm{kJ} / \mathrm{mol.}\) The mass of an \({ }^{16} \mathrm{O}\) atom is \(15.99491 \mathrm{u}\).

Short answer

Mass defect: 0.13245 u or 2.1996 x 10^-25 g. Binding energy: 7.713 MeV/nucleon, 1.9145 x 10^13 kJ/mol.

Step-by-step solution

Understanding the Concept of Mass Defect

Mass defect refers to the difference between the mass of the individual nucleons (protons and neutrons) when they are free and the actual mass of the nucleus. This difference represents how much mass is converted into energy to bind the nucleons together.

Calculating the Theoretical Mass of Oxygen-16's Nucleons

An oxygen-16 nucleus consists of 8 protons and 8 neutrons. The mass of a proton ( p) is approximately 1.00728 u, and the mass of a neutron ( n) is approximately 1.00866 u. The theoretical mass of the nucleons is:\[8 \times 1.00728 \, \text{u} + 8 \times 1.00866 \, \text{u} = 16.12736 \, \text{u}\]

Calculating the Mass Defect

Subtract the actual mass of the oxygen-16 nucleus from the theoretical mass of its nucleons:\[\Delta m = 16.12736 \, \text{u} - 15.99491 \, \text{u} = 0.13245 \, \text{u}\]

Converting Mass Defect to Grams

To convert atomic mass units (u) to grams, use the conversion factor:\[\text{1 u} = 1.66054 \times 10^{-24} \, \text{g}\]Thus:\[\Delta m = 0.13245 \, \text{u} \times 1.66054 \times 10^{-24} \, \text{g/u} = 2.1996 \times 10^{-25} \, \text{g}\]

Converting Mass Defect to Grams per Mole

There are Avogadro's number ( N_A = 6.022 \times 10^{23} mol^{-1}) of units in a mole, so:\[\Delta m_{\text{per mole}} = 0.13245 \, \text{u/mol} \times 1.66054 \times 10^{-24} \, \text{g/u} \times 6.022 \times 10^{23} \, \text{mol}^{-1} = 0.022 \times 10^{-1} \, \text{g/mol}\]

Calculating Binding Energy in MeV per Nucleon

The energy equivalent of 1 u is 931.5 MeV. Therefore, the binding energy is:\[E = 0.13245 \, \text{u} \times 931.5 \, \text{MeV/u} = 123.41 \, \text{MeV}\]Divide by the number of nucleons (16):\[E_{\text{per nucleon}} = \frac{123.41 \, \text{MeV}}{16} = 7.713 \, \text{MeV/nucleon}\]

Calculating Binding Energy in kJ per Mole

To convert MeV to joules use 1 MeV = 1.60218 × 10^{-13} J, and further to kJ: \[123.41 \, \text{MeV} \times 1.60218 \times 10^{-13} \, \text{J/MeV} \times 6.022 \times 10^{23} \, \text{mol}^{-1} \times 10^{-3} \, \text{kJ/J} = 1.9145 \times 10^{13} \, \text{kJ/mol}\]

Key concepts

binding energy

Binding energy is an essential concept in nuclear physics that refers to the energy required to separate a nucleus into its individual nucleons: protons and neutrons. This energy is a result of the mass defect, which is the difference between the total mass of these separate particles and the actual mass of the nucleus itself.

When nucleons combine to form a nucleus, part of their mass is converted into binding energy, according to Einstein's famous equation, \(E=mc^2\). This energy conversion explains why the nucleus weighs less than the sum of its parts. The binding energy quantifies the stability of the nucleus: the higher the binding energy, the more stable the nucleus.

The binding energy is often measured in MeV (mega-electron volts) per nucleon or in kJ/mol. This conversion involves several steps, including the calculation of mass defect in atomic mass units (u) and converting this into energy using a known conversion factor of 1 u being equivalent to 931.5 MeV. Finally, it can be calculated per nucleon or per mole to compare the stability of different nuclei universally.

oxygen-16 nucleus

The oxygen-16 nucleus is a specific isotope of oxygen that consists of 8 protons and 8 neutrons. Its stability and particular mass characteristics make it an interesting subject of study in nuclear chemistry and physics.

The mass of an oxygen-16 nucleus is slightly less than the sum of the free nucleons' masses due to the mass defect, which is converted into binding energy. This indicates how tightly the nucleons are bound together, contributing to the nucleus's overall stability.

Understanding the oxygen-16 nucleus provides insights into nuclear structure and reactions. Its study helps researchers explore the principles of nuclear forces that hold a nucleus together. Additionally, it's a stepstone to understanding larger and more complex nuclei structures found in heavier elements.

atomic mass units

Atomic mass units (u) are used to express atomic and molecular masses. It provides a convenient scale for dealing with the very tiny masses of atoms and subatomic particles. One atomic mass unit is defined as one-twelfth of the mass of an isolated carbon-12 atom, which makes it approximately equal to 1.66054 × 10^-24 grams.

In nuclear physics, atomic mass units serve a critical purpose in calculating the mass defect and consequently the binding energy. By knowing the mass of nucleons in atomic mass units, we can derive the theoretical mass of a nucleus and then compute any mass defect when compared to the actual nucleus mass.

This system of measurement is crucial because it allows scientists and students alike to handle atomic-scale phenomena with manageable numbers, facilitating easier calculation and discussion in the field of chemistry and physics.

Avogadro's number

Avogadro's number is a fundamental constant that represents the number of constituent particles, usually atoms or molecules, in one mole of a given substance. It is numerically represented as 6.022 × 10^23 mol^{-1}.

In the context of mass defect and binding energy, Avogadro's number facilitates the conversion of atomic-scale measurements into macroscopic measurements that we can more easily grasp. For instance, when converting a mass defect from atomic mass units to grams per mole, Avogadro's number is used to scale the calculation from a single atom to a bulk quantity, enabling practical use in real-world chemistry and physics applications.

Understanding Avogadro's number is essential not only in nuclear physics but across all fields of chemistry and material science, as it ties the microscopic world of atoms to the macroscopic world we observe and measure.