Question

Calculate the binding energy for \(_{1}^{2} \mathrm{H}\) and \(_{1}^{3} \mathrm{H} .\) The atomic masses are \(_{1}^{2} \mathrm{H}, 2.01410 \mathrm{u} ;\) and \(^{3} \mathrm{H}, 3.01605 \mathrm{u} .\)

Short answer

The binding energy for \(_{1}^{2} \mathrm{H}\) is \(4.632*10^{-14} J\) and for \(_{1}^{3} \mathrm{H}\) is \(2.135*10^{-13} J\).

Step-by-step solution

Calculate the mass defect for \(_{1}^{2} \mathrm{H}\) and \(_{1}^{3} \mathrm{H}\)

Find the difference between the masses of the protons and neutrons in the nucleus and the actual mass of each isotope. The mass of a proton is approximately 1.00728 u and the mass of a neutron is approximately 1.00867 u. - For \(_{1}^{2} \mathrm{H}\): Mass of 1 proton + Mass of 1 neutron - Mass of \(_{1}^{2} \mathrm{H}\) - For \(_{1}^{*>}<*>3>\mathrm{H}\): Mass of 1 proton + 2 * Mass of 1 neutron - Mass of \(_{1}^{3} \mathrm{H}\)

Calculate the mass defect for each isotope

- Mass defect of \(_{1}^{2} \mathrm{H}\): \(\Delta m_1 = 1.00728 \mathrm{u} + 1.00867 \mathrm{u} - 2.01410 \mathrm{u} \Rightarrow \Delta m_1 = 0.00185 \mathrm{u}\) - Mass defect of \(_{1}^{3} \mathrm{H}\): \(\Delta m_2 = 1.00728 \mathrm{u} + 2 * 1.00867 \mathrm{u} - 3.01605 \mathrm{u} \Rightarrow \Delta m_2 = 0.00857 \mathrm{u}\)

Convert mass defect to binding energy

We will now use Einstein's equation to convert the mass defect (\(\Delta m\)) into energy. The equation is \(E=mc^2\) where \(E\) is the energy, \(m\) is the mass, and \(c\) is the speed of light. The speed of light, \(c = 2.998 * 10^8 ms^{-1}\). We need to convert the mass defect to kg first. We know that \(1 \mathrm{u} = 1.66054 * 10^{-27} kg\). Then we will find the binding energy for each isotope by multiplying their mass defect by \(c^2\).

Calculate the binding energy for each isotope

- Binding energy of \(_{1}^{2} \mathrm{H}\): \(E_1 = \Delta m_1 * c^2 \Rightarrow E_1 = (0.00185 \mathrm{u}) * (1.66054 * 10^{-27} kg/u) * (2.998 * 10^8 m/s)^2 \Rightarrow E_1 = 4.632 *10^{-14} J\) - Binding energy of \(_{1}^{3} \mathrm{H}\): \(E_2 = \Delta m_2 * c^2 \Rightarrow E_2 = (0.00857 \mathrm{u}) * (1.66054 * 10^{-27} kg/u) * (2.998 * 10^8 m/s)^2 \Rightarrow E_2 = 2.135 *10^{-13} J\) The binding energy for \(_{1}^{2} \mathrm{H}\) is \(4.632*10^{-14} J\) and for \(_{1}^{3} \mathrm{H}\) is \(2.135*10^{-13} J\).

Key concepts

Mass Defect

The mass defect is a crucial concept in understanding binding energy in nuclear physics. It refers to the discrepancy between the mass of a composite nucleus and the sum of the individual masses of its protons and neutrons. This difference arises because some mass is converted into energy to hold the nucleus together.
To determine the mass defect, we start by calculating the total mass of the protons and neutrons present in the nucleus. Take, for example, deuterium \(_{1}^{2} \mathrm{H}\) comprising 1 proton and 1 neutron, while tritium \(_{1}^{3} \mathrm{H}\) has 1 proton plus 2 neutrons. Each proton and neutron has a mass of about 1.00728 u and 1.00867 u, respectively.

• For \(_{1}^{2} \mathrm{H}\), the mass of components = 1.00728 u (proton) + 1.00867 u (neutron) = 2.01595 u.
• The actual observed atomic mass of \(_{1}^{2} \mathrm{H}\) is 2.01410 u.
• The mass defect is thus 2.01595 u - 2.01410 u = 0.00185 u.
• Similarly, for \(_{1}^{3} \mathrm{H}\), the mass defect is 2.02462 u + 1.00728 u - 3.01605 u = 0.00857 u.

Understanding mass defect helps explain why nuclei weigh less than the sum of their parts and how this relates to binding energy.

Proton and Neutron Mass

The mass of protons and neutrons are fundamental in calculating nuclear properties like the mass defect and binding energy. Protons and neutrons, collectively known as nucleons, are the building blocks of atomic nuclei. The proton mass is approximately 1.00728 atomic mass units (u), while the neutron is slightly heavier at about 1.00867 u. Despite their small mass, these particles are incredibly dense and compact.

• A proton carries a positive electric charge, which is balanced by an electron in neutral atoms.
• Neutrons, on the other hand, are neutral particles with no electric charge.

Because both protons and neutrons have masses around 1 u, their sum for any nucleus typically slightly exceeds the measured mass of the nucleus itself. This mismatch is what creates the mass defect, which is critical for understanding the energy release in nuclear reactions.

Einstein's Equation

Einstein's Equation \(E=mc^2\) is pivotal in converting mass defects into energy values, which contribute to our understanding of nuclear stability. This famous equation states that mass (\(m\)) and energy (\(E\)) are interchangeable; they are two sides of the same coin.

• \(E\) is the energy equivalent of the mass, \(m\).
• \(c\) represents the speed of light in a vacuum, approximately \(2.998 \, \times \, 10^8\) m/s.

To harness this concept in nuclear physics, we convert the mass defect into kilograms (using 1 u = \(1.66054 \, \times \, 10^{-27}\) kg) and then apply \(E=mc^2\) to calculate the binding energy.
For instance, using Einstein's equation allows us to find that the binding energy for deuterium \(_{1}^{2} \mathrm{H}\) is \(4.632 \, \times \, 10^{-14}\) J, highlighting how a small mass change translates into significant energy release. This principle is central to nuclear power generation and atomic weapons.