Question

The binding energy per nucleon for magnesium- 27 is \(1.326\) \(\times 10^{-12} \mathrm{~J} /\) nucleon. Calculate the atomic mass of \({ }^{27} \mathrm{Mg} .\)

Short answer

The atomic mass of Magnesium-27 can be calculated using the given binding energy per nucleon and the mass-energy equivalence equation. First, find the total binding energy by multiplying the binding energy per nucleon (\(1.326 \times 10^{-12}\) J/nucleon) by the total number of nucleons (27). Then, convert the total binding energy to mass difference using the equation \(m = \frac{E}{c^2}\), where c is the speed of light (\(3 \times 10^8\) m/s). Finally, add the mass difference to the total mass of 27 nucleons (12 protons and 15 neutrons) to find the atomic mass of Magnesium-27.

Step-by-step solution

Calculate the total binding energy

To find the total binding energy, multiply the binding energy per nucleon by the total number of nucleons. For Mg-27, there are 27 nucleons (12 protons and 15 neutrons). Total Binding Energy = Binding energy per nucleon x Total number of nucleons Total Binding Energy = \(1.326 \times 10^{-12}\) J/nucleon × 27 nucleons #Step 2: Convert the total binding energy to mass difference#

Convert binding energy to mass difference

Use the mass-energy equivalence equation \(E = mc^2\), where E is the total binding energy, m is the mass difference, and c is the speed of light. First, rearrange the equation to solve for the mass difference: m = \(\frac{E}{c^2}\) Then, plug in the values for total binding energy (E) and the speed of light (c = \(3 \times 10^8\) m/s): m = \(\frac{(1.326 \times 10^{-12}\) J/nucleon × 27 nucleons}{(3 \times 10^8 m/s)^2}\) #Step 3: Calculate the mass of Magnesium-27#

Calculate the mass of Mg-27

Now that we have the mass difference, we can find the mass of Magnesium-27 by adding the mass difference to the mass of the 27 nucleons (the sum of the individual proton and neutron masses). The mass of a proton is approximately \(1.673 \times 10^{-27}\) kg, and the mass of a neutron is approximately \(1.675 \times 10^{-27}\) kg. Since Magnesium-27 has 12 protons and 15 neutrons, we can find the total mass of 27 nucleons: Total mass of nucleons = (12 × mass of a proton) + (15 × mass of a neutron) Total mass of nucleons = (12 × \(1.673 \times 10^{-27}\) kg) + (15 × \(1.675 \times 10^{-27}\) kg) Then, add the mass difference to the total mass of nucleons to find the atomic mass of Magnesium-27: Atomic mass of Mg-27 = Total mass of nucleons + Mass difference This will give us the atomic mass of Magnesium-27 in kg.

Key concepts

Binding Energy

Binding energy is a crucial concept in nuclear physics. It represents the energy needed to separate all nucleons within a nucleus. When nucleons come together to form a nucleus, they release energy due to strong nuclear forces. This energy, known as binding energy, helps keep the nucleus stable. In the case of Magnesium-27, the problem specifies a binding energy per nucleon of \(1.326 \times 10^{-12}\) J.

• Total Binding Energy: To find the total binding energy for Mg-27, you multiply the binding energy per nucleon by the total number of nucleons (27 in this case).
• Significance: A higher binding energy per nucleon often indicates a more stable nucleus.

Understanding these calculations can deepen your knowledge about nuclear stability and the forces at play within atomic nuclei.

Mass-Energy Equivalence

The concept of mass-energy equivalence, articulated by the famous equation \(E = mc^2\), shows that mass can be converted into energy and vice versa. In this equation, \(E\) stands for energy, \(m\) is mass, and \(c\) is the speed of light (approximately \(3 \times 10^8\) m/s). This principle illustrates the profound link between mass and energy.

• Finding Mass Difference: When you rearrange \(E = mc^2\) to solve for mass \(m = \frac{E}{c^2}\), you can determine the mass difference caused by binding energy for Magnesium-27.
• Significance in Nuclear Calculations: It allows us to relate nuclear binding energy to the corresponding mass difference, thus providing insights into fundamental nuclear processes.

This step is critical in calculating the atomic mass of elements like Magnesium-27 after accounting for binding energy effects.

Nucleons

Nucleons are the building blocks of an atomic nucleus. They comprise protons and neutrons. Each isotope of an element has a specific number of nucleons, contributing to its mass and properties. For Magnesium-27, there are 27 nucleons: 12 protons and 15 neutrons.

• Protons and Neutrons: Protons have a positive charge with a mass of approximately \(1.673 \times 10^{-27}\) kg. Neutrons are neutral with a slightly greater mass of around \(1.675 \times 10^{-27}\) kg.
• Total Nucleons: These nucleons collectively define the isotope and determine its characteristics, such as atomic mass.

Understanding the role and calculation of nucleons is essential in nuclear physics to deduce properties like the mass of Magnesium-27.

Magnesium-27

Magnesium-27 is a specific isotope of the element magnesium, identified by its 27 nucleons. The nuclear configuration consists of 12 protons and 15 neutrons, impacting its stability and nuclear characteristics.

• Atomic Mass Calculation: The mass of Magnesium-27 is derived from the combined mass of these nucleons and the adjustments due to binding energy.
• Application: Isotopes like Magnesium-27 are significant in scientific fields like nuclear physics and materials science for their unique nuclear behaviors.

By understanding how to calculate the atomic mass of isotopes, you're gaining insight into the fundamental concepts of nuclear science and isotopic distinctions.