Problem 54 - Chemistry

Question

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

Step-by-Step Solution

Answer

The atomic mass of magnesium-27 is approximately \(4.4335 * 10^{-26} kg\).

Step 1: Calculate the total binding energy

First, we need to determine the total binding energy of magnesium-27. Since there are 27 nucleons in Mg-27, we can multiply the binding energy per nucleon by the number of nucleons: Total binding energy = Binding energy per nucleon × Number of nucleons Total binding energy = \(1.326 * 10^{-12} J/nucleon * 27\) Total binding energy = \(3.5802 * 10^{-11} J\)

Step 2: Calculate the mass defect

Next, we'll calculate the mass defect (i.e., the difference between the sum of nucleon masses and the atomic mass of magnesium-27) using the mass-energy equivalence formula: Mass defect = Total binding energy / \(c^2\) Mass defect = \(\frac{3.5802 * 10^{-11} J}{(3 * 10^8 m / s)^2}\) Mass defect = \(3.989 * 10^{-28} kg\)

Step 3: Calculate the atomic mass of magnesium-27

Now, we'll determine the atomic mass of magnesium-27 by adding the mass defect to the sum of the individual nucleon masses. Since magnesium has 12 protons and 15 neutrons, we need to account for their respective masses: - Mass of proton = \(1.6726 * 10^{-27} kg\) - Mass of neutron = \(1.6749 * 10^{-27} kg\) Sum of nucleon masses = (12 * Mass of proton) + (15 * Mass of neutron) Sum of nucleon masses = (\(12 * 1.6726 * 10^{-27} kg\)) + (\(15 * 1.6749 * 10^{-27} kg\)) Sum of nucleon masses = \(4.4733 * 10^{-26} kg\) Now we can calculate the atomic mass of magnesium-27: Atomic mass of Mg-27 = Sum of nucleon masses - Mass defect Atomic mass of Mg-27 = \(4.4733 * 10^{-26} kg - 3.989 * 10^{-28} kg\) Atomic mass of Mg-27 = \(4.4335 * 10^{-26} kg\) Therefore, the atomic mass of magnesium-27 is approximately \(4.4335 * 10^{-26} kg\).

Key Concepts

Understanding Atomic Mass

Atomic mass is a crucial concept when studying atoms and their components. It is essentially the mass of an atom, typically expressed in atomic mass units (amu) or kilograms for scientific calculations. An atomic mass unit is about one-twelfth the mass of a carbon-12 atom, which helps in comparing different elements.

To calculate the atomic mass of magnesium-27, we first need to understand that it includes the sum of the masses of its protons, neutrons, and electrons. However, since the mass of electrons is negligible compared to nucleons, they can be ignored in precision calculations.

• Atomic mass formula: It includes the masses of protons and neutrons in the nucleus.
• Calculation: For Mg-27, you would sum up 12 proton masses and 15 neutron masses, then adjust this value by the mass defect.

By understanding atomic mass, one can better grasp how elements react and bond with each other.

Exploring Mass Defect

Mass defect is an intriguing concept that explains why the sum of an atom's parts seems greater than its whole. It represents the apparent "loss" of mass when nucleons bind to form a nucleus. This difference manifests because some mass is converted to energy, the "binding energy," which holds the nucleus together.

This phenomenon is explained by Einstein's mass-energy equivalence principle, where the binding energy released can be expressed as a drop in mass:

• Calculate mass defect: Use the formula \( \text{Mass defect} = \frac{\text{Binding energy}}{c^2} \), where \( c \) is the speed of light.
• Importance: It helps to understand the stability of an atom; a greater mass defect generally means a more stable nucleus.

The mass defect shows how energy transformations occur at an atomic level and gives insight into nuclear reactions.

Nucleons: The Building Blocks of the Nucleus

Nucleons are the particles residing within an atom's nucleus, consisting of protons and neutrons. These particles are pivotal to the structure and stability of atoms.

The protons carry a positive charge, while neutrons are neutral. Their presence and ratio in the nucleus determine the identity and stability of an element.

• Protons: They determine the atomic number and thus the element's identity. For magnesium, 12 protons mean it is element number 12.
• Neutrons: These contribute to the mass of an atom and influence its stability; more neutrons can lead to isotope formation.

The concept of nucleons helps us understand atomic mass, nuclear reactions, and element properties, underscoring their foundational role in chemistry and physics.