Problem 41 - Chemistry: Principles and Reactions

Question

Which has the larger binding energy, fluorine- 19 or oxygen-17?

Step-by-Step Solution

Answer

Answer: Fluorine-19 (F-19) has a larger binding energy than oxygen-17 (O-17).

Step 1: Find the mass defect for F-19 and O-17

To find the mass defect, we need to know the masses of F-19, O-17, protons, and neutrons. The atomic masses of F-19 and O-17 are approximately 18.9984u and 16.9991u, respectively. The masses of a proton and a neutron are approximately 1.0073u and 1.0087u, respectively (u stands for atomic mass units). For F-19, there are 9 protons and 10 neutrons. The total mass is: Total mass of F-19 constituents = (9 × 1.0073u) + (10 × 1.0087u) = 19.1534u The mass defect of F-19 is: Mass defect of F-19 = Total mass of F-19 constituents - mass of F-19 = 19.1534u - 18.9984u = 0.155u For O-17, there are 8 protons and 9 neutrons. The total mass is: Total mass of O-17 constituents = (8 × 1.0073u) + (9 × 1.0087u) = 17.1308u The mass defect of O-17 is: Mass defect of O-17 = Total mass of O-17 constituents - mass of O-17 = 17.1308u - 16.9991u = 0.1317u

Step 2: Convert mass defect to energy

To convert the mass defect to energy, we use Einstein's equation: E = mc^2. In this case, we only need the energy conversion factor for atomic mass units (u): 1u = 931.5 MeV/c^2. For F-19, the binding energy is: Binding energy of F-19 = Mass defect of F-19 × energy conversion factor = 0.155u × 931.5 MeV/c^2 = 144.3825 MeV For O-17, the binding energy is: Binding energy of O-17 = Mass defect of O-17 × energy conversion factor = 0.1317u × 931.5 MeV/c^2 = 122.69055 MeV

Step 3: Compare the binding energies of F-19 and O-17

Based on our calculations, the binding energies are as follows: - Binding energy of F-19: 144.3825 MeV - Binding energy of O-17: 122.69055 MeV Since the binding energy of F-19 is greater than the binding energy of O-17, fluorine-19 has a larger binding energy than oxygen-17.

Key Concepts

Mass Defect

In nuclear physics, the concept of the mass defect is crucial in understanding nuclear binding energy. The mass defect refers to the difference between the sum of the individual masses of a nucleus’s protons and neutrons, and the actual mass of the nucleus itself. This discrepancy arises because some mass is converted into energy to hold the nucleus together, according to Einstein's equation. To calculate the mass defect, you add up the masses of all protons and neutrons in a nucleus and subtract the measured atomic mass of the isotope. For example:

• For Fluorine-19: With 9 protons and 10 neutrons, the theoretical mass is approximated as 19.1534u.
• The actual mass is 18.9984u, leading to a mass defect of 0.155u.

Understanding mass defects helps in calculating the nuclear binding energy, which indicates the stability of a nucleus. A larger mass defect generally means that more energy is needed to hold the nucleus together, signifying higher nuclear stability.

Isotopes

Isotopes are different forms of the same element, with the same number of protons but varying numbers of neutrons. This difference in neutron count results in different atomic masses and, consequently, different nuclear properties such as binding energy. For instance, both Fluorine-19 and Oxygen-17 are isotopes of their respective elements, fluorine and oxygen. While Fluorine-19 has 9 protons and 10 neutrons, Oxygen-17 has 8 protons and 9 neutrons. Their differing compositions affect their mass defects and nuclear binding energies. This characteristic imparts isotopes with unique nuclear behaviors and applications, such as:

• In medical imaging through radioactive isotopes.
• In dating archaeological samples using carbon isotopes.

Isotopes allow scientists to explore the nuanced behaviors of elements beyond their standard forms, unlocking doors to various scientific areas.

Einstein's Equation

Einstein’s famous equation, \( E=mc^2 \), plays a pivotal role in understanding nuclear binding energy. It expresses the concept that mass and energy are interchangeable, hence affecting how we interpret nuclear stability.The "\( E \)" represents energy, "\( m \)" is mass, and "\( c \)" is the speed of light in a vacuum, a constant. In nuclear physics, this equation allows us to convert the mass defect from atomic mass units (u) into energy (MeV).Applying Einstein's equation in our example, for Fluorine-19:

• The mass defect of 0.155u is converted into energy by multiplying with the conversion factor, resulting in a binding energy of 144.3825 MeV.

This framework not only helps calculate nuclear binding energy but also reveals how seemingly small mass changes in the nuclear level equate to significant energy transformations, elucidating the potential of nuclear reactions for both energy generation and understanding fundamental physics.