Problem 24 - Fundamentals of Physics

Question

A penny has a mass of 3.0 g. Calculate the energy that would be required to separate all the neutrons and protons in this coin from one another. For simplicity, assume that the penny is made entirely of \({ }^{63} \mathrm{Cu}\) atoms (of mass \(62.92960 \mathrm{u}\) ). The masses of the proton-plus-electron and the neutron are \(1.00783 \mathrm{u}\) and \(1.00866 \mathrm{u}\), respectively.

Step-by-Step Solution

Answer

The energy required to separate all the neutrons and protons in the penny is calculated using the binding energy per copper atom and the total number of copper atoms in 3.0 g.

Step 1: Calculate the number of Copper Atoms

First, determine the number of copper atoms in the penny by converting its mass to atomic mass units (amu). Given the mass of the penny is 3.0 grams and the mass of a single copper atom is 62.92960 amu, first convert grams to amu using the conversion factor 1 amu = 1.66053906660 × 10^{-24} g. Thus, the total number of copper atoms is calculated as follows:\[ \text{Number of copper atoms} = \frac{3.0 \text{ g}}{62.92960 \text{ u} \times 1.66053906660 \times 10^{-24} \text{ g/u}} \approx \text{number of moles} \times \text{Avogadro's number}. \]

Step 2: Determine Atomic Structure of Copper

Assume that each {}^{63} Cu atom consists of 29 protons and (63 - 29 = 34) neutrons. This gives the total nucleon number as 63, as specified by the atomic and mass number of {}^{63} Cu.Thus:\[ \text{Protons per atom} = 29 \]\[ \text{Neutrons per atom} = 34 \]

Step 3: Calculate Mass Defect for One Copper Atom

The mass defect can be calculated using the sum of the individual nucleon masses minus the given atomic mass of the copper atom. The mass defect \(\Delta m\) is:\[ \Delta m = \left(29 \times 1.00783 \text{ u} + 34 \times 1.00866 \text{ u}\right) - 62.92960 \text{ u} \]Calculate \(\Delta m\) to find the mass defect.

Step 4: Calculate Binding Energy for One Copper Atom

The binding energy \(E\) per copper atom can be found using Einstein’s equation \(E = \Delta m \times c^2\), where \(c\) is the speed of light \(3.00 \times 10^8 \text{ m/s}\):\[ E = \Delta m \cdot (931.5 \text{ MeV/u}) \]Note: to convert from amu to MeV, multiply by 931.5.

Step 5: Calculate Total Binding Energy for All Atoms in the Penny

Multiply the binding energy per copper atom by the total number of copper atoms from Step 1 to get the total binding energy for the penny:\[ \text{Total Energy} = E \times \text{Number of Copper Atoms} \]Insert the derived values to obtain the total energy in MeV.

Key Concepts

Mass Defect

Mass defect is the difference between the calculated mass of an atom's protons and neutrons and its actual atomic mass. When we break down an atom's nuclear formation, we find that the sum of all its individual nucleons (protons and neutrons) has a slightly higher mass than the actual measured atomic mass. This missing mass is crucial because it has been converted into binding energy. Hence, mass defect reflects the energy needed to keep the nucleons bound in the nucleus.To calculate the mass defect for a single copper atom, we sum the mass of its 29 protons and 34 neutrons, each with their respective masses. The expression for mass defect \( \Delta m \) is:\[ \Delta m = (29 \times 1.00783 \text{ u} + 34 \times 1.00866 \text{ u}) - 62.92960 \text{ u} \]This value is indicative of how much mass has been converted into the energy that holds the nucleus together.

Atomic Structure

At the heart of atomic theory lies the atomic structure, a detailed arrangement of protons, neutrons, and electrons within an atom. For {}^{63}Cu, understanding its structure aids in solving binding energy problems. It comprises 63 nucleons, 29 of which are protons, presenting a positively charged environment within the nucleus. The remaining 34 neutrons contribute to the stability of the atom by providing a neutral charge that helps offset potential proton repulsion. Copper, being a transition metal, holds significant importance in both chemistry and physics due to its well-defined isotopic enrichment. This particular isotope, {}^{63}Cu, is the predominant form of copper and helps in understanding typical atomic interactions and energy calculations. Knowing the structure is vital for calculating not only mass defect but also the total energy required to disassemble the nucleus into its proton and neutron components.

Binding Energy Calculation

Binding energy is the energy required to disassemble a nucleus into its separate protons and neutrons. It is a direct outcome of the mass defect since the 'missing' mass is converted into energy that holds the nucleus together. We calculate the binding energy of a copper atom using Einstein's famous equation \( E = \Delta m \times c^2 \).However, the conversion involves units typically expressed in terms of energy per nucleon. Therefore, in practice, we use:\[ E = \Delta m \cdot (931.5 \text{ MeV/u}) \]This equation simplifies our task of relating atomic mass units (amu) to mega-electron volts (MeV). Each copper atom's binding energy indicates how tightly the nucleus is held together, reflecting the energy required to forcefully disassemble it into constituent particles.

Energy Conversion

Energy conversion is the process of converting one form of energy into another. In nuclear physics, mass and energy are interchangeable, a concept famously expressed in Einstein's equation \( E = mc^2 \). For nuclear binding energy calculations, this principle allows us to convert the mass defect into energy, typically in units of MeV (mega-electron volts).The conversion factor plays a crucial role: \( 1 \text{ amu} = 931.5 \text{ MeV/c}^2 \). This tells us the amount of energy equivalent to one atomic mass unit at the speed of light squared, making it universally applicable across nuclear reactions.In practical scenarios, for instance, when calculating the total energy required to separate all copper atoms in a penny, we account for this conversion with precision. By understanding energy conversion, students grasp how minuscule mass defects amount to significant energy in nuclear science, illuminating the potential power harnessed in nuclear reactions.