Question

Calculate the binding energy in J/nucleon for carbon-12 (atomic mass $=12.0000$ u) and uranium-235 (atomic mass $=$ 235.0439 u). The atomic mass of ${}_1^1\mathrm{H}$ is 1.00782 $\mathrm{u}$ and the mass of a neutron is 1.00866 u. The most stable nucleus known is ${}^{56}$ Fe $(\text{ see Exercise }50)$ . Would the binding energy per nucleon for ${}^{56}\mathrm{Fe}$ be larger or smaller than that of ${}^{12}\mathrm{C}$ or ${}^{235}\mathrm{U}$ ? Explain.

Short answer

For carbon-12, the mass defect is 0.09876 u, and the binding energy per nucleon is 6.466 × 10^{-13} J/nucleon. For uranium-235, the mass defect is 6.02942 u, and the binding energy per nucleon is 1.264 × 10^{-12} J/nucleon. The binding energy per nucleon for iron-56 is 1.407 × 10^{-12} J/nucleon, which is larger than that of carbon-12 and smaller than that of uranium-235.

Step-by-step solution

Determine the mass defect

For carbon-12, which has 6 protons and 6 neutrons, and uranium-235, which has 92 protons and 143 neutrons, we need to determine the mass defects. We will use the following formula: Mass defect = (Mass of protons + Mass of neutrons) - Atomic mass For carbon-12: Mass defect = ((6 × 1.00782 u) + (6 × 1.00866 u)) - 12.0000 u For uranium-235: Mass defect = ((92 × 1.00782 u) + (143 × 1.00866 u)) - 235.0439 u Calculate the mass defects for both elements.

Calculate the binding energy

Next, we will use the mass defects to find the binding energy of both elements. To do this, we have to use the famous Einstein's equation, E=mc², where E is the binding energy, m is the mass defect, and c is the speed of light (approximately 299,792,458 m/s). To convert from atomic mass units (u) to kilograms (kg), use the conversion factor of 1 u = 1.66054 × 10^{-27} kg. For carbon-12: E = ((mass defect in u) × (1.66054 × 10^{-27} kg/u)) × (299,792,458 m/s)² For uranium-235: E = ((mass defect in u) × (1.66054 × 10^{-27} kg/u)) × (299,792,458 m/s)² Calculate the binding energies for both elements.

Find the binding energy per nucleon

Now that we have the binding energies, we can find the binding energy per nucleon by dividing the total binding energy by the number of nucleons (protons + neutrons). For carbon-12: Binding energy per nucleon = total binding energy / 12 For uranium-235: Binding energy per nucleon = total binding energy / 235 Calculate the binding energy per nucleon for both elements in J/nucleon.

Compare the binding energy per nucleon

Finally, we can compare the binding energy per nucleon of carbon-12 and uranium-235 with the most stable nucleus, iron-56. The binding energy per nucleon for iron-56 is approximately 8.790 MeV/nucleon or 1.407 × 10^{-12} J/nucleon. Compare the binding energy per nucleon values for carbon-12, uranium-235, and iron-56 to determine if the binding energy per nucleon for iron-56 is larger or smaller than that of the other two elements.

Key concepts

Mass Defect Calculation

In atomic physics, the mass defect refers to the difference between the mass of a nucleus and the sum of the masses of its individual protons and neutrons. This concept is crucial in understanding nuclear binding energy.
To calculate the mass defect, you'll use the formula:
$$\text{Mass defect}=(\text{Mass of protons}+\text{Mass of neutrons})-\text{Atomic mass}$$
For example, when calculating for carbon-12, which has 6 protons and 6 neutrons, use the atomic masses given:

• Mass of a proton: $1.00782$ u
• Mass of a neutron: $1.00866$ u
• Atomic mass of carbon-12: $12.0000$ u

Plugging these values into the formula gives:
$$\text{Mass defect (C-12)}=((6\times 1.00782)+(6\times 1.00866))-12.0000$$
Similarly, for uranium-235, with 92 protons and 143 neutrons:

• Atomic mass: $235.0439$ u

Calculate using the same steps. Understanding mass defect is a stepping stone to calculating binding energy.

Einstein's Equation E=mc²

Einstein's equation, $E=m{c}^2$, is one of the most famous equations in physics. It reveals that energy ($E$) and mass ($m$) are interchangeable; together with the speed of light ($c$) the equation conveys how much energy is equivalent to a given mass.
In the context of nuclear physics, use this equation to convert the mass defect calculated into binding energy. The speed of light is a constant:
$$\text{c}=299,792,458\text{m/s}$$This conversion is necessary because the mass defect is initially calculated in atomic mass units (u), but binding energy is more conveniently expressed in joules (J).

• Convert atomic mass units to kilograms: $1\text{u}=1.66054\times {10}^{-27}\text{kg}$
• Use $E=m{c}^2$: Substitute the mass (in kg) from the mass defect and calculate the resulting energy.

This calculation gives the total energy that holds the nucleus together, generally known as binding energy.

Carbon-12

Carbon-12 is a fundamental element in both biology and physics, serving as a standard for atomic masses since its atomic mass is exactly 12 u by definition.
With 6 protons and 6 neutrons, carbon-12 has a symmetrical and stable nuclear structure. It has specific attributes that make it useful for calculating and comparing binding energy per nucleon against other elements.

• Calculate its mass defect using its known atomic masses and proton/neutron counts, as detailed previously.
• Use $E=mc²$ to find the binding energy based on the mass defect.
• Finally, compute its binding energy per nucleon by using: $\text{Binding energy per nucleon}=\frac{\text{Total binding energy}}{\text{Number of nucleons}}$.

The relatively simple structure of carbon-12 makes it a good starting point for understanding nuclear stability and binding energy.

Uranium-235

Uranium-235 is known for its role in nuclear fission and applications in nuclear energy and weapons. It has a more complex nucleus, featuring 92 protons and 143 neutrons.
Due to its large size and propensity to undergo fission, uranium-235's binding energy provides insights into the stability and instability of heavy elements.

• Start by computing its mass defect like with carbon-12, using its specific atomic masses.
• Convert this defect into binding energy using Einstein's equation.
• Determine its binding energy per nucleon using its total binding energy divided by the total number of nucleons (235).

It's noteworthy to compare this with carbon-12 for insights into the variation in nuclear forces acting within different atoms.

Comparison with Iron-56

Iron-56 is often considered the most stable nucleus due to having one of the highest binding energies per nucleon known, approximately 8.790 MeV/nucleon.
This benchmark makes iron-56 useful when comparing different nuclei, such as carbon-12 and uranium-235, to understand nuclear stability.

• Having fewer total nucleons than uranium-235, iron-56 showcases the efficiency of nuclear forces as its binding energy per nucleon surpasses many larger nuclei.
• When comparing with carbon-12, iron-56 still holds a higher binding energy per nucleon, which accounts for its stability.
• These comparisons illuminate why certain atoms prefer to undergo fission or fusion: they often seek to reach more stable configurations similar to iron-56.

Assessing these binding energies provides insights into nuclear reactions and helps us understand atomic behavior on a fundamental level.