Question

The nuclear masses of \({ }^{7} \mathrm{Be},{ }^{9} \mathrm{Be}\), and \({ }^{10} \mathrm{Be}\) are \(7.0147,9.0100\), and 10.0113 amu, respectively. Which of these nuclei has the largest binding energy per nucleon?

Short answer

The largest binding energy per nucleon is for the \({ }^{7} \mathrm{Be}\) nucleus, with a value of approximately -1.954 MeV/nucleon.

Step-by-step solution

Calculate the mass number for each nucleus

To calculate the mass number for each nucleus, simply use the superscript in the given symbols: $${ }^{7} \mathrm{Be} \Rightarrow A_{1} = 7$$ $${ }^{9} \mathrm{Be} \Rightarrow A_{2} = 9$$ $${ }^{10} \mathrm{Be} \Rightarrow A_{3} = 10$$

Calculate the mass defect for each nucleus

The mass defect can be calculated using the mass number (A) and the nuclear mass (M) provided in the problem. We use the formula: Mass defect = [Number of nucleons (A)] x [mass of the individual nucleons (m)] - [Total mass of nucleus (M)]. The individual mass of a nucleon is approximately 1 amu. $$\Delta m_{1} = A_{1} \times m - M_{1} = 7 \times 1 - 7.0147 \approx -0.0147 \text{ amu}$$ $$\Delta m_{2} = A_{2} \times m - M_{2} = 9 \times 1 - 9.0100 \approx -0.0100 \text{ amu}$$ $$\Delta m_{3} = A_{3} \times m - M_{3} = 10 \times 1 - 10.0113 \approx -0.0113 \text{ amu}$$

Calculate the binding energy for each nucleus

Use the mass defect to calculate the binding energy (BE) in units of mega electron volts (MeV), by using the formula: Binding energy = Mass defect (Δm) x c², where c represents the speed of light, and c² = 931.49 MeV/amu. $$\text{BE}_{1} = \Delta m_{1} \times c^{2} \approx -0.0147 \times 931.49 \approx -13.68 \text{ MeV}$$ $$\text{BE}_{2} = \Delta m_{2} \times c^{2} \approx -0.0100 \times 931.49 \approx -9.31 \text{ MeV}$$ $$\text{BE}_{3} = \Delta m_{3} \times c^{2} \approx -0.0113 \times 931.49 \approx -10.53 \text{ MeV}$$

Calculate the binding energy per nucleon for each nucleus

Divide the binding energy of the nucleus by the total number of nucleons (mass number) to find the binding energy per nucleon. $$\frac{\text{BE}_{1}}{A_{1}} = \frac{-13.68}{7} \approx -1.954 \text{ MeV/nucleon}$$ $$\frac{\text{BE}_{2}}{A_{2}} = \frac{-9.31}{9} \approx -1.034 \text{ MeV/nucleon}$$ $$\frac{\text{BE}_{3}}{A_{3}} = \frac{-10.53}{10} \approx -1.053 \text{ MeV/nucleon}$$

Determine which nucleus has the largest binding energy per nucleon

Compare the binding energy per nucleon values we calculated in the previous step to determine which nuclei have the largest value. $${ }^{7} \mathrm{Be} \rightarrow -1.954 \text{ (MeV/nucleon)}$$ $${ }^{9} \mathrm{Be} \rightarrow -1.034 \text{ (MeV/nucleon)}$$ $${ }^{10} \mathrm{Be} \rightarrow -1.053 \text{ (MeV/nucleon)}$$ The largest binding energy per nucleon is for the \({ }^{7} \mathrm{Be}\) nucleus, with a value of approximately -1.954 MeV/nucleon.

Key concepts

Mass Defect Calculation

Let's start by understanding what a mass defect is. It's the difference between the calculated mass of separated nucleons and the actual mass of a nucleus. Essentially, it accounts for the 'missing' mass that's converted into binding energy - the energy holding the nucleus together. To calculate the mass defect, we use a simple formula:
Mass defect = (Total mass of separated nucleons) - (Actual mass of the nucleus)
For a single nucleon, the approximate mass is 1 atomic mass unit (amu). Thus, if we have a nucleus with a mass number A (the sum of protons and neutrons), we expect its mass to be approximately A amu if there were no binding energy. However, due to the strong nuclear force joining these nucleons, some mass is converted into energy, resulting in the actual mass being slightly less.
In the exercise provided, we calculate the mass defect with given masses of various isotopes of Beryllium. By subtracting the nuclear mass from the product of the number of nucleons and the mass of an individual nucleon, we get a negative number which signifies a mass defect, indicating the binding energy of the nucleus.

Binding Energy Calculation

Now, let's explore the binding energy calculation. The binding energy is vital because it provides insight into the stability of a nucleus. This energy correlates with the mass defect through Albert Einstein’s famous equation, E=mc², where E is energy, m is mass, and c is the speed of light. In nuclear physics, we convert the mass defect (in amu) into energy (in mega electron volts, MeV) using the conversion factor 931.49 MeV/amu. The formula to calculate the binding energy from the mass defect is:
Binding energy (BE) = Mass defect (Δm) × c²
The results give us the energy that would be required to disassemble the nucleus into separate protons and neutrons. It's a profound indicator of the nucleus' stability; the higher the binding energy, the more energy needed to break apart the nucleus, signifying greater stability. By applying this formula to the mass defects of Beryllium isotopes, we’ve determined their respective binding energies.

Nuclear Mass

The term nuclear mass refers to the total mass of a nucleus, and it's lower than the sum of individual masses of protons and neutrons due to the mass defect. When we look at a periodic table, the mass of an atom is reported in atomic mass units (amu); however, this includes the mass of electrons which is negligible when considered on the nuclear scale. The nuclear mass is significant in nuclear physics as it helps us understand nuclear reactions and stability. The nuclear mass is crucial in calculating the mass defect and, subsequently, the binding energy. In our example, the nuclear masses of Beryllium isotopes are given. These values are essential for computation as they represent the actual measured masses of the isotopes and are the starting point for determining both the mass defect and the binding energy.