Problem 54 - Chemistry: The Central Science in SI Units, Global Edition

Question

The isotope \({ }_{28}^{62} \mathrm{Ni}\) has the largest binding energy per nucleon of any isotope. Calculate this value from the atomic mass of nickel-62 \((61.928345 \mathrm{u})\) and compare it with the value given for iron- 56 in Table 21.7 .

Step-by-Step Solution

Answer

The binding energy per nucleon for nickel-62 can be calculated using the formula: \(Binding \ Energy = (Z * m_p + (A - Z) * m_n - M) * c^2\), where Z = 28, A = 62, m_p = 1.007276 u, m_n = 1.008664 u, and M = 61.928345 u. After calculating the binding energy, we can find the binding energy per nucleon by dividing it by the total number of nucleons (A). Finally, compare this value to the binding energy per nucleon for iron-56 given in Table 21.7.

Step 1: Recall the binding energy formula and the relevant constants

To calculate the binding energy, we need to use the formula: Binding Energy = (Z * m_p + (A - Z) * m_n - M) * c^2 where Z = number of protons, A = number of nucleons, m_p = mass of a proton (1.007276 u), m_n = mass of a neutron (1.008664 u), M = atomic mass of the isotope (given as 61.928345 u for nickel-62), c = speed of light (approximately 2.99792458 * 10^8 m/s, but we will only require the square of c which is c^2 = 8.98755179 * 10^16 m^2/s^2), and 1 u = 931.494 MeV/c^2.

Step 2: Calculate the number of protons and neutrons for nickel-62

The isotope notation indicates that nickel-62 has an atomic number (Z) of 28, which means it has 28 protons. To find the number of neutrons (A - Z), we subtract the number of protons from the total number of nucleons, which is also given in the isotope notation as 62. Number of neutrons = A - Z = 62 - 28 = 34

Step 3: Calculate the binding energy of nickel-62

Now that we have all the relevant values, we can plug them into the binding energy formula: Binding Energy = (Z * m_p + (A - Z) * m_n - M) * c^2 = (28 * 1.007276 + 34 * 1.008664 - 61.928345) * 8.98755179 * 10^16 * (931.494)

Step 4: Calculate the binding energy per nucleon for nickel-62

To find the binding energy per nucleon, divide the binding energy by the total number of nucleons (A): Binding Energy per nucleon = Binding Energy / A

Step 5: Compare the binding energy per nucleon of nickel-62 with that of iron-56

Refer to Table 21.7 provided for the binding energy per nucleon of iron-56. Compare the calculated binding energy per nucleon of nickel-62 with this value.

Key Concepts

Nucleons

In the nucleus of an atom, nucleons are the particles that bind together thanks to the strong force. They consist of protons, which are positively charged, and neutrons, which have no charge. The number of nucleons in an atom is crucial because it defines the mass number, represented as \(A\).
Nickel-62, for example, has 62 nucleons. This means a combination of protons and neutrons that add up to this number. Knowing the number of protons (atomic number, \(Z\)) allows you to find the number of neutrons by subtracting \(Z\) from \(A\).

• Protons: Determine the element's chemical properties.
• Neutrons: Affect the mass and stability of an atom.

Understanding nucleons helps to calculate the binding energy, which measures how strongly these particles are held together. A greater binding energy per nucleon often indicates more nuclear stability.

Isotopes

Isotopes are variations of the same element, having the same number of protons but different numbers of neutrons. This means they have different mass numbers. For instance, nickel-62 is one isotope among other isotopes of nickel. All isotopes of nickel have 28 protons, but nickel-62 differs due to its 34 neutrons.
Isotopes are fascinating because they often have similar chemical behavior but distinct nuclear properties. The stability of an isotope can vary significantly, affecting its abundance and use in various applications, such as dating geological samples or medical imaging.

• Stable isotopes: Do not undergo radioactive decay.
• Radioactive isotopes: Decay over time, releasing particles and energy.

The concept of isotopes is pivotal in nuclear chemistry and physics, influencing our understanding of elements and allowing us to harness nuclear energy.

Atomic Mass

Atomic mass, often denoted as \(M\) in formulas, is the mass of an atom expressed in atomic mass units (u). It's crucial for understanding atomic and nuclear properties. Atomic mass includes both protons and neutrons within the nucleus, while electrons have negligible mass in this context.
Nickel-62 has an atomic mass of 61.928345 u, indicating the total mass of its protons and neutrons. Atomic mass is essential for calculating binding energy, which requires precise values.
Here's why atomic mass is important:

• Determines the element's characteristics and stability.
• Vital for calculations involving nuclear reactions and processes.

Atomic mass varies slightly between isotopes, contributing to their unique identities and uses in science and technology.