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Chapter 21: Problem 54

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 .

Short Answer

Expert verified
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 by step solution

01

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.
02

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
03

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)
04

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
05

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.

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