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 ${}^{\mathbf{63}}\mathit{C}\mathit{u}$atoms (of mass62.92960u).The masses of the proton-plus- electron and the neutron are 1.00783uand 1.00867u, respectively.

The mass of the penny, m = 3.0 g

Atomic mass of the atom ${}^{63}\mathrm{Cu}$, $m_{Cu}=62.92960u$

The atomic mass of the hydrogen,$m_H=1.00783u$

The atomic mass unit of neutron, $m_n=1.00867u$

The binding energy of an element is defined as the amount of energy required to separate a particle from a system of particles or to disperse all the particles of the system. To remove all the protons and neutrons, we require the total binding energy of all the atoms in the nuclei. For that, you need to calculate the total number of atoms present in the given mass of the copper coin using the Avogadro number.

Formulae:

The binding energy of an atom is,

$∆E_{be}=\left[Z{M}_H+\left(A-Z\right)M_H-M_{atom}\right]c^{2}or∆m{c}^2$ ….. (i)

Where, Z is the atomic number (number of protons), A is the mass number (number of nucleons), $M_H$is the mass of a hydrogen atom, $M_n$is the mass of a neutron, and $M_{atom}$ is the mass of an atom.

The number of atoms in a given mass of an atom is,

$N_{Cu}=\frac{m}{A}{N}_A$ ….. (ii)

Here, the Avogadro number is,$N_A=6.022\times {10}^{23}atoms/mol$

First “separate” all the nucleons in one copper nucleus (which amounts to simply calculating the nuclear binding energy) and then figure the number of nuclei in the penny (so that we can multiply the two numbers and obtain the result). At first, we note that the copper - 63 nucleus has 29 protons and 34 neutrons. Thus, the binding energy of the copper atom using equation (i) as follows:

$$\begin{gathered} ∆E_{be}=\left[29\left(1.00783u\right)+34\left(1.00867u\right)-62.92960u\right]\left(931.5MeV/u\right) \\ =\left[29.22707u+34.29478u-62.92960u\right]\left(931.5MeV/u\right) \\ =\left[29.22707u+34.29478u-62.92960u\right]\left(931.5MeV/u\right) \\ =551.4MeV \end{gathered}$$

The number of atoms present in the copper penny can be calculated using equation (ii) as follows:

$$\begin{gathered} N_{Cu}=\frac{3.0g}{62.92960g/mol}\left(6.022\times {10}^{23}atoms/mol\right) \\ =2.9\times {10}^{22}atoms \end{gathered}$$

The total binding energy of the copper penny or the energy that would be required to separate all the neutrons and protons in this coin from one another is can be calculated as follows:

$$\begin{gathered} E_{total}=N_{Cu}\times ∆E_{be} \\ =2.9\times {10}^{22}atoms\times 551.4MeV \\ =1.6\times {10}^{25}MeV \end{gathered}$$

Hence, the value of the energy is $1.6\times {10}^{25}MeV$.