Evaluate the minimum energy required to separate a proton from a...

The separation of a proton from a nucleus with $Z$ protons and $N$ neutrons can be represented as:

Nucleus$(Z, N) \rightarrow$ Nucleus$(Z-1, N) +$ proton

The minimum energy required for this process is the proton-separation energy, given by the mass difference between the products and the reactant:

$\Delta E = [m_{nuc}(Z-1, N) + m_p - m_{nuc}(Z, N)] c^2$

The nuclear mass $m_{nuc}$ can be expressed in terms of the atomic mass $M$ by subtracting the mass of the electrons:

$m_{nuc}(Z, N) = M_{Z, N} - Z m_e$

$m_{nuc}(Z-1, N) = M_{Z-1, N} - (Z-1) m_e$

Substituting these into the energy equation:

$\Delta E = [M_{Z-1, N} - (Z-1) m_e + m_p - (M_{Z, N} - Z m_e)] c^2$

$\Delta E = [M_{Z-1, N} + m_p + m_e - M_{Z, N}] c^2$

Since the mass of a hydrogen atom is $M_H = m_p + m_e$ (neglecting the small electronic binding energy), we get:

$\Delta E = (M_{Z-1, N} + M_H - M_{Z, N}) c^2$