Final answer:
The minimum energy required to free an electron in the twenty-second energy level around a hydrogen nucleus is approximately -0.027 eV.
Step-by-step explanation:
The minimum energy required to free an electron in the twenty-second energy level around a hydrogen nucleus can be determined using the equation:

where n is the principal quantum number. In this case, n = 22. Substituting the value of n into the equation, we get:

Solving for E, we find that the minimum energy required is approximately -0.027 eV.