Explanation:

1. Introduction:
The potential energy of an electron in the first Bohr orbit in the He ion can be determined using the Bohr model of the atom. The Bohr model describes the electron's energy levels in terms of quantized orbits around the nucleus.

2. Bohr Model:
The Bohr model states that the electron in a hydrogen-like atom, such as He ion, can only occupy specific energy levels. The energy of each level is given by the equation:

E = -13.6 eV / n^2

where E is the energy, -13.6 eV is the ionization energy of hydrogen, and n is the principal quantum number.

3. Energy of the First Bohr Orbit:
For the first Bohr orbit (n = 1), the energy can be calculated as:

E = -13.6 eV / (1^2) = -13.6 eV

So, the energy of the electron in the first Bohr orbit of the He ion is -13.6 eV.

4. Effective Nuclear Charge:
In the case of the He ion, the effective nuclear charge experienced by the electron is different from that of hydrogen. The effective nuclear charge (Zeff) can be calculated using the equation:

Zeff = Z - S

where Z is the atomic number (2 for He) and S is the screening constant.

5. Calculation of Zeff:
The screening constant (S) for the He ion can be determined using Slater's rules. According to these rules, the electrons in the inner shells contribute less to the screening effect than the electrons in the outer shells.

For the He ion, the screening constant can be calculated as:

S = 1 * (1 - 0.35) = 0.65

where the factor 0.35 represents the screening effect of the electrons in the 1s orbital.

Therefore, the effective nuclear charge for the He ion is:

Zeff = 2 - 0.65 = 1.35

6. Revised Energy Calculation:
Taking the effective nuclear charge into account, the energy of the electron in the first Bohr orbit of the He ion can be calculated as:

E = -13.6 eV / (1.35^2) = -13.6 eV / 1.8225 ≈ -7.47 eV

So, the potential energy of an electron in the first Bohr orbit of the He ion is approximately -7.47 eV.

7. Comparing with the Given Range:
The given range for the potential energy is between -109 and -108.5 eV. As the calculated energy (-7.47 eV) is significantly different from the given range, it seems that there might be an error in the question or the range provided.

Conclusion:
The potential energy of an electron in the first Bohr orbit of the He ion is approximately -7.47 eV, which is significantly different from the given range. It is possible that there is an error in the question or the provided range.