Solution:

Understanding Energy States

The energy states of an atom or ion are defined by the arrangement of its electrons. Electrons in an atom or ion can occupy different energy levels or orbitals. The energy of an electron in an orbital is quantized, meaning it can only have certain discrete values. The energy of an atom or ion is the sum of the energies of its electrons.

Calculating Energy of He ion in First Excited State

The energy of a hydrogen atom in the ground state is -13.6 eV. This means that it takes 13.6 eV of energy to remove the electron from the hydrogen atom.

A He ion has two protons in its nucleus, compared to one in hydrogen, which means it has a stronger attraction for its electrons, and its energy levels are different from those of hydrogen.

The first excited state of a He ion has one electron in an orbital with n=2, which is higher in energy than the n=1 orbital of the ground state. To calculate the energy of the He ion in the first excited state, we need to use the equation:

E = -Rhc/n^2

where E is the energy, R is the Rydberg constant (2.18×10^-18 J), h is Planck's constant (6.63×10^-34 J s), c is the speed of light (3.0×10^8 m/s), and n is the principal quantum number.

For the first excited state of He ion, n = 2. Plugging these values into the equation, we get:

E = -Rhc/n^2 = -2.18×10^-18 J × 6.63×10^-34 J s × (3.0×10^8 m/s)^-1 × (2)^-2 = -54.4 eV

Therefore, the energy of a He ion in the first excited state is -54.4 eV.

Conclusion

The energy of a He ion in the first excited state is -54.4 eV. This is higher than the energy of a hydrogen atom in the ground state, which is -13.6 eV, due to the stronger attraction of the He ion's nucleus for its electrons.