To determine the final state energy (Ef) and the principal quantum number of the final state (nf), we need to use the formula for the energy of a photon:

E = hc/λ

Where:
E = energy of the photon
h = Planck's constant (6.626 x 10^-34 J·s)
c = speed of light (3.00 x 10^8 m/s)
λ = wavelength of the photon

Given that the wavelength of the emitted photon is 470nm, we can substitute this value into the formula to find the energy of the photon:

E = (6.626 x 10^-34 J·s * 3.00 x 10^8 m/s) / 470 x 10^-9 m
E = 4.23 x 10^-19 J

Now, let's consider the energy levels of a singly ionized Helium atom (He+). The energy levels are determined by the principal quantum number (n). In the ground state, the principal quantum number is n = 1. When an electron transitions from a higher energy level (initial state) to a lower energy level (final state), the energy of the emitted photon is equal to the difference in energy between the two levels.

Since the question states that the helium atom is in the 4 state (initial state), we need to find the final state with a principal quantum number (nf) that results in the calculated energy of the photon (4.23 x 10^-19 J).

Using the formula for the energy levels of hydrogen-like atoms:

E = -13.6 eV / n^2

Where:
E = energy of the level
-13.6 eV = ionization energy of hydrogen (1 electron volt = 1.602 x 10^-19 J)
n = principal quantum number

We can solve for nf by rearranging the equation:

Ef = -13.6 eV / nf^2

Substituting the calculated energy of the photon:

4.23 x 10^-19 J = -13.6 eV / nf^2

To convert the energy of the photon from joules to electron volts, we can use the conversion factor:

1 eV = 1.602 x 10^-19 J

4.23 x 10^-19 J * (1 eV / 1.602 x 10^-19 J) = 2.64 eV

Now, let's solve the equation for nf:

2.64 eV = -13.6 eV / nf^2

Dividing both sides of the equation by -13.6 eV:

2.64 eV / -13.6 eV = 1 / nf^2

Simplifying the left side:

-0.1941 = 1 / nf^2

Taking the reciprocal of both sides of the equation:

-1 / 0.1941 = nf^2

Solving for nf^2:

nf^2 = -1 / -0.1941

nf^2 = 5.151

Taking the square root of both sides of the equation:

nf = √5.151

nf ≈ 2.27

Since the principal quantum number must be a whole number, we can conclude that the final state has nf = 2.

Therefore, the correct answer is:

a) nf