In his explanation of the threshold frequency in the photoelectric effect, Einstein reasoned that the absorbed photon must have the minimum energy required to dislodge an electron from the metal surface. This energy is called the work function (Φ) of that metal. What is the longest wavelength of radiation (in nm) that could cause the photoelectric effect in each of the following metals?
(a) calcium, Φ=4.60 x 10^-19J
(b) titanium, Φ=6.94 x 10^19J
(c) sodium, Φ=4.41 x 10^19J

The minimum energy to dislodge an electron froma metal has to be greater than the work function. We know the longer the wavelength the lower the energy of radiation, hence what we are looking in this problem is the maximum wavelengtrh which will produce a photoelectric effect for these metals.

The calculation is done by utilizing the formula

E = ( h x c )/ λ ⇒ λ = ( h x c ) / E

where E is the work function given in the question and h is Planck's constant ( 6.626 x 10 ⁻³⁴ Js) and C is the speed of light ( 3 x 10 ^8 m/s)

After plugging our values for h and c we get

λ = 6.626 x 10⁻³⁴ Js x 3 x 10 ^8/ E = 2 x 10⁻²⁵ m/ E

and then we can calculate for each of our three work functions.

For example for the calcium metal

λ = 2 x 10⁻²⁵ m / 4.60 x 10⁻¹⁹ = 4 x 10⁻⁷ m x 10⁹ nm/m = 400 nm

Note: the work functions for the titanium and sodium have and error in the exponents in the scientific notation for the two values. It should be

titanium 6.9 x 10⁻¹⁹ J and 4.41 x 10⁻¹⁹ J. The wrong values will be so high that the photoelectric effect will be impossible.