3) the amount of energy it takes to eject an electron from the surface of a metal is 4.15 x 10-19j. if light with a wavelength of 625nm is shined on the metal, what will happen? prove your answer mathematically by finding the frequency and energy of the wave.

We can use two equations to solve this.
(1) - E = hf
        E = Energy (J)
        h = plank's constant (6.63 × 10⁻³⁴ J s)
        f  = frequency (Hz)

(2) - v = fλ
        v = velocity of the wave (m/s)
       f  = frequency (Hz)
       λ = wavelength (m)

the λ for the given wave is 625nm (625 x 10⁻⁹ m) and the velocity of the wave is equal to speed of light (3 x 10⁸ m/s) since the wave is a light.

hence we can find the frequency of the wave by using (2) formula.
3 x 10⁸ m/s = f x 625 x 10⁻⁹ m
                f  = 4.8 x 10¹⁴ Hz

by using calculated frequency and (1) formula, we can find the energy of the wave.
 E = 6.63 × 10⁻³⁴ J s x 4.8 x 10¹⁴ Hz
 E = 3.18 x 10⁻¹⁹ J

The energy of wave is less than energy needed to eject an electron.
Hence, the metal does not eject electron.