Question

Question #a9596

Answer 1

WARNING! Long answer! The minimum energy needed is `2.31 × 10^5color(white)(l) "J·mol"^"-1"` and the maximum wavelength is 518 nm.

Explanation:

The photoelectric effect

When a photon of sufficient energy hits the surface of a metal, some of the energy is used to eject an electron from the surface of the metal (the work function) and the rest goes into the kinetic energy of the electron.

`"Energy of photon" = "work function + kinetic energy"`

`color(blue)(bar(ul(|color(white)(a/a)E = φ + KEcolor(white)(a/a)|)))" "`

Minimum energy

The energy `E` of a photon is given by the formula

`color(blue)(bar(ul(|color(white)(a/a) E = hf color(white)(a/a)|)))" "`

where

`h` = Planck's constant
`f` = the frequency of the light.

Another important formula is

`color(blue)(bar(ul(|color(white)(a/a) fλ = c color(white)(a/a)|)))" "`

where

`λ` = the wavelength of the light
`c` = the speed of light

We can combine these two expressions and get the formula

`color(blue)(bar(ul(|color(white)(a/a) E = (hc)/λ color(white)(a/a)|)))" "`

For a mole of photons,

`E = N_"A"(hc)/λ`

where

`N_"A"` = Avogadro's number

The minimum energy will be that for which the photon has just enough energy to remove the electron with nothing left over (i.e. the work function)

`E = φ + KE`

`φ = E - KE`

`E = (N_"A"hc)/λ = (6.022 × 10^23color(white)(l) "mol"^"-1" × 6.626 × 10^"-34" color(white)(l)"J"·color(red)(cancel(color(black)("s"))) × 2.998 × 10^8 color(red)(cancel(color(black)("m·s"^"-1"))))/(300 × 10^"-9" color(red)(cancel(color(black)("m")))) = 3.99 × 10^5color(white)(l) "J·mol"^"-1"`

`KE = 1.68 × 10^5color(white)(l) "J·mol"^"-1"`

∴ `φ = 3.99 × 10^5color(white)(l) "J·mol"^"-1" - 1.68 × 10^5color(white)(l) "J·mol"^"-1" = 2.31 × 10^5color(white)(l) "J·mol"^"-1"`

Maximum wavelength

A photon with just enough energy to equal the work function will have the maximum wavelength.

`E = N_"A"(hc)/λ = φ`

`λ = (N_"A"hc)/φ = (6.022 × 10^23 color(red)(cancel(color(black)("mol"^"-1"))) × 6.626 × 10^"-34" color(red)(cancel(color(black)("J·s"))) × 2.998 × 10^8 "m"·color(red)(cancel(color(black)("s"^"-1"))))/(2.31 × 10^5 color(red)(cancel(color(black)("J·mol"^"-1")))) = 5.18 × 10^"-7"color(white)(l) "m" = 518 × 10^"-9"color(white)(l) "m" = "518 nm"`