Question #a9596

1 Answer
Oct 29, 2016

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.

www.daviddarling.info

#"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"#