The formula for the de Broglie wavelength λ is
color(blue)(bar(ul(|color(white)(a/a)λ = h/(mv)color(white)(a/a)|)))" "
where
h = Planck's constant
m = the mass of the electron
v = the speed of the electron.
Calculate the speed of the electron
The energy E of a hydrogen electron in an orbit is
E = -R_text(H)/n^2
where
R_text(H) = the Rydberg energy constant (2.180 × 10^"-18"color(white)(l) "J")
Since n = 1,
E_1 = "-2.180 × 10"^"-18"color(white)(l) "J"
KE = "-"E_1 = 2.180 × 10^"-18"color(white)(l) "J"
KE = 1/2mv^2
v = sqrt((2KE)/m) = sqrt((2 × 2.180 × 10^"-18" color(red)(cancel(color(black)("J"))))/(9.11 × 10^"-31" color(red)(cancel(color(black)("kg")))) × (1 color(red)(cancel(color(black)("kg")))·"m"^2"s"^"-2")/(1 color(red)(cancel(color(black)("J"))))) = sqrt(4.786 × 10^12color(white)(l) "m"^2"s"^"-2") = 2.188 × 10^6color(white)(l) "m·s"^"-1"
Calculate the de Broglie wavelength
λ = h/(mv) = (6.626 × 10^"-34" color(red)(cancel(color(black)("J·s"))))/(9.109 × 10^"-31" color(red)(cancel(color(black)("kg"))) × 2.188 × 10^6 color(red)(cancel(color(black)("m·s"^"-1")))) × (1 color(red)(cancel(color(black)("kg")))·"m"^color(red)(cancel(color(black)(2)))·color(red)(cancel(color(black)("s"^"-2"))))/(1 color(red)(cancel(color(black)("J")))) = 3.325× 10^"-10"color(white)(l) "m" = "332.5 pm"
