Question

How to Calculate the de Broglie wavelength for each of the following ? a. an electron with a velocity 10.% of the speed of light b. a tennis ball (55 g) served at 35 m/s (,80 mi/h)

Answer 1

To calculate the de Broglie wavelength for a particle, or for a tennis ball for that matter, just use the equation

`p = h/(lamda)`, where

`p` - the momentum of the atom;
`h` - Planck's constant - `6.626 * 10^(-34)"m"^(2)"kg s"^(-1)`
`lamda` - wavelength;

Momentum can be expressed as

`p = m* v`, where

`m` - the mass of the particle;
`v` - the speed of the particle.

So, starting with the electron that travels at 10% of the speed of light. The speed of light can be approximated to be

`c = 3 * 10^(8)"m/s "`, which means that the electron's speed will be

`v = 1/10 * c = 3 * 10^(7)"m/s"`

The mass of an electron is `m = 9.1094 * 10^(-31)"kg"`

Now plug your values into the main equation and solve for `lamda`

`p = h/(lamda) => m * v = h/(lamda) => lamda = h/(m * v)`

`lamda_"electron" = (6.626 * 10^(-34)"m"^(cancel(2))cancel("kg")cancel("s"^(-1)))/(9.1094 * 10^(-31)cancel("kg") * 3 * 10^(7)cancel("m")cancel("s"^(-1))`

`lamda_"electron" = color(green)(2.42 * 10^(-11)"m")`

Now for the tennis ball

`lamda_"tennis" = (6.626 * 10^(-34)"m"^(cancel(2))cancel("kg")cancel("s"^(-1)))/(55 * 10^(-3)cancel("kg") * 35cancel("m")cancel("s"^(-1))`

`lamda_"tennis" = color(green)(3.44 * 10^(-34)"m")`