Question

What is the wavelength of an electron with a velocity of `3.85xx10^6" m/s"`?

Answer 1

The wavelength of an electron with a velocity of `3.85xx 10^6` `"m/s"` is `1.89xx10^(-10)` `"m"`.

Explanation:

The de Broglie equation relating the wavelength of an electron to its velocity is:

`lambda` = `h/p`

where

`lambda= "wavelength in m"`

`h =` Planck's constant

`p = "momentum"= mv`,

where:

`m= "kg"`, and `v = "m/s"`

Known/Given:

Planck's constant, `6.626xx10^(-34)"J"*"s"`

velocity of electron, `3.85 xx10^6"m/s"`

mass of electron, `9.109xx10^(-31)"kg"`

Unknown:

momentum `(p)`

wavelength `(lambda)`

Equations:

`p` = `mv`

`lambda` = `h/p`

Solution:

1. Calculate momentum.

`p = (9.109xx10^(-31)"kg")xx(3.85xx10^6"m/s")= 3.507xx10^(-24)"kg"*"m/s"`

2. Calculate wavelength:

`lambda=(6.626xx10^(-34)"J"*"s")/(3.507xx10^(-24)"kg"*"m/s")=1.89xx10^(-10)` `"m"`. (rounded to three significant figures)

Because `"1 Joule"= "1 kg"*"m"^2/("s"^2)`, kilograms and seconds cancel, leaving meters.

Answer:
The wavelength of an electron with a velocity of `3.85xx 10^6` `"m/s"` is `1.89xx10^(-10)` `"m"`.