Question

Calculate the energy of the 1st line of the Hydrogen spectrum.Its wavelength is 410nm?

Answer 1

The energy of the 1st line of the hydrogen spectrum is `4.8xx10^(-19)"J"`.

Explanation:

This question requires a two-part answer. You first need to determine the frequency of the given wavelength using the relationship between the speed of light, frequency, and wavelength. Once you know the frequency, you will determine the energy using the relationship between energy, Planck's constant, and frequency.

Determine frequency.

You have been given the wavelength but not the frequency. You will need to use the following formula to determine the frequency.

`c=lambda*nu`,

where `c` is the speed of light `(3.00xx10^8 "m/s")`, `lambda` is the wavelength `("410 nm")`, and `nu` is the frequency.

The length unit must be the same for `c` and `lambda`. Therefore, you will need to convert nanometers to meters.

`"1 m"` `=` `1xx"10"^9 "nm"`

`410color(red)cancel(color(black)("nm"))xx(1"m")/(1xx10^9color(red)cancel(color(black)("nm")))=4.1xx"10"^(-7) "m"`

To calculate frequency , rearrange the formula to isolate `nu`. Plug in the known values and solve.

`nu=c/lambda`

`nu=(3.00xx10^8color(red)cancel(color(black)("m"))/"s")/(4.1xx10^(-7)color(red)cancel(color(black)("m")))=(7.3xx10^14)/"s"` `=` `7.3xx10^14 "Hz"`

Determine the energy of the electromagnetic wave.

The energy of a wave of electromagnetic radiation is determined by the formula:

`E=hnu`,

where `E` is energy, `h` is Planck's constant `(6.626070040xx10^(-34) "J"*"s"")`, and `nu` is the frequency in `"Hz"`, or `1/s"`.

Plug the known values into the equation and solve.

`E=(6.626070040xx10^(-34) "J"*color(red)cancel(color(black)("s")))xx(7.3xx10^14/color(red)cancel(color(black)("s")))=4.8xx10^(-19)"J"`