Question

Question #18492

Answer 1

`DeltaE_("vis") = (5.094 - 2.838) xx 10^(-19)color(white)(.)"J"`

`Deltanu_("vis") = (7.687 - 4.283) xx 10^(14)color(white)(.)"s"^(-1)" or Hz"`

for the visible light range of `390 - "700 nm"`.

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Visible light is known to be in the region `390 - 700` `"nm"` (although I find it easier to just use `400 - 700`). We can start there.

The relationship between energy and frequency is given by the equation for energy `bbE` as quanta, `hnu` ("packets" of light, each one called a quantum of light):

`bb(E = hnu)`,

where `h = 6.626 xx 10^(-34) "J"cdot"s"` is Planck's constant, and `nu` is the frequency of the light in `"s"^(-1)`.

And the relationship between frequency `nu` and wavelength lambda is through the speed of light `c`:

`bb(nu = c/lambda)`,

with `lambda` in `"m"` and `c = 2.998 xx 10^(8)` `"m/s"`.

Or, combined, we get

`bb(E = (hc)/lambda)`.

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`a)` The lower wavelength bound of `"390 nm"`, or `390 xx 10^(-9) "m"`, corresponds to an energy of:

`E = (hc)/lambda = ((6.626 xx 10^(-34) "J"cdot"s")(2.998 xx 10^(8) "m/s"))/(390 xx 10^(-9) "m")`

`= ul(5.094 xx 10^(-19)color(white)(.)"J")`

Likewise, the upper wavelength bound of `"700 nm"` will give you `E = ul(2.838 xx 10^(-19)color(white)(.)"J")`, and the range shall be:

`color(blue)(barul|stackrel(" ")(" "DeltaE_("vis") = (5.094 - 2.838) xx 10^(-19)color(white)(.)"J"" ")|)`

`b)` The lower wavelength bound of `"390 nm"`, or `390 xx 10^(-9) "m"`, corresponds to a frequency of:

`nu = E/h = c/lambda`

`= (2.998 xx 10^(8) "m/s")/(390 xx 10^(-9) "m")`

`= ul(7.687 xx 10^(14)color(white)(.)"s"^(-1)" or Hz")`

Likewise, the upper wavelength bound of `"700 nm"` will give you `E = ul(4.283 xx 10^(14)color(white)(.)"s"^(-1)" or Hz")`, and the range shall be:

`color(blue)(barul|stackrel(" ")(" "Deltanu_("vis") = (7.687 - 4.283) xx 10^(14)color(white)(.)"s"^(-1)" or Hz"" ")|)`