# Question 18492

Aug 7, 2017

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 $390 - \text{700 nm}$.

Visible light is known to be in the region $390 - 700$ $\text{nm}$ (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 $\boldsymbol{E}$ as quanta, $h \nu$ ("packets" of light, each one called a quantum of light):

$\boldsymbol{E = h \nu}$,

where $h = 6.626 \times {10}^{- 34} \text{J"cdot"s}$ is Planck's constant, and $\nu$ is the frequency of the light in ${\text{s}}^{- 1}$.

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

$\boldsymbol{\nu = \frac{c}{\lambda}}$,

with $\lambda$ in $\text{m}$ and $c = 2.998 \times {10}^{8}$ $\text{m/s}$.

Or, combined, we get

$\boldsymbol{E = \frac{h c}{\lambda}}$.

a) The lower wavelength bound of $\text{390 nm}$, or $390 \times {10}^{- 9} \text{m}$, 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")

$= \underline{5.094 \times {10}^{- 19} \textcolor{w h i t e}{.} \text{J}}$

Likewise, the upper wavelength bound of $\text{700 nm}$ will give you $E = \underline{2.838 \times {10}^{- 19} \textcolor{w h i t e}{.} \text{J}}$, and the range shall be:

$\textcolor{b l u e}{\overline{\underline{|}} \stackrel{\text{ ")(" "DeltaE_("vis") = (5.094 - 2.838) xx 10^(-19)color(white)(.)"J"" }}{|}}$

b)# The lower wavelength bound of $\text{390 nm}$, or $390 \times {10}^{- 9} \text{m}$, corresponds to a frequency of:

$\nu = \frac{E}{h} = \frac{c}{\lambda}$

$= \left(2.998 \times {10}^{8} \text{m/s")/(390 xx 10^(-9) "m}\right)$

$= \underline{7.687 \times {10}^{14} \textcolor{w h i t e}{.} \text{s"^(-1)" or Hz}}$

Likewise, the upper wavelength bound of $\text{700 nm}$ will give you $E = \underline{4.283 \times {10}^{14} \textcolor{w h i t e}{.} \text{s"^(-1)" or Hz}}$, and the range shall be:

$\textcolor{b l u e}{\overline{\underline{|}} \stackrel{\text{ ")(" "Deltanu_("vis") = (7.687 - 4.283) xx 10^(14)color(white)(.)"s"^(-1)" or Hz"" }}{|}}$