# The minimum energy needed to dissociate iodine molecules, I2, is 151 kJ/mol. What is the wavelength of photons (in nm) that supplies this energy, assuming each bond dissociated by absorbing one photon?

Mar 24, 2018

$792$ nanometers (or scientifically $7.92 \cdot {10}^{2} \cdot \mu m$.)

#### Explanation:

Equations this solution applied:

1. $N = n \cdot {N}_{A}$ where $N$ is the quantity of $n$ moles of particles and ${N}_{A} = 6.02 \cdot {10}^{23} \cdot {\text{mol}}^{- 1}$ is the Avagordoro's number

2. The Planck's Law $E = h \cdot f$ where $E$ is the energy of a single photon of frequency $f$ and $h$ is the Planck's Constant, h=6.63 × 10^(-34)* "m"^2* "kg" *"s"^(-1)= 6.63*10^(-34) color(blue)("J")*"s" [1]

3. $\lambda = \frac{v}{f}$ where $\lambda$ is the wavelength of a wave or an electromagnetic (EM) radiation of frequency $f$.

From the question, breaking ${N}_{A}$ (one mole of) iodine molecules consumes
E("one mole")=E*N_A=151*color(red)("kJ")*"mol"^(-1)
=1.51*10^5*color(blue)("J") *"mol"^(-1)
where $E$ is the energy input required to break a single molecule.

Thus it takes
E=(E("one mole"))/(N_A)=(1.51*10^5)/(6.02*10^23)=2.51* 10^(-19) * color(blue)("J")
to break a single iodine molecule.

Apply the Planck's Law to find the maximum frequency of the EM radiation capable of breaking one such molecule:
f=E/h= (2.51* 10^(-19) * color(blue)("J"))/(6.63*10^(-34) color(blue)("J")*"s")
$= 3.79 \cdot {10}^{14} {\text{s}}^{- 1}$

*Make sure that you get the unit that corresponds to the quantity after canceling out corresponding pairs. Here we are expecting ${\text{s}}^{-} 1$ for frequency, which does appear to be the case. *

Assuming 3.00*10^8 * color(magenta)("m")* "s"^-1=3.00\cdot 10^(17)* color(green)(mu "m")\cdot "s"^-1 to be the speed of light. A photon of such EM radiation is expected to have the wavelength
lambda=v/f=(3.00*10^17* color(green)(mu"m")*"s"^(-1))/(3.79*10^14* "s"^(-1))=7.92*10^2* color(green)(mu "m)

Sources:
1. Units ("dimensions") of the Planck's Constant: https://www.askiitians.com/forums/General-Physics/find-the-dimension-of-planck-constant-h-from-the-e_74309.htm