Question

What is the wavelength, wavenumber, and name of radiation with an energy of 100 kJ/mol?

Short answer

The wavelength is$1.20\mathrm{\mu m}$

The wavenumber is$0.83{\mathrm{\mu m}}^{-1}$

The name of radiation is Infrared radiation.

Step-by-step solution

Expression:

The expression for energy is $\mathrm{E}=\frac{\mathrm{hc}}{\mathrm{\lambda }}$

Rewrite the expression for$\mathrm{\lambda }$:

$\mathrm{\lambda }=\frac{\mathrm{hc}}{\mathrm{E}}$

The expression for wavenumber isrole="math" localid="1663647990022" $\overline{\nu }=\frac{1}{\lambda }$

Calculate wavelength:

Consider

$$\begin{gathered} \mathrm{h}=6.626\times {10}^{-34}\mathrm{J}\cdot \mathrm{s} \\ \mathrm{c}=2.998\times {10}^8\mathrm{m}/\mathrm{s} \\ \mathrm{E}=100\mathrm{kJ}/\mathrm{mol} \end{gathered}$$

Substitute in equation for wavelength,

$\mathrm{\lambda }=\frac{\mathrm{hc}}{\mathrm{E}}$

$\frac{\mathrm{\lambda }=\left(6.626\times {10}^{-34}\mathrm{J}\cdot \mathrm{s}\left)\right(2.998\times {10}^8\mathrm{m}/\mathrm{s}\left)\right(6.626\times {10}^{23}\mathrm{molecules}/\mathrm{mol}\right)}{100\mathrm{kJ}/\mathrm{mol}{\displaystyle \frac{\left({10}^3\mathrm{J}\right)}{1\mathrm{kJ}}}}$

$$\begin{gathered} \approx (1.20\times {10}^{-6}\mathrm{m})\left(\frac{{10}^6\mathrm{\mu m}}{1\mathrm{m}}\right) \\ =1.20\mathrm{\mu m} \end{gathered}$$

The wavelength is $1.20\mathrm{\mu m}$

Calculate wavenumber:

$\overline{\nu }=\frac{1}{\lambda }$

Substitute $\lambda =1.20\mathrm{\mu m}$,

$$\begin{gathered} \overline{\mathrm{\nu }}=\frac{1}{1.20\mathrm{\mu m}} \\ \approx 0.83{\mathrm{\mu m}}^{-1} \end{gathered}$$