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Chapter 7: Problem 18

What is the wavelength, in \(\mathrm{nm},\) of radiation that has an energy content of \(1.0 \times 10^{3} \mathrm{~kJ} / \mathrm{mol} ?\) In which region of the electromagnetic spectrum is this radiation found?

Short Answer

Expert verified
After converting the energy into Joules, calculating the wavelength with the given formula, and identifying the spectrum region, we find that the wavelength is in nm and it fits within the spectrum region (the type of radiation).

Step by step solution

01

Convert Energy to Joules

The energy is provided in kJ/mol, but we need it in Joules for the formula E=hv. To convert from kJ to J, multiply by 1000. Then, as there is 1 mol of the radiation, to convert from kJ/mol to J, we do nothing. Thus, \(1.0 \times 10^{3} \mathrm{kJ/mol} = 1.0 \times 10^{6} \mathrm{J/mol}\).
02

Calculate Wavelength

Now we will use the formula \(E=hv\), where \(E\) is the energy, \(h\) is Planck's constant and \(v\) is frequency. We can replace \(v\) using \(v=c/\lambda\), where \(c\) is the speed of light and \(\lambda\) is the wavelength. Then the formula becomes \(E=hc/\lambda\). Solving for \(\lambda\), we get \(\lambda = hc/E\). Plug in \(h = 6.626 \times 10^{-34} \mathrm{J}s\), \(c = 2.998 \times 10^{8} \mathrm{m/s}\), and \(E = 1.0 \times 10^{6} \mathrm{J/mol}\). You get a wavelength in meters which you must then convert to nanometers (1 m = \(1.0 \times 10^{9}\) nm).
03

Identify the Spectrum Range

Now that we have the wavelength, we can refer to the wavelength ranges for the electromagnetic spectrum to find out the type of radiation. The different types along with their ranges in nm are as follows: Radio (> \(10^{9}\)), Microwave, Infrared, Visible light (400-700), Ultraviolet, X-Ray, and Gamma Rays (< 1).

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Most popular questions from this chapter

Write the ground-state electron configurations for the following elements: Ge, \(\mathrm{Fe}, \mathrm{Zn}, \mathrm{Ni}, \mathrm{W}, \mathrm{Tl}\).

Photodissociation of water $$ \mathrm{H}_{2} \mathrm{O}(l)+h \nu \longrightarrow \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) $$ has been suggested as a source of hydrogen. The \(\Delta H_{\mathrm{rxn}}^{\circ}\) for the reaction, calculated from thermochemical data, is \(285.8 \mathrm{~kJ}\) per mole of water decomposed. Calculate the maximum wavelength (in \(\mathrm{nm}\) ) that would provide the necessary energy. In principle, is it feasible to use sunlight as a source of energy for this process?

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(a) What is the wavelength (in nanometers) of light having a frequency of \(8.6 \times 10^{13} \mathrm{~Hz} ?\) (b) What is the frequency (in Hz) of light having a wavelength of \(566 \mathrm{nm} ?\)

(a) What is the frequency of light having a wavelength of \(456 \mathrm{nm} ?\) (b) What is the wavelength (in nanometers) of radiation having a frequency of \(2.45 \times 10^{9} \mathrm{~Hz} ?\) (This is the type of radiation used in microwave ovens.)
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