Problem 26 - Chemistry: The Central Science

Question

The energy from radiation can be used to cause the rupture of chemical bonds. A minimum energy of \(941 \mathrm{~kJ} / \mathrm{mol}\) is required to break the nitrogen-nitrogen bond in \(\mathrm{N}_{2}\). What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

Step-by-Step Solution

Answer

The longest wavelength of radiation that possesses the necessary energy to break the nitrogen-nitrogen bond in N2 is 1.274 x 10^(-7) m or 127 nm. This type of electromagnetic radiation is Ultraviolet (UV) radiation.

Step 1: Convert the energy from kJ/mol to Joules/Photon

First, we need to convert the given energy from kJ/mol to Joules/Photon. We have 941 kJ/mol of energy and we know that 1 mol contains Avogadro's number of molecules (6.022 x 10^23 molecules/mol). To convert the energy, we will carry out the following calculation: Energy in Joules/Photon = (Energy in kJ/mol) x (\(\dfrac{1000 J}{1 kJ}\)) x (\(\dfrac{1 mol}{6.022 \times 10^{23} molecules}\))

Step 2: Calculate the energy (E) per photon in Joules

By plugging the given values into the equation above, we will get the energy per photon in Joules: Energy in Joules/Photon = \(\dfrac{941\,\mathrm{kJ/mol} \times 1000\,\mathrm{J/kJ}}{6.022\,\mathrm{\times 10^{23}\, molecules/mol}}\) = 1.56 x 10^(-19) J/Photon

Step 3: Use Planck's equation to find the wavelength (λ)

Now that we have the energy per photon in Joules, we can find the wavelength using Planck's equation: \(E = h \times \dfrac{c}{\lambda}\) Rearrange Planck's equation to solve for λ: \(\lambda = \dfrac{h \times c}{E}\) We know Planck's constant (h) = 6.63 x 10^(-34) Js, the speed of light (c) = 3 x 10^(8) m/s, and the energy (E) = 1.56 x 10^(-19) J/Photon.

Step 4: Calculate the longest wavelength (λ)

Plug in the values into the rearranged Planck's equation and solve for λ: \(\lambda = \dfrac{(6.63\,\mathrm{\times 10^{-34}\, Js})\,(3\,\mathrm{\times 10^{8}\, m/s})}{1.56\,\mathrm{\times 10^{-19}\, J/Photon}}\) = 1.274 x 10^(-7) m The longest wavelength is 1.274 x 10^(-7) m, which can also be expressed as 127 nm.

Step 5: Identify the type of electromagnetic radiation

To classify the type of electromagnetic radiation for this wavelength, we refer to the electromagnetic spectrum: - Gamma rays: < 0.01 nm - X-rays: 0.01 to 10 nm - Ultraviolet rays: 10 to 380 nm - Visible light: 380 to 700 nm - Infrared rays: 700 nm to 1 mm - Microwaves: 1 mm to 1 m - Radio waves: > 1 m Since the calculated wavelength is 127 nm, it falls in the Ultraviolet (UV) range. Therefore, the type of electromagnetic radiation needed to break the nitrogen-nitrogen bond in N2 is ultraviolet radiation.