Question

The wavelength at which the \(\mathrm{O}_{2}\) molecule most strongly absorbs light is approximately 145 \(\mathrm{nm}\) . (a) In which region of the electromagnetic spectrum does this light fall? (b) Would a photon whose wavelength is 145 nm have enough energy to photodissociate \(\mathrm{O}_{2}\) whose bond energy is 495 \(\mathrm{kJ} / \mathrm{mol} ?\) Would it have enough energy to photoionize \(\mathrm{O}_{2} ?\)

Short answer

(a) The given wavelength of 145 nm falls in the ultraviolet (UV) region of the electromagnetic spectrum. (b) A photon with a wavelength of 145 nm has an energy of about \(1.37 \times 10^{-18} \mathrm{J}\) per photon, which is enough to photodissociate O2 since it requires about \(8.22 \times 10^{-19} \mathrm{J}\) per photon. We cannot determine if the photon has enough energy to photoionize O2 without information on the ionization energy of O2.

Step-by-step solution

(Step 1: Identify the region of the electromagnetic spectrum)

Given the wavelength of 145 nm, we can identify the region of the electromagnetic spectrum it belongs to. Refer to any electromagnetic spectrum chart, and you'll see that a wavelength of 145 nm falls in the ultraviolet (UV) region of the spectrum.

(Step 2: Calculate the energy of the photon)

To compare the energy of the photon with the bond energy and ionization energy of the O2 molecule, we can use the equation for the energy of a photon: \[E = \frac{hc}{\lambda}\] where \(E\) is the energy, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), \(c\) is the speed of light (\(3 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the wavelength (145 nm, but we need to convert it to meters, which is \(1.45 \times 10^{-7} \mathrm{m}\)). Solving for \(E\): \[E = \frac{(6.626 \times 10^{-34} \mathrm{Js})(3 \times 10^{8} \mathrm{m/s})}{1.45 \times 10^{-7} \mathrm{m}}\] \[E \approx 1.37 \times 10^{-18} \mathrm{J}\]

(Step 3: Convert bond energy to energy per photon)

We are given the bond energy of O2 as 495 kJ/mol. First, we need to convert it to J/mol: \(495 \mathrm{kJ/mol} \times \frac{1000 \mathrm{J}}{1 \mathrm{kJ}} = 495,000 \mathrm{J/mol}\) Now, we need to convert the bond energy from J/mol to J/photon. We use Avogadro's number (\(6.022 \times 10^{23}\)) to do this: \(\frac{495,000 \mathrm{J/mol}}{6.022 \times 10^{23}\ \mathrm{particles/mol}} \approx 8.22 \times 10^{-19} \mathrm{J/photon}\)

(Step 4: Determine if the photon has enough energy to photodissociate O2)

Now, we can compare the energy of the photon (\(1.37 \times 10^{-18} \mathrm{J}\)) to the energy required to photodissociate O2 (\(8.22 \times 10^{-19} \mathrm{J}\)). Since the energy of the photon is greater than the energy required to photodissociate O2, the photon indeed has enough energy to photodissociate O2.

(Step 5: Photoionization of O2)

The exercise does not provide the ionization energy of O2, so we cannot answer whether the photon has enough energy to photoionize O2. However, if given the ionization energy, one can follow the same procedure as in steps 3 and 4 and compare the energy of the photon with the ionization energy to determine if it has enough energy to photoionize the O2 molecule. In conclusion, the given wavelength of 145 nm belongs to the ultraviolet region of the electromagnetic spectrum, and it has enough energy to photodissociate O2. However, without the ionization energy information, we cannot determine if it has enough energy to photoionize O2.

Key concepts

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all types of electromagnetic radiation, ranging from long-wavelength radio waves to short-wavelength gamma rays.

Each type of radiation within the spectrum is characterized by its wavelength or frequency. The spectrum is usually divided into seven regions, in order of increasing frequency: Radio waves, Microwaves, Infrared, Visible light, Ultraviolet (UV), X-rays, and Gamma rays.

The ultraviolet region, in particular, sits between visible light and X-rays and typically covers wavelengths from about 10 nm to 400 nm. This is the region where the ozone molecule has a strong absorption at a wavelength of 145 nm, indicating that it's susceptible to being affected by UV radiation. UV radiation itself is further categorized into UVA, UVB, and UVC based on varying ranges of wavelengths.

Ultraviolet (UV) Radiation

Ultraviolet radiation is a form of energy traveling through space, just like other components of the electromagnetic spectrum. It has shorter wavelengths than visible light, which means it has higher energy and can be harmful in excessive amounts.

UV radiation is divided into three main types, based on its wavelength: UVA (320-400 nm), UVB (280-320 nm), and UVC (100-280 nm). Each type has different effects on biological materials. For instance, UVB is responsible for causing sunburn and skin cancers, while UVC is the most harmful but luckily is mostly absorbed by the Earth's atmosphere and does not reach the ground in significant amounts.

The 145 nm wavelength that causes photodissociation of O2 falls into the UVC category, highlighting the inherent energy that is high enough to break chemical bonds within the oxygen molecule.

Photon Energy Calculation

The energy of a photon is inversely proportional to its wavelength. To calculate this energy, we use the equation: \[E = \frac{hc}{\lambda}\]where E is the energy per photon, h is Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), c is the speed of light (\(3 \times 10^{8} \mathrm{m/s}\)), and \(\lambda\) is the wavelength in meters.

For a photon with a wavelength of 145 nm, the energy is calculated to be about \(1.37 \times 10^{-18} \mathrm{J}\). This is the photon's own quantum pack of energy, and if it's enough to surpass the bond energy of O2 (\(8.22 \times 10^{-19} \mathrm{J}\) for example), it could cause photodissociation, separating the oxygen into individual atoms.

Understanding the relationship between wavelength, frequency, and energy is crucial when studying the interactions of light with matter, especially in processes like photodissociation where precise energy thresholds are required to initiate chemical changes.