Question

(a) Ozone formation in the upper atmosphere starts when oxygen molecules absorb UV radiation with wavelengths less than or equal to \(242 \mathrm{nm}\). Find the frequency and energy of the least energetic of these photons. (b) Ozone absorbs radiation with wavelengths in the range \(2200-2900\) A thus protecting organisms from this radiation. Find the frequency and energy of the most energetic of these photons.

Short answer

(a) Frequency: \( 1.24 \times 10^{15} \ \mathrm{Hz} \), Energy: \( 8.22 \times 10^{-19} \ \mathrm{J} \). (b) Frequency: \( 1.36 \times 10^{15} \ \mathrm{Hz} \), Energy: \( 9.01 \times 10^{-19} \ \mathrm{J} \).

Step-by-step solution

- Understand the Problem

We need to find the frequency and energy of photons based on given wavelengths. The two parts address different wavelength ranges: (a) for the least energetic photon with wavelength ≤ 242 nm and (b) for the most energetic photon with wavelength in the range 2200-2900 Å.

- Convert Wavelength to Frequency

Use the formula for the relationship between wavelength \( \lambda \) and frequency \( u \): \[ u = \frac{c}{\lambda} \] Here, \( c \) is the speed of light (approximately \( 3.00 \times 10^8 \ \mathrm{m/s} \)).

- Calculate Frequency for Part (a)

For \( \lambda = 242 \ \mathrm{nm} = 242 \times 10^{-9} \ \mathrm{m} \): \[ u = \frac{3.00 \times 10^8 \ \mathrm{m/s}}{242 \times 10^{-9} \ \mathrm{m}} = 1.24 \times 10^{15} \ \mathrm{Hz} \]

- Calculate Energy for Part (a)

Use Planck's equation to find the energy \( E \) of the photon: \[ E = h u \] Here, \( h \) is Planck's constant (approximately \( 6.626 \times 10^{-34} \ \mathrm{Js} \)): \[ E = (6.626 \times 10^{-34} \ \mathrm{Js}) (1.24 \times 10^{15} \ \mathrm{Hz}) = 8.22 \times 10^{-19} \ \mathrm{J} \]

- Convert Wavelength to Frequency for Part (b)

The most energetic photon in the range 2200-2900 Å has the shortest wavelength (2200 Å = 220 nm = 220 \times 10^{-9} m): \[ u = \frac{3.00 \times 10^8 \ \mathrm{m/s}}{220 \times 10^{-9} \ \mathrm{m}} = 1.36 \times 10^{15} \ \mathrm{Hz} \]

- Calculate Energy for Part (b)

Using Planck's equation for the highest frequency: \[ E = h u \] \[ E = (6.626 \times 10^{-34} \ \mathrm{Js}) (1.36 \times 10^{15} \ \mathrm{Hz}) = 9.01 \times 10^{-19} \ \mathrm{J} \]

Key concepts

Wavelength to Frequency Conversion

Understanding how to convert wavelength to frequency is a fundamental skill in physics. The relationship between wavelength (\lambda) and frequency (u) is defined by the equation: \[ u = \frac{c}{\lambda} \] where c is the speed of light (approximately \[ 3.00 \times 10^8 \mathrm{m/s} \]). This means that as the wavelength of light decreases, its frequency increases. To apply this, you simply need to divide the speed of light by the given wavelength. In our problem, to find the frequency of a photon with a wavelength of 242 nm, we first convert it to meters (1 nm = \[10^{-9} \mathrm{m} \]), then use the formula to get the frequency.

Planck's Equation

Planck's equation connects the energy (E) of a photon to its frequency (u). It is given by: \[ E = hu \] where h is Planck's constant (approximately \[6.626 \times 10^{-34} \mathrm{Js} \]). This means that the higher the frequency of a photon, the more energy it carries. To find the energy of a photon after calculating its frequency, you simply multiply the frequency by Planck's constant. For instance, having already determined the frequency of the least energetic photon at 1.24 \times 10^{15} \mathrm{Hz}, we use Planck's equation to find its energy.

Ultraviolet Radiation Absorption

Ultraviolet (UV) radiation is a type of electromagnetic radiation with a wavelength shorter than that of visible light. When oxygen molecules in the upper atmosphere absorb UV radiation, it results in chemical reactions vital for life on Earth, like the formation of ozone. UV radiation has wavelengths ranging from 10 nm to 400 nm. The absorption of UV radiation by molecules like ozone help protect living organisms by filtering out harmful UV light from the sun. This photochemical reaction initiates crucial processes.

Ozone Formation

Ozone (O3) forms in the upper atmosphere when oxygen molecules (O2) absorb UV radiation. This absorption splits O2 into two individual oxygen atoms. These single oxygen atoms then react with other O2 molecules to form ozone: \[ O_2 + UV \rightarrow 2O \] \[ O + O_2 \rightarrow O_3 \] This process is crucial for creating the ozone layer, which shields the Earth from most of the sun's harmful ultraviolet radiation. Without this ozone layer, life on Earth would be exposed to far more intense UV radiation.

Energy of Photons

The energy of a photon is crucial in understanding many physical and chemical processes. Photon energy depends on its frequency (or inversely on its wavelength). Higher energy photons have higher frequencies and lower wavelengths. This is critical in explaining why different types of radiation have different impacts. For instance, UV photons have enough energy to cause chemical reactions or damage biological tissues, while visible light generally does not. Calculations of photon energy using Planck's equation are fundamental in quantum mechanics and help elucidate many phenomena in physics and chemistry.