Question

The quantum yield for \(\mathrm{CO}(g)\) production in the photolysis of gaseous acetone is unity for wavelengths between 250 and \(320 \mathrm{nm} .\) After 20.0 min of irradiation at \(313 \mathrm{nm}\) \(18.4 \mathrm{cm}^{3}\) of \(\mathrm{CO}(g)\) (measured at \(1008 \mathrm{Pa}\) and \(22^{\circ} \mathrm{C}\) ) is produced. Calculate the number of photons absorbed and the absorbed intensity in \(\mathrm{J} \mathrm{s}^{-1}\)

Short answer

Solution: Step 1: Calculate the moles of CO n = (1008 × 18.4 × 10⁻⁶) / (8.314 × 295.15) Step 2: Calculate the number of absorbed photons Number of photons = n × Nₐ Step 3: Calculate the energy per photon ν = c / λ E = hν Step 4: Calculate the absorbed intensity I = (Total energy absorbed) / (1200 s)

Step-by-step solution

Calculate the moles of CO

First, we need to convert the produced CO gas volume into moles. For this, we can use the Ideal Gas Law equation: PV=nRT where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin. Given: P = 1008 Pa V = 18.4 cm³ = 18.4 × 10⁻⁶ m³ R = 8.314 J mol⁻¹ K⁻¹ T = 22°C + 273.15 = 295.15 K Rearrange the equation to find n: n = PV/(RT) n = (1008 × 18.4 × 10⁻⁶) / (8.314 × 295.15)

Calculate the number of absorbed photons

Since the quantum yield is unity, the number of absorbed photons is equal to the number of CO molecules produced. Use Avogadro's number (Nₐ = 6.022 × 10²³ mol⁻¹) to find the total number of CO molecules produced, which equals the total number of absorbed photons. Number of photons = n × Nₐ

Calculate the energy per photon

Now, we need to calculate the energy per absorbed photon using Planck's equation: E = hν where E is energy, h is Planck's constant (6.63 × 10⁻³⁴ J s), and ν is the frequency of light. We are given the wavelength as 313 nm. We need to convert it to meters and then calculate the frequency using the speed of light (c = 3 × 10⁸ m/s): ν = c / λ Calculate the energy per photon using Planck's equation.

Calculate the absorbed intensity

Now, we have all the information needed to calculate the absorbed intensity. The absorbed intensity (I) is the total energy absorbed per unit time (20 minutes = 1200 seconds). First, calculate the total energy absorbed by multiplying the number of photons by the energy per photon. Then, divide the total energy by the irradiation time (1200 s) to find the absorbed intensity in J s⁻¹. Once all the calculations are complete, you will have the number of photons absorbed and the absorbed intensity in J s⁻¹.

Key concepts

Understanding the Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics, providing a clear relationship between the pressure, volume, temperature, and number of moles of an ideal gas. Represented by the equation PV = nRT, it describes the behavior of ideal gases under various conditions.

Let's break down the components of this equation:

• P stands for pressure, which is the force exerted by the molecules of the gas on the walls of its container.
• V is the volume that the gas occupies.
• n represents the number of moles of the gas.
• R is the ideal gas constant, which is 8.314 J∙mol⁻¹∙K⁻¹.
• T is the temperature in Kelvin, which must be used for this equation to be accurate.

To solve for any one of these variables, you simply rearrange the equation accordingly. In the context of our exercise, we used the Ideal Gas Law to find the number of moles (n) of CO gas produced during the photolysis of acetone, demonstrating the practical application of the law in a real-world scenario.

Its usefulness extends to many areas such as calculating gas densities, molar masses, and is also essential in stoichiometry - particularly when dealing with reactions involving gases.

Planck's Equation and the Energy of Photons

Planck's equation is pivotal to quantum mechanics, as it makes it possible to determine the energy of a single photon. The equation is articulated as E = hν, where E represents the energy of the photon, h is Planck's constant (6.626 x 10⁻³⁴ J∙s), and ν is the frequency of the electromagnetic wave associated with the photon.

In our exercise, we convert a given wavelength to frequency through the relationship ν = c / λ, where c is the speed of light and λ is the wavelength. Then, we use Planck’s equation to discover the energy associated with photons of this frequency. This calculation is essential when studying photochemical reactions, like the photolysis of acetone, to fathom the energy absorbed and the subsequent chemical changes. Understanding this allows us to delve into concepts such as the Photoelectric Effect and the dual nature of light - showcasing light's particle-like properties alongside its wave-like characteristics.

Avogadro's Number and the Mole Concept

Avogadro's number is a cornerstone of chemical quantification, bridging the gap between the microscopic scale of atoms and molecules and the macroscopic world we measure in the lab. It is defined as the number of constituent particles (usually atoms or molecules) in one mole of a substance, which is 6.022 x 10²³ particles/mol.

This immense number allows chemists to work with the mole, a unit that equates a set number of particles to a measurable amount that can be handled and experimented upon. In the context of our photolysis problem, Avogadro's number is used to convert the number of moles of CO gas produced into the actual number of CO molecules - and by extension, photons - since the quantum yield informs us that each photon absorbed will yield one molecule of CO.

Such a conversion is vital in helping us conceptualize and quantify chemical reactions on a scale that makes sense, from the particles involved to comparing different substances through the mole concept. Avogadro's number essentially provides the link that turns abstract molecular encounters into tangible chemical outcomes.