Problem 40 - Chemistry: An Atoms-Focused Approach

Question

The first ionization energy of a gas-phase atom of a particular element is \(6.24 \times 10^{-19} \mathrm{J} .\) What is the maximum wavelength of electromagnetic radiation that could ionize this atom?

Step-by-Step Solution

Answer

Answer: The maximum wavelength of electromagnetic radiation capable of ionizing this atom is 100 nm.

Step 1: Understand the energy-wavelength relationship

Planck's equation states that the energy of a photon of electromagnetic radiation (E) is directly proportional to its frequency (ν) and can be expressed as: E = hν where h is Planck's constant, approximately \(6.63 \times 10^{-34} \mathrm{Js}\). The frequency of the electromagnetic radiation is related to its wavelength (λ) by the formula: ν = c/λ where c is the speed of light, approximately \(3.00 \times 10^8 \mathrm{m/s}\). We can combine these two equations to express the energy in terms of the wavelength: E = h(c/λ)

Step 2: Rearrange the equation for the wavelength, λ

To find the maximum wavelength, we need to rearrange the equation for λ: λ = hc/E

Step 3: Plug in the given values and constants

Now we plug in the given values and constants into the equation to find the maximum wavelength: λ = (\(6.63 \times 10^{-34} \mathrm{Js}\))(\(3.00 \times 10^8 \mathrm{m/s}\)) / (\(6.24 \times 10^{-19} \mathrm{J}\))

Step 4: Calculate the maximum wavelength

Calculate the maximum wavelength: λ = \(1.00 \times 10^{-7} \mathrm{m}\) Since we need to express the wavelength in nanometers (nm), we convert it: \(λ = 1.00 \times 10^{-7} \mathrm{m} \times \frac{10^9 \, nm}{1 \, m} = 100 \, nm\) So, the maximum wavelength of electromagnetic radiation that could ionize this atom is 100 nm.