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Chapter 3: Problem 40

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?

Short Answer

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

Step by step solution

01

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 (谓) and can be expressed as: E = h谓 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 (位) by the formula: 谓 = c/位 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/位)
02

Rearrange the equation for the wavelength, 位

To find the maximum wavelength, we need to rearrange the equation for 位: 位 = hc/E
03

Plug in the given values and constants

Now we plug in the given values and constants into the equation to find the maximum wavelength: 位 = (\(6.63 \times 10^{-34} \mathrm{Js}\))(\(3.00 \times 10^8 \mathrm{m/s}\)) / (\(6.24 \times 10^{-19} \mathrm{J}\))
04

Calculate the maximum wavelength

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

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