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Chapter 7: Problem 121

Cesium was discovered in natural mineral waters in 1860 by R. W. Bunsen and G. R. Kirchhoff using the spectroscope they invented in \(1859 .\) The name came from the Latin caesius ("sky blue") because of the prominent blue line observed for this element at \(455.5 \mathrm{~nm} .\) Calculate the frequency and energy of a photon of this light.

Short Answer

Expert verified
The frequency of the photon of light with a wavelength of 455.5 nm is approximately \(6.58 \times 10^{14} \mathrm{Hz}\), and its energy is approximately \(4.36 \times 10^{-19} \mathrm{J}\).

Step by step solution

01

Convert the wavelength to meters

First, we need to convert the given wavelength from nanometers to meters, as this is the unit that we need for further calculations. To do this, we can use the conversion factor: \(1 \text{ nm} = 10^{-9} \text{ m}\) So, the wavelength in meters is: \(λ = 455.5 \times 10^{-9} \mathrm{m}\) Step 2: Calculate the frequency using the speed of light equation
02

Calculate the frequency using the speed of light equation

The speed of light equation is given as follows: \(c = λν\) where \(c\) is the speed of light (approximately \(3 \times 10^8 \mathrm{m/s}\)), λ is the wavelength (in meters), and \(ν\) is the frequency. Using the given wavelength, we can solve for the frequency: \(ν = \frac{c}{λ} = \frac{3 \times 10^8 \mathrm{m/s}}{455.5 \times 10^{-9} \mathrm{m}}\) Step 3: Calculate the value of the frequency
03

Calculate the value of the frequency

Using a calculator, we can find the value of the frequency: \(ν = 6.58 \times 10^{14} \mathrm{Hz}\) So, the frequency of the photon of light is \(6.58 \times 10^{14} \mathrm{Hz}\). Step 4: Calculate the energy using Planck's equation
04

Calculate the energy using Planck's equation

Planck's equation, which relates the energy of a photon to its frequency, is given as follows: \(E = hν\) where \(E\) is the energy of the photon, \(h\) is Planck's constant (approximately \(6.63 \times 10^{-34} \mathrm{Js}\)), and \(ν\) is the frequency. Using the calculated frequency, we can find the energy of the photon: \(E = (6.63 \times 10^{-34} \mathrm{Js}) (6.58 \times 10^{14} \mathrm{Hz})\) Step 5: Calculate the value of the energy
05

Calculate the value of the energy

Using a calculator, we can find the value of the energy: \(E = 4.36 \times 10^{-19} \mathrm{J}\) So, the energy of the photon of light is \(4.36 \times 10^{-19} \mathrm{J}\). In conclusion, the frequency of the photon of light with a wavelength of 455.5 nm is approximately \(6.58 \times 10^{14} \mathrm{Hz}\), and its energy is approximately \(4.36 \times 10^{-19} \mathrm{J}\).

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