Problem 129 - Chemistry An Atoms First Approach

Question

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.

Step-by-Step Solution

Answer

The frequency of the photon is approximately \(6.585 \times 10^{14} \, Hz\) and the energy of the photon is approximately \(4.359 \times 10^{-19} \, J\).

Step 1: Convert wavelength to meters

First, we need to convert the given wavelength (455.5 nm) to meters. We know that 1 nm is equal to \(1 \times 10^{-9}\) m. So, we can convert the wavelength to meters as follows: \(455.5 \, nm \times \frac{1\,m}{1 \times 10^9\, nm} = 4.555 \times 10^{-7} \, m\)

Step 2: Calculate the frequency

Next, we will use the equation \(ν = \frac{c}{λ}\) to calculate the frequency of the photon. The speed of light (c) is approximately \(3 \times 10^8 \, m/s\). So, the frequency will be: \(ν = \frac{3 \times 10^8\, m/s}{4.555 \times 10^{-7}\, m} ≈ 6.585 \times 10^{14} \, Hz\)

Step 3: Calculate the energy

Finally, we will calculate the energy of the photon using Planck's constant (h) and the frequency (ν). Planck's constant is approximately \(6.626 \times 10^{-34} \, Js\). The energy of the photon can be calculated as follows: \(E = hν = (6.626 \times 10^{-34}\, Js)(6.585 \times 10^{14}\, Hz) ≈ 4.359 \times 10^{-19} \, J\) So, the frequency of the photon is approximately \(6.585 \times 10^{14} \, Hz\) and the energy of the photon is approximately \(4.359 \times 10^{-19} \, J\).

Key Concepts

Wavelength Conversion

When dealing with wavelengths, it is crucial to convert them into consistent units for easy computation. In scientific calculations, wavelength is commonly measured in meters. However, it might also be given in nanometers (nm). One nanometer is equal to one billionth of a meter. This means that 1 nm equals \(1 \times 10^{-9}\) meters. To convert a wavelength given in nanometers to meters, you can multiply the number of nanometers by \(1 \times 10^{-9}\).

Let's walk through an example conversion. Suppose we have a wavelength of \(455.5\) nm, as in our exercise. We convert this to meters by multiplying:

• \(455.5\, \text{nm} \times \frac{1\, \text{m}}{1 \times 10^9\, \text{nm}} = 4.555 \times 10^{-7}\, \text{m}\)

This conversion is an essential first step in photon energy calculations, allowing us to use formulas that require measurements in meters.

Frequency Calculation

Frequency is a measure of how many waves pass a point in one second, and it's expressed in hertz (Hz). It is inversely related to wavelength through the speed of light. The formula to calculate frequency is \(u = \frac{c}{\lambda}\), where:

• \(u\) is the frequency
• \(c\) is the speed of light (approximately \(3 \times 10^8\, \text{m/s}\))
• \(\lambda\) is the wavelength in meters

Let's calculate the frequency using the wavelength we converted to meters (\(4.555 \times 10^{-7}\) m):

• \(u = \frac{3 \times 10^8\, \text{m/s}}{4.555 \times 10^{-7}\, \text{m}} \approx 6.585 \times 10^{14}\, \text{Hz}\)

This calculation is straightforward if the wavelength is correctly converted to meters first. The result gives us the frequency of the photon's light, which is key in determining other properties, such as energy.

Planck's Constant

Planck's constant is a fundamental value in physics that plays a key role in the field of quantum mechanics. It relates the energy of a photon to its frequency and is denoted by \(h\). The usual value for Planck's constant is approximately \(6.626 \times 10^{-34}\, \text{Js}\). With these units, energy can be calculated with the formula \(E = hu\), where:

• \(E\) is the energy of the photon in joules (J)
• \(h\) is Planck's constant
• \(u\) is the frequency calculated earlier

Using our frequency from the exercise (\(6.585 \times 10^{14}\, \text{Hz}\)):

• \(E = (6.626 \times 10^{-34}\, \text{Js})(6.585 \times 10^{14}\, \text{Hz}) \approx 4.359 \times 10^{-19}\, \text{J}\)

By multiplying Planck's constant with frequency, we derive the energy of a single photon of the light, revealing the quantum nature of light through simple arithmetic.