Question

The bright yellow light emitted by a sodium vapor lamp consists of two emission lines at 589.0 and 589.6 nm. What are the frequency and the energy of a photon of light at each of these wavelengths? What are the energies in kJ/mol?

Short answer

The frequency and energy of a photon of light at \(589.0 nm\) are \(5.093 \times 10^{14} Hz\) and \(203.0 kJ/mol\), respectively. The frequency and energy of a photon of light at \(589.6 nm\) are \(5.088 \times 10^{14} Hz\) and \(202.9 kJ/mol\), respectively.

Step-by-step solution

Convert Wavelengths to Meters

First, we need to convert the wavelengths from nanometers (nm) to meters (m) by multiplying by \(10^{-9}\). This is because the speed of light is in meters/second. \(589.0 nm = 589.0 \times 10^{-9}m = 5.89 \times 10^{-7}m\) \(589.6 nm = 589.6 \times 10^{-9}m = 5.896 \times 10^{-7}m\)

Calculate Frequency

Now, we will calculate the frequency of each wavelength using the formula: \(v = \frac{c}{\lambda}\) For wavelength \(589.0 nm\): \(v = \frac{3.0 \times 10^8 m/s}{5.89 \times 10^{-7} m} = 5.093 \times 10^{14} Hz\) For wavelength \(589.6 nm\): \(v = \frac{3.0 \times 10^8 m/s}{5.896 \times 10^{-7} m} = 5.088 \times 10^{14} Hz\)

Calculate Energy

Using the frequencies we just found, we can now calculate the energy of a photon of light at each wavelength using the formula \(E = h \times v\): For wavelength \(589.0 nm\): \(E = (6.626 \times 10^{-34} Js)(5.093 \times 10^{14} Hz) = 3.372 \times 10^{-19} J\) For wavelength \(589.6 nm\): \(E = (6.626 \times 10^{-34} Js)(5.088 \times 10^{14} Hz) = 3.370 \times 10^{-19} J\)

Convert Energies to kJ/mol

Finally, we need to convert the energies from joules to kJ/mol using Avogadro's number: For wavelength \(589.0 nm\): \(\frac{3.372 \times 10^{-19} J}{1 photon} \times \frac{10^3 kJ}{1 J} \times \frac{6.022 \times 10^{23} photons}{1 mol} = 203.0 kJ/mol\) For wavelength \(589.6 nm\): \(\frac{3.370 \times 10^{-19} J}{1 photon} \times \frac{10^3 kJ}{1 J} \times \frac{6.022 \times 10^{23} photons}{1 mol} = 202.9 kJ/mol\) So, the frequency and energy of a photon of light at \(589.0 nm\) are \(5.093 \times 10^{14} Hz\) and \(203.0 kJ/mol\), respectively. The frequency and energy of a photon of light at \(589.6 nm\) are \(5.088 \times 10^{14} Hz\) and \(202.9 kJ/mol\), respectively.

Key concepts

Wavelength to Frequency Conversion

Understanding how to convert a wavelength to frequency is a crucial concept in the world of physics. Wavelength represents the distance between successive peaks of a wave and is usually measured in nanometers for light waves. To find the frequency, denoted as \(v\), you use the relationship through the formula:

• \[ v = \frac{c}{\lambda} \]

where \(c\) is the speed of light \( (3.0 \times 10^8 \, \text{m/s}) \) and \(\lambda\) is the wavelength. Conversion between units is essential here. Always convert nanometers to meters, as the speed of light uses meters per second. This conversion ensures the two units are compatible in equations.
By following this process, you can calculate the frequency for any wavelength, giving insight into the properties of the light involved. For example, light at 589.0 nm has a frequency of \(5.093 \times 10^{14} \, \text{Hz}\).
Remember, understanding both the concepts and the math will help you navigate more complex scenarios in wave physics.

Energy per Photon

Energy per photon is another foundational concept in understanding light's properties. Each photon carries energy, and the amount can be determined using the formula:

• \[ E = h \times v \]

where \(E\) is the energy in joules (J), \(h\) is Planck's constant \( (6.626 \times 10^{-34} \, \text{Js}) \), and \(v\) is the frequency, which we already calculated earlier.
If you have the frequency, it's simple to find the energy. For instance, using the frequency \(5.093 \times 10^{14} \, \text{Hz}\), the calculated energy will be \(3.372 \times 10^{-19} \, \text{J}\).
Understanding the energy per photon helps explain phenomena such as the photoelectric effect and forms the basis for technologies like photovoltaic cells and other light-related applications.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry and physics, used to relate macroscopic measurements to atomic scale calculations. This number, \(6.022 \times 10^{23}\), represents the quantity of molecules or atoms in one mole of a substance.
In the context of photon energy calculations, Avogadro's number is vital when converting the energy of individual photons (in joules) to energy per mole of photons (in kJ/mol). By multiplying the energy per photon by Avogadro's number, you can find how much energy is present in a mole of photons. For sodium vapor light at 589.0 nm, this conversion gives us insight into the effective energy when large numbers of photons are involved.

Unit Conversion from Joules to kJ/mol

Converting energy values from joules to kilojoules per mole is critical for understanding practical energy values in chemical and physical processes. When you multiply the energy \( E \) of a single photon by Avogadro's number, you'll be converting from a microscopic to a macroscopic scale.
For instance, you might have found the energy per photon, like \(3.372 \times 10^{-19} \, \text{J}\). By multiplying by Avogadro's number and converting joules to kilojoules, the practical energy level becomes discernible.

• Formula to remember: \[ \frac{E \, (\text{in J})}{1 \, \text{photon}} \times 10^3 \frac{\text{kJ}}{\text{J}} \times 6.022 \times 10^{23} \frac{\text{photons}}{\text{mol}} = \text{Energy in kJ/mol} \]

Understanding these conversions is key to studying reactions and processes in chemistry where energy scales from individual particles to bulk quantities of material.