Question

Consider the following approximate visible light spectrum: Barium emits light in the visible region of the spectrum. If each photon of light emitted from barium has an energy of \(3.59 \times\) \(10^{-19} \mathrm{~J}\), what color of visible light is emitted?

Short answer

The emitted visible light from barium is green with a wavelength of 553 nm.

Step-by-step solution

Determine the energy of the photon

Given the information, each photon of light emitted from barium has an energy of \(3.59 \times 10^{-19}\) J.

Calculate the wavelength of the emitted light

To calculate the wavelength, use the Planck's formula, which relates the energy of a photon to its frequency: \[E = h \nu\] Where: - E = energy of the photon (\(3.59 \times 10^{-19}\) J) - \(\nu\) = frequency of the photon (in Hz) - h = Planck's constant (\(6.626 \times 10^{-34}\) Js) Since the speed of light (\(c\)) is given by the product of the wavelength (\(\lambda\)) and the frequency (\(\nu\)), the formula can be rewritten as: \[E = h \frac{c}{\lambda}\] Solve for the wavelength (\(\lambda\)): \[\lambda = \frac{hc}{E}\] Substitute the values of h, c, and E: \[\lambda = \frac{(6.626 \times 10^{-34}\ \text{Js})(3.00 \times 10^8\ \text{m/s})}{3.59 \times 10^{-19}\ \text{J}}\]

Solve for the wavelength

Perform the calculations to find the wavelength: \[\lambda = \frac{(6.626 \times 10^{-34}\ \text{Js})(3.00 \times 10^8\ \text{m/s})}{3.59 \times 10^{-19}\ \text{J}} = 5.53 \times 10^{-7}\ \text{m} = 553\ \text{nm}\]

Determine the color of the light

Compare the wavelength obtained in the previous step with the range of wavelengths associated with different colors of visible light. The approximate visible light spectrum (in nanometers) is: - Violet: 380 - 450 nm - Blue: 450 - 495 nm - Green: 495 - 570 nm - Yellow: 570 - 590 nm - Orange: 590 - 620 nm - Red: 620 - 750 nm Since the wavelength we calculated is 553 nm, it falls within the green color range.

Conclusion

Therefore, the emitted visible light from barium is green with a wavelength of 553 nm.

Key concepts

Photon Energy

Photon energy is a fundamental concept in understanding electromagnetic radiation, such as light. It determines the amount of energy carried by a single photon, the basic unit of light. The energy of a photon is directly proportional to its frequency, which means higher frequency light, like ultraviolet (UV), has more energy compared to lower frequency light, like infrared (IR).
This relationship is given by the equation:

• \(E = h u\)

Where:

• \(E\) - Energy of the photon (in Joules)
• \(h\) - Planck's constant, which is approximately \(6.626 \times 10^{-34}\) Js
• \(u\) - Frequency of the photon (in Hertz)

The importance of photon energy lies in its application across different fields, such as determining the color of light, photovoltaic systems, and energy transfer processes.

Wavelength Calculation

Calculating the wavelength of light is crucial to understanding how different colors and energies are represented in the visible spectrum. The wavelength of light determines its color and is inversely proportional to its frequency, meaning that as frequency increases, the wavelength decreases.
To calculate the wavelength, we can use the relationship between energy, Planck’s constant, and the speed of light. The formula is given as:

• \(\lambda = \frac{hc}{E}\)

Where:

• \(\lambda\) - Wavelength (in meters)
• \(h\) - Planck's constant (\(6.626 \times 10^{-34}\) Js)
• \(c\) - Speed of light (approximately \(3.00 \times 10^8\) m/s)
• \(E\) - Energy of the photon

By substituting the values given, you can determine the wavelength of light emitted, which will help in identifying its color on the visible spectrum.

Planck's Constant

Planck's constant is a fundamental principle in quantum mechanics and plays a key role in the quantization of energy levels. It acts as a bridge between the macro and micro worlds, linking the energy of photons to their frequency.
Planck's constant ( \(h\)) is approximately \(6.626 \times 10^{-34}\) Js. This tiny value indicates that the energy changes at the quantum level are very small.

• Planck's constant enables the calculation of energy for different electromagnetic waves.
• It is a fundamental component in the formula \(E = h u\), showing its critical role in photon energy determination.

Planck's constant is essential for studying quantum phenomena and has numerous practical applications, including in the fields of astronomy, nuclear physics, and even holography.

Photon Frequency

Photon frequency is a key factor that determines the energy and color of light. It is the rate at which photons oscillate per second and is measured in Hertz (Hz). The frequency is directly proportional to the energy, meaning photons with higher frequencies carry more energy.
The relationship with photon energy is expressed as:

• \(E = h u\)

Where:

• \(u\) - Frequency of the photon
• \(h\) - Planck's constant
• \(E\) - Energy of the photon

Frequency also has an inverse relationship with wavelength, described by:

• \(c = \lambda u\)

Where \(c\) is the speed of light. This means higher frequency results in a shorter wavelength and is critical in determining the band of the visible spectrum a specific color of light falls into. Understanding photon frequency is essential for identifying colors and energies of different spectral regions.