Problem 119 - Chemistry Matter and Change

Question

Calculate the energies of photons related to the two lines using the relationships expressed in the following equations. \(E_{\text { photon }}=h v ; c=\lambda v ; E=h c / \lambda\)

Step-by-Step Solution

Answer

To calculate the energies of the photons for the two lines, use the equation \(E=h\cdot c/ \lambda\). Plug in the values of Planck's constant (h), speed of light (c) and the given wavelengths (λ) for each line separately, and calculate the energies accordingly. Please provide the wavelengths of the two lines to complete the calculations.

Step 1: Understand the given equations

The three relationships given are: 1. \(E_{photon} = h \cdot v\): The energy of a photon is the product of Planck's constant (h) and its frequency (v). 2. \(c = \lambda \cdot v\): The speed of light (c) is equal to the product of the wavelength (λ) and frequency (v). 3. \(E=h\cdot c/ \lambda\): The energy of a photon can be expressed as the product of Planck's constant (h) and the speed of light (c) divided by the wavelength (λ).

Step 2: Relate equations 1 and 2 to find a new equation for energy

Since we have \(E_{photon} = h \cdot v\) and \(c = \lambda \cdot v\), we can solve for frequency (v) in Equation 2 in terms of wavelength (λ). Divide both sides of Equation 2 by λ: \(v = \frac{c}{\lambda}\) Now substitute this expression for v in Equation 1: \(E_{photon} = h \cdot \frac{c}{\lambda}\) This gives us the same expression as Equation 3, confirming its validity.

Step 3: Calculate the energies of photons for the two lines

To calculate the energy of a photon for each line, we need to use Equation 3: \(E=h\cdot c/ \lambda\). However, we are not given any values for wavelengths, so we cannot continue with further calculations. If we had the values for the wavelengths, we could follow these steps to calculate the energy of the photons for the two lines: 1. Plug in the values of Planck's constant (h) and speed of light (c) and the given wavelengths (λ) for each line separately. 2. Calculate the energy of the photons for each line accordingly. Please provide the wavelengths of the two lines to complete the calculations.

Key Concepts

Planck's Constant

Planck's Constant, typically denoted as \( h \), is a fundamental constant in physics, playing a crucial role in the realm of quantum mechanics. It allows us to understand the quantization of energy levels in various systems. The constant has a value of approximately \(6.62607015 \times 10^{-34} \text{ Js}\). This value is essential when we're looking at phenomena on a microscopic scale, particularly when discussing the relationship between the energy of photons and their frequency.
\[ \]The key formula that involves Planck's constant is:

• \(E_{photon} = h \cdot v\)

This equation tells us that the energy \(E_{photon}\) of a photon can be determined by multiplying Planck's constant \(h\) by its frequency \(v\). Understanding this relationship helps in explaining how energy is transferred at the atomic and subatomic levels, bridging the gap between classical concepts and quantum physics.

Speed of Light

The Speed of Light in a vacuum, commonly symbolized as \( c \), is an essential constant in physics, with a value of approximately \(3.00 \times 10^8 \text{ m/s}\). It represents the highest speed at which information or matter can travel, serving as a fundamental limit in the universe.
\[ \]When it comes to photon energy calculations, the speed of light connects with both the wavelength \(\lambda\) and frequency \(v\) of electromagnetic radiation. This relationship is captured in the equation:

• \(c = \lambda \cdot v\)

This formula suggests that as the frequency of a wave increases, the wavelength must decrease for the speed of light to remain constant. This correlation is particularly valuable in discussing light's behavior and properties, such as when photons transition between energy levels or travel through different mediums.

Wavelength-Frequency Relationship

The Wavelength-Frequency Relationship is a critical concept in understanding electromagnetic waves. Wavelength \(\lambda\) and frequency \(v\) are inversely related through the speed of light \(c\).
This relationship is mathematically expressed by the equation:

• \(\lambda = \frac{c}{v}\)

This indicates that as the frequency of a wave increases, its wavelength must decrease, ensuring that their product equals the unchanging speed of light. This inverse relationship is key to understanding the behavior of light across different spectrums.
\[ \]Additionally, in photon energy calculations, this relationship helps us derive the formula \(E = h \cdot \frac{c}{\lambda}\). By combining Planck's constant \(h\), the speed of light \(c\), and the wavelength \(\lambda\), we can understand how the energy of a photon is dependent on its characteristics. This helps explain phenomena such as color variations in light and spectral emissions from different sources.