Problem 12 - Chemistry The Molecular Nature of Matter

Question

Show that the energy of a photon is given by the equation $$ E=\frac{h c}{\lambda} $$

Step-by-Step Solution

Answer

The energy of a photon is given by the equation \( E = \frac{h c}{\lambda} \) by relating frequency to wavelength and substituting into the Planck-Einstein relation.

Step 1: Understand the Relationship Between Energy and Frequency

The energy, E, of a photon is directly proportional to its frequency, f, according to the Planck-Einstein relation: \( E = h f \), where \( h \) is Planck's constant.

Step 2: Relate Frequency to Wavelength and Speed of Light

The speed of light, c, is related to the frequency and wavelength, \( \lambda \), of the photon by the equation \( c = \lambda f \). Therefore, you can express frequency as \( f = \frac{c}{\lambda} \).

Step 3: Substitute Frequency in the Energy Equation

Now substitute the expression for frequency from Step 2 into the initial energy equation from Step 1 to get \( E = h \frac{c}{\lambda} \).

Key Concepts

Planck-Einstein Relation

The Planck-Einstein relation is a fundamental concept in quantum mechanics that links the energy of a photon to its frequency. The equation for this relationship is:
\[ E = hf \
where:\]

• \( E \) is the energy of the photon,
• \( h \) is Planck's constant, approximately \(6.626 x 10^{-34} J\cdot s\), and
• \( f \) is the frequency of the photon.

This equation implies that the energy contained within a single photon is proportional to its frequency: the higher the frequency, the higher the energy. This principle is essential for understanding how light interacts with matter. For instance, it helps explain why ultraviolet light can cause sunburns—its higher frequency (compared to visible light) means it carries more energy that can damage skin cells.

Speed of Light

The speed of light, denoted \( c \), is one of the most important constants in physics. Its value is approximately \(3.00 x 10^8 m/s\) in a vacuum, and it represents the maximum speed at which all energy, matter, and information in the universe can travel. The constancy of the speed of light is a key component of Einstein's theory of relativity. Given that this speed is always the same, regardless of the observer’s frame of reference, it has far-reaching implications for our understanding of space and time. When working with the Planck-Einstein relation, the speed of light allows us to relate the energy of a photon to its wavelength, \( \lambda \), providing a comprehensive picture of the photon's characteristics.

Why is \( c \) Crucial in the Photon Energy Equation?

Using the speed of light, one can calibrate instruments to accurately measure the distance light travels over time, which is vital for experiments and calculations in both quantum mechanics and astrophysics. It is c that underpins the photon energy equation, helping to translate wavelength into frequency, and thus into energy.

Wavelength and Frequency Relationship

Wavelength (\( \lambda \)) and frequency (\( f \)) are inversely related properties of a photon, connected through the speed of light with the equation:
\[ c = \lambda f \
This equation illustrates that as the wavelength of a photon increases, its frequency decreases, and vice versa, while the speed of light remains constant. This wavelength-frequency relationship is crucial when analyzing the energy of a photon. It allows the substitution of frequency in the Planck-Einstein relation with a term involving wavelength, hence the energy equation becomes:\]

• E = \frac{hc}{\lambda}

Practical Implications

Understanding the relationship between wavelength and frequency has practical applications in various fields such as spectroscopy, telecommunications, and medical imaging. For example, different wavelengths are used in optical fibers to transmit data over long distances, and different frequencies are chosen for magnetic resonance imaging (MRI) to obtain clear images of different types of tissues.