The Energy of Light: Videos & Practice Problems

Light is composed of particles known as photons, a concept introduced by physicists Max Planck and Albert Einstein. The energy of a photon can be calculated using Planck's constant (h), which is 6.626 x 10^-34 joules seconds. There are two key formulas for determining the energy of a photon: one involving its frequency:

$E=hf$

where frequency is measured in hertz, and the other its wavelength:

$E=\frac{hc}{\lambda }$

with wavelength measured in meters. Energy is directly proportional to frequency and inversely proportional to wavelength. To find the energy of a mole of photons, Avogadro's number (6.022 x 10²³ photons/mole) is used as a conversion factor, allowing the transition from energy per photon to energy per mole of photons. This relationship is crucial for understanding the behavior of light and its quantized nature in the realm of quantum mechanics.

Photons and Planck's Constant

Max Planck and Albert Einstein contributed significantly to our understanding of light, proposing that it consists of packets or particles known as photons. When discussing a collection of these light particles, the term "quantum" is often used. To determine the energy of a photon, we utilize Planck's constant, denoted by the symbol h, which is valued at approximately 6.626 × 10^-34 joules·seconds.

When calculating the energy of light, the relationship can be expressed through the formula:

E = hν

In this equation, E represents the energy of the photon, h is Planck's constant, and ν (nu) is the frequency of the light. This formula highlights the direct correlation between the energy of a photon and its frequency, emphasizing that higher frequency light carries more energy. Understanding this relationship is crucial for further exploration of electromagnetic radiation and its applications in various fields of physics.

Photon Energy Formulas

When calculating the energy of light in the form of a photon, two key formulas are essential. The first formula relates energy to frequency, while the second connects energy to wavelength. The more commonly used formula, which is important to memorize, is:

$$E = h \cdot f$$

In this equation, \(E\) represents the energy of a photon in joules, \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{J s}\)), and \(f\) is the frequency measured in hertz (Hz), which is equivalent to seconds inverse.

The second formula is used when calculating energy based on wavelength:

$$E = \frac{h \cdot c}{\lambda}$$

Here, \(c\) is the speed of light (\(3.00 \times 10^8 \, \text{m/s}\)), and \(\lambda\) is the wavelength in meters. This formula highlights the relationship between energy and wavelength.

Understanding these equations reveals that energy is directly proportional to frequency; as energy increases, frequency also increases. Conversely, energy is inversely proportional to wavelength, meaning that as energy or frequency rises, the wavelength must decrease. This relationship is crucial for grasping the behavior of photons in various contexts.

Photon Energy Example

To calculate the energy of a photon based on its wavelength, we can use the formula:

E = \frac{hc}{\lambda}

In this equation, E represents the energy of the photon, h is Planck's constant, c is the speed of light, and λ (lambda) is the wavelength of the photon.

For this specific calculation, we have:

• Planck's constant, h = 6.626 × 10^-34 joules·seconds
• Speed of light, c = 3.00 × 10⁸ meters/second
• Wavelength, λ = 293.7 meters

Substituting these values into the formula gives:

E = \frac{(6.626 \times 10^{-34} \, \text{J·s})(3.00 \times 10^{8} \, \text{m/s})}{293.7 \, \text{m}}

When performing the calculation, the units of seconds and meters cancel out, leaving us with joules as the final unit. The result of this calculation is:

E ≈ 6.768 × 10^-28 joules per photon

It's important to note that the significant figures in the final answer should reflect the precision of the given wavelength, which has four significant figures. Thus, the energy of the photon is expressed as 6.768 × 10^-28 joules per photon.

Moles and Photon Energy

To calculate the energy of a mole of photons, we start by determining the energy of a single photon, which is expressed in joules per photon. The energy of a photon can be calculated using the formula:

E = h \cdot f

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

Once we have the energy of one photon, we can find the energy for a mole of photons by utilizing Avogadro's number, which is \(6.022 \times 10^{23}\) photons per mole. The conversion from joules per photon to joules per mole of photons is straightforward:

E (per mole) = E (per photon) \times 6.022 \times 10^{23}

This relationship allows us to easily convert the energy calculated for a single photon into the total energy for a mole of photons, reinforcing the connection between microscopic and macroscopic scales in chemistry. Understanding this conversion is essential for applications in thermodynamics and quantum chemistry, where energy calculations are frequently required.

Moles and Photon Energy Example

To calculate the energy of photons based on their frequency, we utilize the relationship defined by the equation for photon energy:

E = h × ν

In this equation, E represents the energy of a photon, h is Planck's constant, and ν (nu) is the frequency of the photon. For this calculation, Planck's constant is given as:

h = 6.626 × 10^-34 J·s

The frequency provided is:

ν = 4.29 × 10¹⁵ s^-1

Substituting these values into the equation, we find:

E = (6.626 × 10^-34 J·s) × (4.29 × 10¹⁵ s^-1)

Upon performing the multiplication, we obtain:

E ≈ 2.842554 × 10^-18 J per photon

To express this energy in terms of moles of photons, we apply Avogadro's number, which is:

N_A = 6.022 × 10²³ photons/mol

Thus, to convert the energy per photon to energy per mole, we multiply the energy by Avogadro's number:

E_mole = E × N_A

Calculating this gives:

E_mole ≈ (2.842554 × 10^-18 J) × (6.022 × 10²³ photons/mol)

This results in:

E_mole ≈ 1.71 × 10⁶ J/mol

Finally, it is important to note that the significant figures in the final answer should reflect those of the initial frequency value, which had three significant figures. Therefore, the energy of the photons is approximately:

1.71 × 10⁶ J/mol

Calculate the energy (in nJ) of a photon emitted by a mercury lamp with a frequency of 6.88 x 10¹⁴ Hz.

• A. 5.46 x 10^-9 nJ
• B. 9.89 x 10^-9 nJ
• C. 2.89 x 10^-10 nJ
• D. 4.56 x 10^-10 nJ

Answer: (D)

A light ray has a wavelength that is 835 µm contains 6.32 x 10^-3 J of energy. How many photons does this light ray have?

• A. 7.98 x 10¹⁸ photons
• B. 4.85 x 10¹⁹ photons
• C. 9.48 x 10¹⁸ photons
• D. 2.65 x 10¹⁹ photons

Answer: (D)

How much energy (in kJ) do 4.50 moles of photons contain at a wavelength of 705 nm?

• A. 2.58 x 10² kJ
• B. 7.64 x 10² kJ
• C. 8.45 x 10² kJ
• D. 5.89 x 10² kJ

Answer: (B)

Here’s what students ask on this topic:

What is a photon?

A photon is a fundamental particle that represents the quantum of light or other electromagnetic radiation. It is the carrier of the electromagnetic force, even when static via virtual photons. Photons are unique because they have no mass and always move at the speed of light in a vacuum, which is approximately 299,792 kilometers per second (or about 186,282 miles per second).

Photons exhibit both wave-like and particle-like properties, a duality that is a cornerstone of quantum mechanics. When considering light as a wave, it is characterized by its wavelength or frequency. However, when considered as particles, interactions of light with matter, such as absorption and emission, are best described as discrete packets of energy, or quanta, which we call photons.

The energy of a photon is directly proportional to its frequency and inversely proportional to its wavelength. This relationship is expressed by the equation:

$E=hf$

where E is energy, h is Planck's constant, and f is the frequency of the photon. This quantized nature of light plays a critical role in understanding phenomena such as the photoelectric effect, lasers, and the behavior of atoms and molecules.

What is Planck's constant?

Planck's constant is a fundamental physical constant denoted by the symbol 'h' that relates the energy of a photon to the frequency of its electromagnetic wave. It is a key quantity in quantum mechanics, a branch of physics that deals with the behavior of particles at the atomic and subatomic levels.

The constant is named after Max Planck, the physicist who proposed it in 1900. Planck discovered that energy is not exchanged continuously but rather in discrete amounts called quanta. The value of Planck's constant is approximately 6.62607015 x 10^-34 joule-seconds. This small number indicates that the energy quanta are very tiny, which is why quantum effects are not noticeable in everyday life.

The equation $E=hf$ illustrates the direct proportionality between the energy (E) of a photon and its frequency (f), with Planck's constant (h) being the proportionality factor. This relationship is pivotal for understanding phenomena such as the photoelectric effect, and it laid the groundwork for the development of quantum theory.

What does Planck's constant represent?

Planck's constant is a fundamental physical constant denoted by the symbol 'h' that is crucial in the field of quantum mechanics. It represents the proportionality constant that relates the energy of a photon (the smallest quantum of light) to its frequency. The exact value of Planck's constant is approximately 6.62607015 x 10^-34 joule seconds. This constant is named after the physicist Max Planck, who first proposed it in 1900.

In the equation $E=hf$, 'E' stands for energy, 'h' is Planck's constant, and 'f' is the frequency of the photon. The introduction of Planck's constant marked a significant departure from classical physics, indicating that energy, at a microscopic level, is quantized and can only exist in discrete amounts rather than a continuous range. This quantization is the cornerstone of quantum theory, which governs the behavior of particles at atomic and subatomic scales.