Question

What is the energy of red light with a wavelength of \(660.5 \mathrm{~nm}\) ?

Short answer

The energy of the red light with a wavelength of 660.5 nm is approximately \(2.9987 \times 10^{-19}\) joules.

Step-by-step solution

Convert wavelength to meters

We are given the wavelength of the red light as 660.5 nm. We need to convert this value to meters in order to use it in our calculations. 1 nm = 10^-9 meters So, 660.5 nm = 660.5 x 10^-9 meters

Calculate the energy of the red light

Now, let's plug in the values into the formula for the energy of a photon: Energy (E) = \( \dfrac{hc}{λ} \) Where: h = 6.626 x 10^-34 J·s (Planck's constant) c = 2.998 x 10^8 m/s (speed of light) λ = 660.5 x 10^-9 m (wavelength in meters) Energy (E) = \( \dfrac{(6.626 \times 10^{-34} \text{J} \cdot \text{s})(2.998 \times 10^{8} \text{m}/\text{s})}{660.5 \times 10^{-9} \text{m}} \)

Simplify and find the energy

Now, we will simplify this equation and find the energy of the red light. Energy (E) ≈ \( \dfrac{(6.626 \times 10^{-34})(2.998 \times 10^{8})}{660.5 \times 10^{-9}} \) J Energy (E) ≈ 2.9987 x 10^-19 J The energy of the red light with a wavelength of 660.5 nm is approximately 2.9987 x 10^-19 joules.

Key concepts

Planck's constant

Planck's constant is a fundamental physical constant that is imperative to the field of quantum mechanics. It is denoted by the symbol \( h \) and has a value of \(6.626 \times 10^{-34}\) Joule-seconds. This constant was introduced by Max Planck, the founder of quantum theory, and it essentially quantifies the discrete and smallest possible unit of energy change at the quantum level. In simpler terms, it helps relate the energy of photons to their frequency.
When calculating photon energy, Planck's constant plays a key role within the formula:

• Energy (\( E \)) = \( \dfrac{hc}{λ} \)

Here, \( h \) represents Planck's constant, \( c \) is the speed of light, and \( λ \) is the wavelength of light. This equation shows that energy is directly proportional to frequency (or inversely proportional to wavelength), highlighting the significance of Planck's constant in energy-related calculations within quantum mechanics.
Understanding the role of Planck's constant allows us to bridge classical and modern physics, enabling calculations that are essential for technology such as lasers, semiconductors, and even our daily-use devices.

Wavelength conversion

Wavelength conversion is a critical step in many physics calculations, particularly when dealing with photon energy. Understanding and executing this step accurately is important as it allows for consistent units across formulas.
In our given problem, the wavelength is provided in nanometers (nm) and needs to be converted to meters, since SI units are typically required for these types of calculations. The conversion factor for nanometers to meters is:

• 1 nm = \(10^{-9}\) meters

This implies that a wavelength of \(660.5\) nm converts to \(660.5 \times 10^{-9}\) meters.
Such conversions are vital because they standardize calculations, ensuring that all components in an equation are compatible. When performing scientific calculations, adhering to consistent units helps prevent errors and maintains accuracy, which is crucial for deriving meaningful results, like calculating the energy of a photon.

Speed of light

The speed of light, denoted by \( c \), is a fundamental constant of nature that plays a critical role in the equations of physics. It is defined as \(2.998 \times 10^{8}\) meters per second (m/s). This value is instrumental when analyzing the behavior of photons and electromagnetic waves.
In the context of photon energy calculations, the speed of light links the wavelength and frequency of a photon through the relationship:

• Speed of light (\( c \)) = wavelength (\( λ \)) \( \times \) frequency (\( ν \))

By manipulating this equation, one can derive relationships such as \( u = \dfrac{c}{\lambda} \), which connects it to Planck's formula for energy.
The speed of light is not just a parameter in equations; it represents a limit on how fast information can travel in the universe. Understanding its place in physics and its integration with quantities like wavelength and frequency allows us to comprehend the broader implications regarding time, space, and energy.