The electronic transition in Na from \(3 p^{1}\) to \(3 s^{1}\) gives rise to a bright yellow-orange emission at \(589.2 \mathrm{nm}\). What is the energy of this transition?

To calculate the energy of a photon, we use a fundamental equation in quantum mechanics. The equation is \( E = \frac{hc}{abla}\), and it links several important constants and variables together, providing a direct way to determine the energy of light from its wavelength.

Here, \( E \) represents the energy of the photon. This is the value we are trying to find.
The term \( h \) stands for Planck’s constant, which we'll discuss more later, and \( c \) is the speed of light.
Lastly, \( abla \) is the wavelength of the photon. The wavelength is essential because it tells us how long the waves of light are, and from this, we can determine its energy.

Plugging these values into the equation involves substituting the constants \( h \) and \( c \), and the specific wavelength \( abla \) of the light we are examining.

Before using the wavelength in our calculations, it must often be converted to meters, the standard unit of length in the International System of Units (SI).

Wavelengths are frequently given in nanometers (nm), especially in the context of visible light. One nanometer is equal to \( 10^{-9} \) meters.

• To convert nanometers to meters, multiply the wavelength by \( 10^{-9} \).
• For example, for our problem, the wavelength \( 589.2 \textrm{ nm} \) becomes \( 589.2 \times 10^{-9} = 5.892 \times 10^{-7} \textrm{m} \).

This step ensures that all our values are in compatible units before performing any further calculations.

The Planck constant is a fundamental physical constant used in quantum mechanics. Denoted by \( h \), it has a value of \( 6.626 \times 10^{-34} \) joule-seconds (\textrm{Js}).

It describes the quantization of energy, in simple terms, how the energy of a photon is related to its frequency. In our photon energy equation, Planck's constant helps establish this relationship.

By multiplying Planck’s constant by the speed of light and dividing by the wavelength of light, we are using its significance in the context of energy transmission in quantized form. Remember, \( h \) is a tiny number, indicating that even light waves can carry very small, discrete packets of energy.

The speed of light is another essential constant in these calculations. Represented by \( c \), its value is \( 3.00 \times 10^{8} \) meters per second (\textrm{m/s}).

This is the speed at which light travels in a vacuum, and it’s incredibly important in physics and engineering.

When we use the speed of light in our photon energy equation, it helps translate the length of the light wave (the wavelength) into a corresponding energy value.

• This means light of different wavelengths moves at the same speed but carries different amounts of energy proportional to those wavelengths.

The wavelength of light, denoted by the Greek letter lambda \( abla \), is the distance between successive peaks of a wave.

The wavelength determines the color of light and its energy; light with shorter wavelengths (like blue light) carries more energy, while light with longer wavelengths (like red light) carries less energy.

Understanding wavelength is crucial, as it connects directly to how we calculate the energy of photon emissions.

• The unit of measurement for wavelength in our calculations is often meters, even if it's initially given in other units like nanometers.

Knowing the wavelength allows us to plug it into our photon energy equation and find out how much energy a single photon of that specific wavelength holds.