Problem 24 - Chemistry The Central Science

Question

(a) A red laser pointer emits light with a wavelength of \(650 \mathrm{nm}\). What is the frequency of this light? (b) What is the energy of one of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of \(650 \mathrm{nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Step-by-Step Solution

Answer

\(a)\) The frequency of the red laser light is approximately \(4.62 \times 10^{14} \ Hz\). \(b)\) The energy of one photon of the red laser light is approximately \(3.06 \times 10^{-19} \ J\). \(c)\) The energy gap between the ground state and the excited state in the laser material is approximately \(3.06 \times 10^{-19} \ J\).

Step 1: Write the given wavelength in meters.

The given wavelength is 650 nm, which is equivalent to \(6.50 \times 10^{-7} \ m\) in meters.

Step 2: Use the speed of light formula to find the frequency.

We can find the frequency using the formula, \(c = \lambda \nu\). Since the speed of light (c) is approximately \(3.0 \times 10^8 \ m/s\), we get: \(\nu = \frac{c}{\lambda} = \frac{3.0 \times 10^8 \ m/s}{6.50 \times 10^{-7} \ m} = 4.62 \times 10^{14} \ Hz\) The frequency of the red laser light is approximately \(4.62 \times 10^{14} \ Hz\). #b. Calculating the energy of one photon#

Step 1: Use Planck's constant to find the energy of a single photon.

The energy of a photon can be found using the formula, \(E = h \nu\). Planck's constant (h) is approximately \(6.63 \times 10^{-34} \ Js\). By substituting the values, we get: \(E = (6.63 \times 10^{-34} \ Js)(4.62 \times 10^{14} \ Hz) = 3.06 \times 10^{-19} \ J\) The energy of one photon of the red laser light is approximately \(3.06 \times 10^{-19} \ J\). #c. Finding the energy gap between the ground state and excited state#

Step 1: Analyze the energy conservation.

The energy gap between the ground state and excited state is equal to the energy of a single photon. Thus, the energy gap is also \(3.06 \times 10^{-19} \ J\). The energy gap between the ground state and the excited state in the laser material is approximately \(3.06 \times 10^{-19} \ J\).

Key Concepts

Wavelength and Frequency Relationship

The relationship between wavelength and frequency is a fundamental concept in understanding light and other electromagnetic waves. Wavelength (\( \lambda \)) is the distance between successive crests of a wave, while frequency (\( u \)) is the number of crests that pass a point in one second. These two properties are inversely related, meaning that as the wavelength increases, the frequency decreases, and vice versa.

This relationship can be mathematically described by the formula:

• \[c = \lambda u\]

where \( c \) is the speed of light, approximately \( 3.0 \times 10^8 \, \text{m/s} \). By rearranging the formula, you can find the frequency:

• \[u = \frac{c}{\lambda}\]

In the example of a red laser pointer with a wavelength of \( 650 \, \text{nm} \), this translates to:

• \( u = \frac{3.0 \times 10^8 \, \text{m/s}}{6.50 \times 10^{-7} \, \text{m}} \approx 4.62 \times 10^{14} \, \text{Hz} \)

Understanding this relationship is crucial in fields like optics, astronomy, and telecommunications.

Planck's Constant

Planck's constant (\( h \)) is a key element in quantum physics, crucial for calculating the energy of photons. It has a fixed value of approximately \( 6.63 \times 10^{-34} \, \text{Js} \). This constant allows us to understand how light and matter interact at the smallest scales.

The energy (\( E \)) carried by a photon is directly proportional to its frequency (\( u \)), and this can be expressed through Planck's equation:

• \[E = h u\]

For the red laser light with a frequency of \( 4.62 \times 10^{14} \, \text{Hz} \), the energy of a single photon would be:

• \[E = (6.63 \times 10^{-34} \, \text{Js})(4.62 \times 10^{14} \, \text{Hz}) = 3.06 \times 10^{-19} \, \text{J}\]

This calculation is not only vital for lasers but also for understanding electromagnetic emissions in various scientific disciplines.

Laser Emission

Laser emission is a fascinating process, involving energy transitions in atoms or molecules. A laser emits light through a process where electrons are excited from a lower energy state (ground state) to a higher energy state (excited state). This can be achieved through external energy sources, like a battery in the case of a laser pointer.

Once the electrons return to their ground state, they release energy as photons. The energy of these emitted photons corresponds to the energy difference between the excited and ground states. In our red laser example, with light of wavelength \( 650 \, \text{nm} \), the energy difference or the energy gap is equivalent to the photon energy:

• \[3.06 \times 10^{-19} \, \text{J}\]

This process ensures that lasers produce light that is coherent, monochromatic, and highly directional, making them indispensable in many applications like medicine, communications, and industry.