Problem 57 - Chemistry Matter and Change

Question

Eye Surgery The argon fluoride (ArF) laser used in some refractive eye surgeries emits electromagnetic radiation of 193.3 nm wavelength. What is the frequency of the ArF laser's radiation? What is the energy of a sin- gle quantum of the radiation?

Step-by-Step Solution

Answer

Frequency: \(1.552 \times 10^{15}\) Hz; Energy: \(1.029 \times 10^{-18}\) J.

Step 1: Convert wavelength to meters

The wavelength provided is in nanometers (nm), and we need to convert it to meters for further calculations. Since 1 nm = \(10^{-9}\) meters, we have:\[ \text{Wavelength in meters} = 193.3 \times 10^{-9} = 1.933 \times 10^{-7} \text{ m} \]

Step 2: Calculate the frequency

Use the speed of light equation, where the speed of light \(c = 3 \times 10^8\) m/s, to find the frequency \(f\) using the formula:\[ c = \lambda f \]Solving for \(f\):\[ f = \frac{c}{\lambda} = \frac{3 \times 10^8 \text{ m/s}}{1.933 \times 10^{-7} \text{ m}} \approx 1.552 \times 10^{15} \text{ Hz} \]

Step 3: Calculate the energy of a single quantum

Using Planck's equation, where the energy \(E\) of a single photon is given by:\[ E = hf \]Here \(h\) is Planck's constant, approximately \(6.626 \times 10^{-34}\) J·s. Plug in the frequency calculated in Step 2:\[ E = (6.626 \times 10^{-34} \text{ J·s})(1.552 \times 10^{15} \text{ Hz}) \approx 1.029 \times 10^{-18} \text{ J} \]

Key Concepts

Wavelength Conversion

To begin with, the wavelength of the ArF laser light used in eye surgeries is given in nanometers (nm). It is essential to convert this value into meters, as most scientific calculations use meters as a standard unit. The conversion factor from nanometers to meters is straightforward: 1 nanometer equals \(10^{-9}\) meters.

• For the given wavelength of 193.3 nm, converting it to meters involves multiplying by this conversion factor, resulting in \(193.3 \times 10^{-9} = 1.933 \times 10^{-7}\) meters.

Calculating in meters is crucial when dealing with equations involving speed of light or frequency to maintain consistency with the units used for other constants like the speed of light.

Frequency Calculation

Frequency is one of the fundamental properties of electromagnetic waves, and it is inversely related to wavelength. To find the frequency of the laser's radiation, we use the formula linking speed of light \(c\), frequency \(f\), and wavelength \(\lambda\):

• The formula is: \( c = \lambda f \)

The speed of light is approximately \(3 \times 10^8\) meters per second. Using the already converted wavelength \(1.933 \times 10^{-7}\) meters, frequency \(f\) can be calculated as follows:

• Rearrange the formula to solve for frequency: \( f = \frac{c}{\lambda} \)
• Substitute the values: \( f = \frac{3 \times 10^8 \text{ m/s}}{1.933 \times 10^{-7} \text{ m}} \approx 1.552 \times 10^{15} \text{ Hz}\)

This frequency signifies how many waves pass a point per second and is expressed in hertz (Hz). Understanding this helps in depicting the usage of such lasers in medical applications like eye surgeries.

Planck's Equation

Planck's equation is a pivotal tool in quantum mechanics, allowing us to calculate the energy of a photon. In the context of electromagnetic waves, energy is proportional to frequency. Planck’s equation is given by:

• \( E = hf \), where \(E\) is energy, \(h\) is Planck's constant \(~6.626 \times 10^{-34}~\) joule-seconds, and \(f\) is the frequency.

Using the frequency calculated previously through:

• \( f = 1.552 \times 10^{15} \text{ Hz} \),

we find the energy of a single quantum or photon:

• Substitute the values: \( E = (6.626 \times 10^{-34} \text{ J·s}) \times (1.552 \times 10^{15} \text{ Hz}) \approx 1.029 \times 10^{-18} \text{ J} \)

This indicates each photon of the ArF laser carries about \(1.029 \times 10^{-18}\) joules of energy, explaining its effectiveness in precise medical devices like lasers used in eye surgery.