Question

Pulsed lasers are powerful sources of nearly monochromatic radiation. Lasers that emit photons in a pulse of 5.00 ns duration with a total energy in the pulse of \(0.175 \mathrm{J}\) at \(875 \mathrm{nm}\) are commercially available. a. What is the average power (energy per unit time) in units of watts \((1 \mathrm{W}=1 \mathrm{J} / \mathrm{s})\) associated with such a pulse? b. How many photons are emitted in such a pulse?

Short answer

a. The average power associated with the pulse is \(3.50 \times 10^{10} Watts\). b. Approximately \(7.69 \times 10^{17}\) photons are emitted in such a pulse.

Step-by-step solution

a. Average power calculation

To find the average power, we divide the total energy in the pulse by the duration of the pulse: Average power = \(\frac{Total\ energy}{Duration}\) We are given the total energy in the pulse as \(0.175\) Joules and the duration as \(5.00 ns\) (nanoseconds). Convert the time to seconds: \(5.00 ns \cdot \frac{1 s}{10^9 ns} = 5.00 \times 10^{-9} s\) Now, we can plug in the values and calculate the average power: Average power = \(\frac{0.175 J}{5.00 \times 10^{-9} s}\) = \(3.50 \times 10^{10} W\) So, the average power associated with the pulse is \(3.50 \times 10^{10} Watts\).

b. Number of photons calculation

First, we need to find the energy of a single photon. To do this, we use the equation: Energy of a photon = \(hc \div \lambda\) where \(h\) is the Planck's constant (\(6.626 \times 10^{-34} Js\)), \(c\) is the speed of light (\(3.00 \times 10^8 ms^{-1}\)), and \(\lambda\) is the wavelength of the photons. We are given the wavelength as \(875 nm\). Convert the wavelength to meters: \(875 nm \cdot \frac{1 m}{10^9 nm} = 875 \times 10^{-9} m\) Now we can plug in the values and calculate the energy of a single photon: Energy of a photon = \(\frac{6.626 \times 10^{-34} Js \cdot 3.00 \times 10^8 ms^{-1}}{875 \times 10^{-9} m}\) = \(2.275 \times 10^{-19} J\) Next, we find the number of photons by dividing the total energy in the pulse by the energy of a single photon: Number of photons = \(\frac{Total\ energy}{Energy\ of\ a\ single\ photon}\) Number of photons = \(\frac{0.175 J}{2.275 \times 10^{-19} J}\) = \(7.69 \times 10^{17}\) Hence, there are approximately \(7.69 \times 10^{17}\) photons emitted in such a pulse.

Key concepts

Average Power Calculation

Understanding how to calculate the average power of a pulsed laser is crucial in physical chemistry. Average power quantifies the rate at which energy is emitted by the laser per unit of time. It is essentially the total energy divided by the time during which the energy is released. This can be formulated as:

\textbf{Average Power} = \( \frac{Total\ Energy}{Time\ Duration} \)

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To get this number into a more comprehensible format, divide the energy value (in joules) by the pulse duration (in seconds). For instance, if a pulse laser emits 0.175 Joules over 5 nanoseconds, you convert the nanoseconds to seconds by dividing by \(10^9\), hence \(5\ ns\) becomes \(5 \times 10^{-9}\ s\). Using the total energy and converted time duration, the average power in watts can then be calculated. Remember that 1 watt equals 1 joule per second (1 W = 1 J/s).

Photon Energy Calculation

The energy of individual photons from a laser pulse can be calculated using the equation:\( E = \frac{hc}{\lambda} \), where \(E\) represents the energy of the photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the photon's wavelength.

Due to the direct relationship between energy and wavelength, the shorter the wavelength, the higher the energy of the photons. In calculating photon energy, converting the wavelength from nanometers to meters is necessary by multiplying the given wavelength by \(10^{-9}\). With the constants for Planck's constant and the speed of light, and the converted wavelength, you determine the energy carried by a single photon. This step is fundamental in understanding the properties of laser pulses and their applications.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics. Represented by \(h\), with a value of approximately \(6.626 \times 10^{-34} \) joule seconds (Js), it relates the energy of a photon to the frequency of its associated electromagnetic wave. This constant is named after the physicist Max Planck who discovered the quantization of energy.

Planck's constant is crucial when dealing with phenomena at atomic and subatomic scales, as it sets the scale for the quantum of action or the quantum of energy. Its use in calculating the energy of photons emitted by lasers, as illustrated in the above problem, demonstrates its importance in the analysis of laser interactions and energy quantization in physical chemistry.

Wavelength Conversion

In problems involving photonic energy calculations, wavelengths typically need to be converted from one unit to another—most commonly from nanometers to meters—to utilize the standard units of measurement in formulas. Wavelength conversion is straightforward: simply multiply the given nanometer value by \(10^{-9}\) to find the equivalent in meters.

The significance of this conversion cannot be overstated, as using consistent units is crucial to obtaining correct calculations in physics and chemistry. Moreover, since the energy of a photon is inversely related to its wavelength (a shorter wavelength means higher energy), understanding how to accurately perform this conversion is essential in interpreting the behavior and properties of lights and lasers.