Problem 66 - College Physics

Question

\(\bullet\) Removing vascular lesions. A pulsed dye laser emits light of wavelength 585 nm in 450\(\mu\) s pulses. Because this wave- length is strongly absorbed by the hemoglobin in the blood, the method is especially effective for removing various types of blemishes due to blood, such as port-wine- colored birth-marks. To get a reasonable estimate of the power required for such laser surgery, we can model the blood as having the same specific heat and heat of vaporization as water \((4190 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}\) , \(2.256 \times 10^{6} \mathrm{J} / \mathrm{kg} ) .\) Suppose that each pulse must remove 2.0\(\mu g\) of blood by evaporating it, starting at \(33^{\circ} \mathrm{C}\) . (a) How much energy must each pulse deliver to the blemish? (b) What must be the power output of this laser? (c) How many photons does each pulse deliver to the blemish?

Step-by-Step Solution

Answer

The laser delivers ~4.51 mJ per pulse, needs ~10.03 W power, and emits ~1.33 × 10¹⁶ photons per pulse.

Step 1: Calculating the Energy Required to Evaporate the Blood

To determine the energy required, we need to consider two parts: heating the blood from 33°C to 100°C and then vaporizing it. First, find the mass of the blood in kilograms: \(2.0 \mu g = 2.0 \times 10^{-9} \ \text{kg}\). For the energy to heat, use the formula:\[ Q_1 = mc\Delta T \]where \( m = 2.0 \times 10^{-9} \ \text{kg} \), \( c = 4190 \ \text{J/kg} \cdot \text{K} \), and \( \Delta T = 100 - 33 = 67 \ \text{K} \). Substitute to find \( Q_1 \).\[ Q_1 = (2.0 \times 10^{-9} \ \text{kg})(4190 \ \text{J/kg} \cdot \text{K})(67 \ \text{K}) = 5.6166 \times 10^{-7} \ \text{J}\]Next, calculate the energy required to vaporize the blood using:\[ Q_2 = mL \]where \( L = 2.256 \times 10^6 \ \text{J/kg} \).\[ Q_2 = (2.0 \times 10^{-9} \ \text{kg})(2.256 \times 10^6 \ \text{J/kg}) = 4.512 \times 10^{-3} \ \text{J} \]The total energy is:\[ Q = Q_1 + Q_2 = 5.6166 \times 10^{-7} \ \text{J} + 4.512 \times 10^{-3} \ \text{J} = 4.51256166 \times 10^{-3} \ \text{J} \]

Step 2: Determining the Power Output

Power is the energy delivered per unit time. Since each pulse lasts for \( 450 \mu s = 450 \times 10^{-6} \ \text{s}\), the power output \( P \) is:\[ P = \frac{Q}{t} = \frac{4.51256166 \times 10^{-3} \ \text{J}}{450 \times 10^{-6} \ \text{s}} \]Calculate the result:\[ P = 10.028 \ \text{W} \]

Step 3: Calculating the Number of Photons

To find the number of photons, first find the energy of a single photon. The energy of a photon \( E_p \) can be calculated using:\[ E_p = \frac{hc}{\lambda} \]where \( h = 6.626 \times 10^{-34} \ \text{J} \cdot \text{s}\) is Planck's constant, \( c = 3.00 \times 10^8 \ \text{m/s}\) is the speed of light, and \( \lambda = 585 \times 10^{-9} \ \text{m}\) is the wavelength. Substitute to find \( E_p \):\[ E_p = \frac{(6.626 \times 10^{-34} \ \text{J} \cdot \text{s})(3.00 \times 10^8 \ \text{m/s})}{585 \times 10^{-9} \ \text{m}} \approx 3.394 \times 10^{-19} \ \text{J} \]Next, calculate the number of photons \( N \) by dividing the total energy by the energy of one photon:\[ N = \frac{Q}{E_p} = \frac{4.51256166 \times 10^{-3} \ \text{J}}{3.394 \times 10^{-19} \ \text{J}} \approx 1.33 \times 10^{16} \]

Step 4: Summary and Short Answer

a) Each pulse must deliver approximately \(4.51 \times 10^{-3}\) J of energy to evaporate the blood. b) The laser must have a power output of about 10.03 W. c) Each pulse delivers approximately \(1.33 \times 10^{16}\) photons to the blemish.

Key Concepts

Pulsed Dye Laser

A pulsed dye laser is a specialized type of laser commonly used in medical treatments, including dermatological applications. This laser emits light in short, intense bursts or pulses. The wavelength of the light produced by a pulsed dye laser can be adjusted by altering the dye used in the laser medium. For treatments such as removing vascular lesions, the laser emits light at a wavelength of 585 nm. This specificity allows the laser to target certain compounds within the skin effectively.
Pulsed dye lasers are highly favored for their precision and minimal damage to surrounding tissues. By delivering energy in controlled, short pulses, the laser can focus its effects on targeted areas. This precision is crucial when dealing with sensitive or delicate tissues such as those involved in vascular lesions.

Hemoglobin Absorption

Hemoglobin, the protein in blood responsible for carrying oxygen, has specific absorption characteristics. At certain wavelengths, hemoglobin absorbs light more effectively. The wavelength of 585 nm used by the pulsed dye laser is strongly absorbed by hemoglobin. This characteristic makes these lasers particularly suitable for targeting blood vessels.
When the laser light is absorbed by hemoglobin, it heats the blood, causing it to coagulate and seal off the vessel. This process can lead to the removal or reduction of visible blood vessels, such as those seen in port-wine stains or spider veins. The absorption of the laser energy by hemoglobin is central to the efficient treatment of vascular lesions.

Evaporation Energy

When using a pulsed dye laser for medical treatments, evaporation energy plays a crucial role. The treatment goal is often to remove unwanted tissue, such as blemishes, by heating and evaporating the blood within. To achieve this, it's necessary to calculate the energy required to heat the blood from an initial temperature to the vaporization point, and then provide additional energy to convert the liquid blood into vapor.
The calculation involves determining the energy to raise the temperature of the blood from its starting temperature (e.g., 33°C) to 100°C, using the specific heat capacity of water. Once it reaches 100°C, the blood must be vaporized, which requires additional energy called the heat of vaporization. This total energy ensures the effective removal of targeted tissues.

Power Calculation

The power output of a laser defines how much energy is delivered in a specific period of time. In laser surgery, each pulse of a laser must deliver a sufficient amount of energy to achieve the desired medical outcome. Using a 450 microsecond pulse length, we can calculate the necessary power output of the laser by dividing the required energy to evaporate the blood by the pulse duration.
The calculation allows practitioners to determine the amount of power their device must output to be effective in treatment. For instance, with the values provided in the original problem, the laser's power output is found to be approximately 10.03 watts, which is suitable for evaporating a tiny mass of blood.

Photon Calculation

Laser light is composed of photons, particles that carry energy. To understand how many photons are involved in a singular pulse of the laser, it is necessary to calculate the energy of a single photon at the laser's wavelength using Planck's constant and the speed of light.
Once the energy of a singular photon is known, you can determine the total number of photons by dividing the total energy required by the energy of each photon. This calculation is vital for understanding the microscopic delivery of energy during treatments. For example, based on the given photon energy and total energy calculations, each pulse from the laser delivers approximately \(1.33 \times 10^{16}\) photons, ensuring efficient energy transfer to the targeted area.