Problem 33 - University Physics with Modern Physics

Question

Cp Blo Removing Vascular Lesions. A pulsed dye laser emits light of wavelength 585 nm in \(450-\mu\) s pulses. Because this wavelength 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 birthmark. 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 \(\left(4190 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, 256 \times 10^{6} \mathrm{J} / \mathrm{kg}\right) .\) Suppose that each pulse must remove 2.0\(\mu \mathrm{g}\) 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

(a) Calculate total energy needed as heating and vaporization; (b) Determine power by dividing energy by pulse duration; (c) Use photon energy to find number of photons per pulse.

Step 1: Determine Energy for Heating

First, calculate the amount of energy needed to heat the blood from its initial temperature of 33°C to its boiling point of 100°C. Use the formula for heat energy: \( Q_1 = mc\Delta T \). Here, \( m = 2.0 \times 10^{-9} \) kg is the mass of blood, \( c = 4190 \) J/kg·K is the specific heat of water, and \( \Delta T = 100 - 33 \) K.

Step 2: Calculate Energy for Vaporization

Next, calculate the energy required to vaporize the blood at 100°C, using the formula: \( Q_2 = mL \), where \( L = 2.56 \times 10^6 \) J/kg is the heat of vaporization for water and \( m = 2.0 \times 10^{-9} \) kg.

Step 3: Total Energy per Pulse

Add the energies calculated in Steps 1 and 2 to find the total energy required per pulse: \( Q_{total} = Q_1 + Q_2 \).

Step 4: Calculate Power Output

To find the power output of the laser, use the relation \( P = \frac{Q_{total}}{t} \), where \( t = 450 \times 10^{-6} \) s is the pulse duration.

Step 5: Calculate Number of Photons per Pulse

To determine the number of photons delivered per pulse, use the formula \( n = \frac{Q_{total}}{E_{photon}} \), where the energy of a single photon \( E_{photon} \) is given by \( E_{photon} = \frac{hc}{\lambda} \). Use \( h = 6.626 \times 10^{-34} \) J·s, \( c = 3.0 \times 10^8 \) m/s, and \( \lambda = 585 \times 10^{-9} \) m.

Key Concepts

Laser Surgery

Laser surgery is a revolutionary technique used in medical procedures to remove unwanted tissues, blemishes, or lesions. In the context of vascular lesion removal, a pulsed dye laser plays a crucial role. This type of laser emits light at a specific wavelength, precisely targeting the hemoglobin in the blood.
The target wavelength is often 585 nm, as it is strongly absorbed by hemoglobin. This characteristic absorption enables the laser to effectively coagulate or vaporize blood vessels without causing damage to the surrounding tissues.

• Pulsed dye lasers are frequently used to treat conditions such as port-wine stains, spider veins, and other vascular-related skin issues.
• The precision of the laser reduces side effects and enhances recovery times, making it a preferred choice in dermatological treatments.

These techniques highlight the significance of wavelength selection and energy calculation in achieving desirable outcomes in laser surgery.

Specific Heat

Specific heat is a physical property of materials that describes how much heat energy is needed to change the temperature of a unit mass by one degree Celsius. In the context of this problem, the specific heat capacity of water, which is 4190 J/kg·K, is utilized.
Modeling blood with the specific heat of water is a reasonable approximation given the high water content in blood. This parameter is crucial when determining how much heat is needed to raise the temperature of blood from its starting point to the boiling point.

• The formula used is: \( Q = mc\Delta T \), where \( m \) is mass, \( c \) is specific heat, and \( \Delta T \) is the change in temperature.
• This calculation helps in understanding the amount of thermal energy necessary for the initial heating phase in laser surgery applications.

Understanding specific heat is pivotal in calculating the thermal dynamics involved in medical laser procedures.

Heat of Vaporization

The heat of vaporization is the amount of energy required to change a unit mass of a substance from liquid to vapor without a temperature change taking place. In laser surgery for blood removal, once the blood reaches the boiling point, additional energy is required to vaporize it.
In this context, the heat of vaporization for water is 2.56 x 10³ J/kg. This value is applicable for calculating how much energy is needed for the phase change of the blood after it reaches 100°C.

• The formula is: \( Q = mL \), where \( m \) is the mass and \( L \) is the heat of vaporization.
• This step is vital as it determines the completeness of the blood removal process, ensuring that the blemish is efficiently treated.

Both specific heat and heat of vaporization play a crucial role in the energy calculations in laser-based medical treatments.

Photon Energy Calculation

Photon energy calculation is a fascinating aspect of how lasers function, especially in precise applications like surgeries. Each photon in a laser beam carries a certain amount of energy which can be calculated using its wavelength.
The energy of a single photon is given by the formula: \( E_{photon} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the laser. For a pulsed dye laser emitting at 585 nm, this formula helps calculate how many photons are needed to deliver the total energy required per pulse.

• Photon energy is fundamental in determining the dosage and effectiveness of the laser treatment.
• The number of photons impacts the precision and intensity of the laser application, crucial for achieving the desired medical outcomes.

Understanding photon energy is imperative for optimizing laser surgeries and ensures that treatments are both safe and efficient.