Problem 57 - University Physics with Modern Physics

Question

Removing Vascular Lesions. A pulsed dye laser emits light of wavelength 585 \(\mathrm{nm}\) 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 \(\left(4190 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, 2.256 \times 10^{6} \mathrm{J} / \mathrm{kg}\right) .\) 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

(a) 5.0731 mJ, (b) 11.29 W, (c) ≈1.49×10¹⁶ photons per pulse.

Step 1: Calculate the energy needed to heat the blood to 100°C

The blood initially at 33°C needs to be heated to 100°C. Use the specific heat capacity of water to calculate this.\[ q_1 = m imes c imes \Delta T \]where- \( m = 2.0\, \mu g = 2.0 \times 10^{-9}\, kg \) (mass of blood),- \( c = 4190 \, J/(kg\cdot K) \) (specific heat of water),- \( \Delta T = 100 - 33 = 67 \, K \).So,\[ q_1 = 2.0 \times 10^{-9} \times 4190 \times 67 = 5.611 \times 10^{-4} \, J \].

Step 2: Calculate the energy needed to vaporize the blood

After heating the blood to 100°C, it needs to be vaporized. Use the heat of vaporization for this calculation.\[ q_2 = m \times L_v \]where- \( L_v = 2.256 \times 10^{6} \, J/kg \) (heat of vaporization of water).\[ q_2 = 2.0 \times 10^{-9} \times 2.256 \times 10^{6} = 4.512 \times 10^{-3} \, J \].

Step 3: Calculate the total energy per pulse

The total energy required is the sum of the energy to heat and vaporize the blood.\[ q_{total} = q_1 + q_2 \]\[ q_{total} = 5.611 \times 10^{-4} + 4.512 \times 10^{-3} = 5.0731 \times 10^{-3} \, J \].

Step 4: Calculate the power output of the laser

Power output can be determined by dividing the energy required by the pulse duration.\[ P = \frac{q_{total}}{t_p} \]where- \( t_p = 450 \times 10^{-6} \, s \) (pulse duration).\[ P = \frac{5.0731 \times 10^{-3}}{450 \times 10^{-6}} = 11.29 \, W \].

Step 5: Calculate the number of photons per pulse

Use the energy of a single photon to determine the number of photons in a pulse.First, calculate the energy of a single photon:\[ E_{photon} = \frac{hc}{\lambda} \]where- \( h = 6.626 \times 10^{-34} \, Js \) (Planck's constant),- \( c = 3 \times 10^{8} \, m/s \) (speed of light),- \( \lambda = 585 \times 10^{-9} \, m \) (wavelength of light).\[ E_{photon} = \frac{6.626 \times 10^{-34} \times 3 \times 10^{8}}{585 \times 10^{-9}} = 3.396 \times 10^{-19} \, J \].Now, calculate the number of photons:\[ n = \frac{q_{total}}{E_{photon}} = \frac{5.0731 \times 10^{-3}}{3.396 \times 10^{-19}} \approx 1.49 \times 10^{16} \text{ photons} \].

Key Concepts

Pulsed Dye Laser

Pulsed dye lasers are innovative tools used in medical treatments, particularly for removing vascular lesions or blemishes. These lasers generate light in short, controlled bursts (or pulses), which allows for precision in treating the targeted area with minimal damage to the surrounding tissues. The pulsing mechanism is crucial because it limits the amount of heat applied over time, reducing the risk of thermal injury to the skin. This makes pulsed dye lasers especially suitable for treating conditions like port-wine stains and other blood-related marks, as they target specific wavelengths absorbed by blood vessels. By emitting highly concentrated beams of light, they can heat and vaporize the target tissue effectively.

Wavelength 585 nm

The 585 nm wavelength used by pulsed dye lasers is significant due to its effectiveness in targeting blood vessels. The wavelength, measured in nanometers (nm), determines the color of light, with 585 nm falling within the yellow light spectrum. This specific wavelength is absorbed well by hemoglobin, the protein in blood. When the laser light at this wavelength penetrates the skin, it is selectively absorbed by the blood vessels in the blemishes rather than the surrounding tissue. This targeted absorption results in precise treatments, minimizing damage to the skin and leading to better cosmetic outcomes. The ability to finely control this wavelength is part of what makes pulsed dye lasers a powerful tool in dermatological treatments.

Specific Heat and Heat of Vaporization

When considering the energy required for laser surgery, two important thermal properties come into play: specific heat and heat of vaporization. Specific heat is the amount of energy needed to raise the temperature of a unit mass of a substance by one degree Kelvin. For blood, which is often approximated with the properties of water, it is 4190 J/kg·K. This property determines how much energy will be needed to heat blood to the boiling point.
On the other hand, the heat of vaporization is the energy required to change a unit mass of a substance from a liquid to a vapor at its boiling point. For blood, again compared to water, this is 2.256 x 10^6 J/kg. Combining these concepts allows us to calculate the total energy needed to both heat and vaporize a small amount of blood, using the laser treatment discussed.

Calculating Energy and Power

Calculating the energy and power used in laser surgery involves understanding the physics of energy transfer. The total energy required, known as the total energy per pulse, is the sum of the energy needed to raise the temperature of the target (blood) to its boiling point and the energy required to vaporize it. This is calculated using:

• Heating energy: \( q_1 = mc\Delta T \), where \( m \) is mass, \( c \) is specific heat capacity, and \( \Delta T \) is the temperature change needed.
• Vaporizing energy: \( q_2 = mL_v \), where \( L_v \) is the heat of vaporization.

The power output of the laser is determined by dividing the total energy by the duration of the laser pulse, as seen in \( P = q_{total}/t_p \). This measurement gives insight into how intense or effective a laser needs to be for successful treatment. Additionally, understanding the energy of individual photons (using Planck's constant, the speed of light, and the given wavelength) allows us to calculate how many photons are involved in each pulse of the laser, which is key for understanding the detailed interaction between laser light and tissue.