Problem 27 - University Physics with Modern Physics

Question

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 birthmarks. 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 J / kg \(\bullet\) K, 2.256 \(\times\) 10\(^6\) J / kg). Suppose that each pulse must remove 2.0 mg of blood by evaporating it, starting at 33\(^\circ\)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) 5.074 J, (b) 11288 W, (c) 1.497 \times 10^{19} photons.

Step 1: Convert mass of blood to kilograms

First, we need to convert the mass of blood from milligrams to kilograms. Given that the mass of blood is 2.0 mg, we have:\[ \text{Mass (kg)} = \frac{2.0 \text{ mg}}{1000 \times 1000} = 2.0 \times 10^{-6} \text{ kg} \]

Step 2: Calculate energy to heat blood to 100°C

The energy required to heat the blood from 33°C to 100°C can be calculated using the specific heat formula. The specific heat (\(c\)) of water is 4190 J/kg·K. The temperature change (\(\Delta T\)) is:\[ \Delta T = 100 - 33 = 67 \text{ K} \]The energy (\(Q_1\)) required is then:\[ Q_1 = m \cdot c \cdot \Delta T = 2.0 \times 10^{-6} \times 4190 \times 67 \]\[ Q_1 = 0.56186 \text{ J} \]

Step 3: Calculate energy to vaporize blood

The energy required to vaporize the blood is calculated using the heat of vaporization, which is 2.256 \(\times 10^6\) J/kg. The energy (\(Q_2\)) required is:\[ Q_2 = m \cdot L_v = 2.0 \times 10^{-6} \times 2.256 \times 10^6 = 4.512 \text{ J} \]

Step 4: Calculate the total energy required per pulse

The total energy required per pulse is the sum of the energy to heat the blood to 100°C and the energy to vaporize it:\[ Q_{\text{total}} = Q_1 + Q_2 = 0.56186 + 4.512 = 5.074 \text{ J} \]

Step 5: Calculate the power output of the laser

The power output of the laser during each pulse can be calculated using the energy from the pulse and the pulse duration (450 \(\mu\)s = 450 \(\times 10^{-6}\) s):\[ P = \frac{Q_{\text{total}}}{\text{pulse duration}} = \frac{5.074}{450 \times 10^{-6}} \approx 11288 \text{ W} \]

Step 6: Calculate the number of photons per pulse

To find the number of photons, we use the relation between energy, the number of photons, and the energy of each photon. The energy of one photon (\(E_p\)) is given by:\[ E_p = \frac{hc}{\lambda} \]where \(h = 6.626 \times 10^{-34}\) Js, \(c = 3 \times 10^8\) m/s, and \(\lambda = 585\) nm = \(585 \times 10^{-9}\) m.\[ E_p = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{585 \times 10^{-9}} = 3.39 \times 10^{-19} \text{ J/photon} \]The number of photons (\(N\)) is then:\[ N = \frac{Q_{\text{total}}}{E_p} = \frac{5.074}{3.39 \times 10^{-19}} \approx 1.497 \times 10^{19} \text{ photons} \]

Key Concepts

Pulsed Dye Laser

A pulsed dye laser is a type of laser with a specific advantage in medical procedures, particularly in dermatology. One of its features is the emission of a short burst of laser light, known as a pulse. This light typically has a wavelength suitable for targeting specific chromophores, such as hemoglobin in blood. Due to its precise targeting ability, pulsed dye lasers are commonly used for procedures like removing port-wine stains—a type of birthmark—by causing local heating and destruction of blood vessels without affecting surrounding tissues.

In laser surgery, pulsed dye lasers utilize the property that the laser's light is absorbed strongly by the blood's hemoglobin. This absorption leads to effective treatment of vascular lesions by converting light energy into thermal energy. The short duration of each pulse reduces thermal damage, allowing for precise surgical actions.

Energy Calculation

Energy calculation involves determining the total energy required to achieve desired physical changes—in this case, removing blood by heating and vaporizing it. The process comprises two main parts:

• Heating: Raise the temperature of the blood from its initial temperature (33°C) to the boiling point (100°C).
• Vaporization: Convert the heated liquid blood to vapor.

To do this calculation, you apply the specific heat equation to find the heating energy and the heat of vaporization equation for the vaporization energy.
For heating, the formula used is: \[ Q_1 = m imes c imes \Delta T \] Where \(m\) is mass, \(c\) is the specific heat, and \(\Delta T\) is the temperature change. For vaporization, the formula is: \[ Q_2 = m \times L_v \] Here \(L_v\) is the heat of vaporization. Together, these calculations offer a complete understanding of the energy needs for the laser pulse during surgery.

Specific Heat

Specific heat is an important concept in thermodynamics. It describes the amount of heat energy needed to change the temperature of a given mass of a substance by 1 degree Celsius. For water, which has a particularly high specific heat of 4190 J/kg·K, the specific heat effectively measures its ability to absorb thermal energy.

In medical laser applications, treating blood like water is convenient because of their similar specific heats. Using water's specific heat allows us to estimate how much energy is needed to increase the temperature of the blood during laser surgery. This calculation is crucial because it helps to gauge the total thermal energy needed to achieve effective treatment without causing damage beyond the targeted area.

Heat of Vaporization

The heat of vaporization is the energy required to convert a substance from a liquid to a vapor at its boiling point without increasing the temperature. For water, this value is quite high, at 2.256 x 10⁶ J/kg. This means that significant energy is necessary to change liquid water into steam.

During laser surgery, the concept is crucial in understanding the energy demands when vaporizing blood. We assume that blood has a similar heat of vaporization to water. Hence, this allows us to calculate the energy required to change the heated blood at 100°C into vapor. Understanding this energy requirement is fundamental in designing precise and effective pulsed dye laser treatments for medical procedures.