Problem 61 - College Physics

Question

Removing vascular lesions. A pulsed dye laser emits light of wavelength \(585 \mathrm{nm}\) in \(450 \mu \mathrm{s}\) 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 \(\left(4190 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, 2.256 \times 10^{6} \mathrm{~J} / \mathrm{kg}\right) .\) Suppose that each pulse must remove \(2.0 \mu \mathrm{g}\) 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) 5.072 J, b) 11271 W, c) 1.493 × 10¹⁹ photons.

Step 1: Convert Mass to Kilograms

The mass of blood to be removed is given as \(2.0 \mu \text{g}\). We need to convert this mass from micrograms to kilograms:\[2.0 \mu \text{g} = 2.0 \times 10^{-6} \text{ kg}\]

Step 2: Calculate Energy Needed for Heating

The blood needs to be heated from \(33\degree \text{C}\) to \(100\degree \text{C}\) (boiling point of water). Calculate the energy required using the specific heat capacity formula. The specific heat capacity \(c\) is \(4190 \text{ J/kg} \cdot \text{K}\), and the temperature change \(\Delta T\) is \(100 - 33 = 67\degree\text{C}\):\[Q_1 = mc\Delta T = (2.0 \times 10^{-6} \text{ kg})(4190 \text{ J/kg} \cdot \text{K})(67 \text{ K})\]Performing the calculation gives us:\[Q_1 = 0.56 \text{ J}\]

Step 3: Calculate Energy Needed for Vaporization

After heating, the blood must be vaporized. The heat of vaporization \(L\) is \(2.256 \times 10^{6} \text{ J/kg}\). Calculate the energy required for vaporization:\[Q_2 = mL = (2.0 \times 10^{-6} \text{ kg})(2.256 \times 10^{6} \text{ J/kg})\]Performing this calculation yields:\[Q_2 = 4.512 \text{ J}\]

Step 4: Calculate Total Energy per Pulse

Add the energy required for heating and vaporization to determine the total energy delivered per pulse:\[Q_{\text{total}} = Q_1 + Q_2 = 0.56 \text{ J} + 4.512 \text{ J} = 5.072 \text{ J}\]

Step 5: Calculate Power Output of Laser

The power output \(P\) of the laser can be found using the formula \(P = \frac{E}{t}\), where \(E\) is the energy per pulse (\(5.072 \text{ J}\)), and \(t\) is the duration of the pulse (\(450 \times 10^{-6} \text{ s}\)):\[P = \frac{5.072 \text{ J}}{450 \times 10^{-6} \text{ s}}\]Carrying out this calculation gives us:\[P = 11271 \text{ W}\]

Step 6: Calculate Number of Photons per Pulse

First, calculate the energy of a single photon using \(E_p = \frac{hc}{\lambda}\), where \(h = 6.626 \times 10^{-34} \text{ Js}\) (Planck's constant), \(c = 3.00 \times 10^8 \text{ m/s}\) (speed of light), and \(\lambda = 585 \times 10^{-9} \text{ m}\) (wavelength):\[E_p = \frac{(6.626 \times 10^{-34}) (3.00 \times 10^8)}{585 \times 10^{-9}}\]This yields an energy of approximately \(3.396 \times 10^{-19} \text{ J/photon}\).Now, calculate the number of photons \(N\) per pulse by dividing the total energy per pulse by the energy of a single photon:\[N = \frac{Q_{\text{total}}}{E_p} = \frac{5.072 \text{ J}}{3.396 \times 10^{-19} \text{ J/photon}}\]This results in approximately \(1.493 \times 10^{19}\) photons per pulse.

Key Concepts

Specific Heat Capacity

Understanding the concept of specific heat capacity is crucial, especially in fields like laser surgery physics. The specific heat capacity (\(c\)) is the amount of heat energy required to raise the temperature of 1 kilogram of a substance by 1 degree Celsius (or 1 Kelvin). In this exercise, blood is modeled with the same specific heat capacity as water, which is \(4190 \, \text{J/kg} \cdot \text{K}\).
Specific heat capacity plays a vital role in calculating how much energy is needed to heat a substance. The formula is \(Q = mc\Delta T\), where \(m\) is the mass and \(\Delta T\) is the change in temperature. Here, the blood temperature is raised from \(33^{\circ} \text{C}\) to \(100^{\circ} \text{C}\). The energy required is calculated by multiplying these values. In this exercise, the specific heat helps us determine the energy needed for this temperature change.
Knowing how much energy is necessary for heating ensures the laser provides enough power to safely and effectively treat the blemish. This is foundational for calculating the total energy required in laser surgery applications.

Heat of Vaporization

Heat of vaporization is another fundamental concept within laser surgery physics. It refers to the amount of energy needed to turn a given quantity of a liquid into a gas at constant temperature and pressure. For water, which serves as our model for blood, this is \(2.256 \times 10^6\, \text{J/kg}\).
Why is this significant in the context of laser surgery? Once the blood is heated to its boiling point, additional energy is needed to vaporize it. This is crucial for removing the mass of blood targeted by the laser. The formula to calculate the required energy for vaporization is \(Q = mL\), where \(m\) is the mass and \(L\) is the heat of vaporization.
This concept ensures you calculate the total energy delivered accurately. Only after considering both warming up and vaporization can the laser's effectiveness be truly maximized, clearing the blemishes thoroughly.

Photon Energy

Photon energy is a vital part of understanding how lasers function. Every laser emits photons, which are particles of light carrying energy. The energy of these photons is determined by their wavelength, following the equation \(E_p = \frac{hc}{\lambda}\). Here, \(h\) is Planck's constant \(6.626 \times 10^{-34} \, \text{Js}\), \(c\) is the speed of light \(3.00 \times 10^8 \, \text{m/s}\), and \(\lambda\) is the wavelength (in meters) of the emitted light.
For the pulsed dye laser discussed in this problem, the wavelength is \(585 \, \text{nm}\). Understanding photon energy helps in calculating how many photons each laser pulse emits. We use the total pulse energy and divide it by the energy of one photon.
Through understanding photon energy, you can assess how effectively the laser delivers energy into the targeted area. This calculation is essential for ensuring the laser's settings are correct and safe.

Power Output Calculation

Calculating power output is your key to understanding how much energy the laser delivers over a period. The power output (\(P\)) is the rate at which energy is discharged, defined as energy divided by time. The formula is \(P = \frac{E}{t}\), where \(E\) is the total energy per pulse and \(t\) is the pulse duration.
For example, here the pulse takes \(450 \, \mu \text{s} \text{ (or } 450 \times 10^{-6} \text{ s)}\) to deliver \(5.072 \, \text{J}\) of energy. By dividing, you find the laser's power output in watts. This measurement is critical as it determines whether the laser can achieve the desired effects in laser surgery efficiently and safely.
Understanding power output ensures you know the intensity of the laser and helps guide decisions about equipment settings and expected outcomes during procedures. It is a must-know for assessing the laser's capability to meet treatment demands.