Question

A Exposure to high doses of microwaves can cause tissue damage. Estimate how many photons, with \(\lambda=12 \mathrm{cm},\) must be absorbed to raise the temperature of your eye by \(3.0^{\circ} \mathrm{C} .\) Assume the mass of an eye is \(11 \mathrm{g}\) and its specific heat capacity is \(4.0 \mathrm{J} / \mathrm{g} \cdot \mathrm{K}\)

Short answer

Approximately \(7.97 \times 10^{25}\) photons are needed.

Step-by-step solution

Calculate the Energy Required to Raise the Temperature

To find out how much energy is needed to increase the temperature of the eye by \( 3.0^{\circ} \mathrm{C} \), we'll use the formula for heat energy: \[Q = mc\Delta T\]where \( m = 11 \mathrm{g} \) is the mass of the eye, \( c = 4.0 \mathrm{J/g \cdot K} \) is the specific heat capacity, and \( \Delta T = 3.0 \mathrm{C} = 3.0 \mathrm{K} \), since the degree increment in Celsius is the same as in Kelvin.Substitute the values:\[ Q = 11 \times 4.0 \times 3.0 = 132 \mathrm{J} \]Thus, 132 Joules of energy are needed.

Calculate the Energy of One Photon

To find the energy of one photon with a wavelength \( \lambda = 12 \mathrm{cm} = 0.12 \mathrm{m} \), we use the formula:\[E_{photon} = \frac{hc}{\lambda}\]where \( h = 6.626 \times 10^{-34} \mathrm{Js} \) is Planck's constant, and \( c = 3.0 \times 10^8 \mathrm{m/s} \) is the speed of light. Substituting the known values:\[E_{photon} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{0.12} = 1.6565 \times 10^{-24} \mathrm{J}\]This is the energy of a single photon.

Calculate the Number of Photons Required

To find out how many photons are needed to supply 132 Joules of energy, we divide the total energy required by the energy per photon:\[N = \frac{Q}{E_{photon}} = \frac{132}{1.6565 \times 10^{-24}}\]Calculating this gives:\[N \approx 7.97 \times 10^{25}\]This is the number of photons needed to raise the temperature of the eye by \( 3.0^{\circ} \mathrm{C} \).

Key concepts

Photon Energy

Photon energy is the amount of energy carried by a single photon, which is a particle of light. Photons have wave characteristics, and their energy depends on their wavelength. The formula to compute the energy of a photon is given by:\[E_{photon} = \frac{hc}{\lambda}\]where:- \(E_{photon}\) is the energy of the photon,- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \text{ Js}\)), which is a fundamental constant that relates energy and frequency,- \(c\) is the speed of light (\(3.0 \times 10^8 \text{ m/s}\)),- \(\lambda\) is the wavelength of the photon.For example, in the original exercise, we calculated the energy of a photon with a wavelength of 12 cm, which is 0.12 meters. Using the formula, we found the energy to be approximately \(1.6565 \times 10^{-24} \text{ J}\). This value is very small, showing that an individual photon carries a tiny amount of energy, which is why so many are needed to impact something like the temperature of the eye.

Specific Heat Capacity

Specific heat capacity is a measure of the amount of heat energy needed to change the temperature of a material by a given amount. It varies for different substances. Given in units of \(\text{J/g} \cdot \text{K}\) or \(\text{J/kg} \cdot \text{K}\), specific heat capacity indicates how much energy is required to raise the temperature of one gram (or kilogram) of a substance by one degree Celsius (or Kelvin).In the problem, the specific heat capacity of the eye is given as \(4.0 \text{ J/g} \cdot \text{K}\). This means that to increase the temperature of 1 gram of the eye material by 1 degree Celsius, 4 Joules of energy is required. For the entire eye mass of 11 grams to increase by 3 degrees Celsius, a total of 132 Joules is needed, as calculated by using the formula:\[Q = mc\Delta T\]where \(m\) is mass, \(c\) is specific heat capacity, and \(\Delta T\) is the temperature change. This illustrates why knowledge of specific heat capacity is crucial in determining how much energy a system absorbs or releases.

Planck's Constant

Planck's constant is a fundamental quantity in quantum mechanics, symbolized by \(h\). It represents the proportional relationship between the frequency of a photon and its energy. The value of Planck's constant is \(6.626 \times 10^{-34} \text{ Js}\), reflecting how discreet and tiny energy levels are at the subatomic level. Planck's constant is integral in calculating the energy of photons, as seen with the formula for photon energy:\[E_{photon} = \frac{hc}{\lambda}\]Here, \(h\) multiplies the speed of light and is divided by the wavelength to determine the photon’s energy. This constant bridges the macroscopic classical physics with quantum mechanics, providing insights into how light and matter interact at very small scales.Understanding Planck's constant helps explain phenomena such as the photoelectric effect, where light can cause the ejection of electrons from a material. It emphasizes the quantized nature of energy transfer in photon interactions, impacting fields like spectroscopy and electronics.