Question

A Exposure to high doses of microwaves can cause 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

About \(7.98 \times 10^{25}\) photons are needed.

Step-by-step solution

Calculate the Energy Required

First, we need to find the energy required to raise the temperature of the eye by 3.0°C. The formula to calculate the energy content (Q) is given by: \[Q = mc\Delta T\]where:- \(m = 11\, \mathrm{g}\) (mass of the eye)- \(c = 4.0\, \mathrm{J/g \cdot K}\) (specific heat capacity)- \(\Delta T = 3.0\, \mathrm{C}\) (change in temperature)Substitute the values into the equation:\[Q = 11 \times 4.0 \times 3.0 = 132 \mathrm{J}\]

Calculate the Energy of a Single Photon

Microwaves have a wavelength \(\lambda = 12\, \mathrm{cm}\). Use the equation for the energy of a single photon:\[E = \frac{hc}{\lambda}\]where:- \(h = 6.626 \times 10^{-34}\, \mathrm{J \cdot s}\) (Planck's constant)- \(c = 3.0 \times 10^8\, \mathrm{m/s}\) (speed of light)- \(\lambda = 0.12\, \mathrm{m}\) (convert cm to m)Substitute these values into the equation:\[E = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{0.12} \approx 1.655 \times 10^{-24}\, \mathrm{J}\]

Calculate the Number of Photons Required

To find the number of photons required to deliver the necessary energy, divide the total required energy by the energy of a single photon:\[N = \frac{Q}{E} = \frac{132}{1.655 \times 10^{-24}}\]Calculate this value:\[N \approx 7.98 \times 10^{25}\] Thus, approximately \(7.98 \times 10^{25}\) photons are required to raise the temperature by 3.0°C.

Key concepts

Microwave Wavelength

Microwave wavelength refers to the distance over which a microwave wave's shape repeats. Wavelength is a key factor in determining the energy of photons produced by microwaves.

When dealing with microwaves, like the ones considered here, the wavelength (\(\lambda\)) is essential. It allows us to calculate the energy carried by a single photon.
Microwaves generally have longer wavelengths compared to visible light, which means they are less energetic per photon.

• The microwave wavelength given in the problem is \(\lambda = 12 \text{ cm}\).
• To perform calculations, convert this wavelength into meters: \(\lambda = 0.12 \text{ m}\).
• This conversion is key to using the formula for photon energy that involves the speed of light in meters per second.

By understanding microwave wavelengths, you gain insight into how these energies are calculated, laying the foundation for the next steps in the problem.

Specific Heat Capacity

Specific heat capacity is a measure of how much energy is required to raise the temperature of a given mass by 1°C (or 1 K).

It essentially tells us how resistant a substance is to temperature changes when it absorbs heat. In this exercise:

• We have a specific heat capacity, \(c = 4.0 \text{ J/g} \cdot \text{K}\), which means that each gram of the eye requires 4.0 joules to increase its temperature by 1°C.
• The mass of the eye is given as 11 grams.

To find the total energy needed to cause a temperature change, we use the equation:\[Q = mc\Delta T\]where \(\Delta T = 3.0 \text{°C}\), resulting in:\[Q = 11 \times 4.0 \times 3.0 = 132 \text{ J}\]This calculation helps us determine that 132 joules of energy are needed to raise the temperature of the eye by 3°C.

Planck's Constant

Planck's constant is a fundamental constant that arises in quantum mechanics. It represents the proportionality between the energy of a photon and the frequency of its electromagnetic wave.

In terms of calculations:

• Symbolized as \(h\), Planck's constant has a value of \(6.626 \times 10^{-34} \text{ J} \cdot \text{s}\).
• It is used in the formula to calculate the energy of a photon:

\[E = \frac{hc}{\lambda}\]Where \(c\) is the speed of light and \(\lambda\) is the wavelength of the microwave.

This equation reveals that the energy of a photon is inversely proportional to its wavelength—larger wavelengths mean less energy per photon. By plugging in the given values:\[E = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{0.12} \approx 1.655 \times 10^{-24} \text{ J}\]we find each microwave photon carries approximately \(1.655 \times 10^{-24}\) joules of energy.

Temperature Change

Temperature change refers to the adjustment in temperature that amounts to how much heat is absorbed or lost by a substance. In this problem, we're considering how the absorption of microwave energy alters the temperature of an eye.
Here:

• The eye's temperature change, or \(\Delta T\), is given as 3.0°C.
• This change is directly tied to the energy input, calculated in the segment on specific heat capacity.

The concept of temperature change is linked to the energy absorbed by the eye, originally measured via specific heat calculations. Once we establish how much energy (132 J) is needed for the rise, we move to determine how many photons (at the calculated energy level per photon) are needed.

We divide the total energy needed by the energy per photon to find:\[N = \frac{132}{1.655 \times 10^{-24}} \approx 7.98 \times 10^{25}\]This immense value shows the sheer number of photons required simply to bring about a relatively small temperature change of 3°C.