Question

Microwave ovens emit microwave radiation that is absorbed by water. The absorbed radiation is converted to heat that is transferred to other components of the food. Suppose the microwave radiation has wavelength \(12.5 \mathrm{~cm} .\) How many photons are required to increase the temperature of \(1.00 \times 10^{2} \mathrm{~mL}\) of water \((d=1.0 \mathrm{~g} / \mathrm{mL})\) from \(20^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) if all the energy of the photons is converted to heat?

Short answer

Answer: About 4.20 x 10^27 photons are required.

Step-by-step solution

Calculate the mass of water

Given that the density \(d\) of water is \(1.0 \mathrm{g/mL}\), and we have \(1.00 \times 10^{2} \mathrm{mL}\) of water, we can find the mass of water by multiplying the volume by the density: \( m = V \times d = 1.00 \times 10^{2} \mathrm{mL} \times 1.0 \mathrm{g/mL} = 100 \mathrm{g}\).

Calculate the energy required to raise the temperature of water

Using the formula \(Q = mc\Delta T\), where the specific heat capacity \(c\) of water is approximately \(4.18 \mathrm{J/(g \cdot K)}\) and the temperature change \(\Delta T\) is \(100^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C} = 80 \mathrm{K}\), we can find the total energy \(Q\) required: \(Q = (100 \mathrm{g}) \times (4.18 \mathrm{J/(g \cdot K)}) \times (80 \mathrm{K}) = 33440 \mathrm{J}\).

Calculate the energy of a single microwave photon

Using the formula \(E = \dfrac{hc}{\lambda}\), where the Planck's constant \(h\) is approximately \(6.63 \times 10^{-34} \mathrm{Js}\), the speed of light \(c\) is approximately \(3.00 \times 10^8 \mathrm{m/s}\), and the wavelength \(\lambda\) is given as \(12.5 \mathrm{cm}\) (or \(0.125 \mathrm{m}\)), we can find the energy \(E\) of a single microwave photon: \(E = \dfrac{(6.63 \times 10^{-34} \mathrm{Js}) \times (3.00 \times 10^8 \mathrm{m/s})}{0.125 \mathrm{m}} = 7.96 \times 10^{-24} \mathrm{J}\).

Calculate the number of photons required

Now that we have the energy required to raise the temperature of the water \(Q\) and the energy of a single photon \(E\), we can calculate the number of photons \(n\) needed by dividing \(Q\) by \(E\): \(n = \dfrac{Q}{E} = \dfrac{33440 \mathrm{J}}{7.96 \times 10^{-24} \mathrm{J}} = 4.20 \times 10^{27} \text{ photons}\). Therefore, about \(4.20 \times 10^{27}\) photons are required to increase the temperature of \(1.00 \times 10^{2} \mathrm{mL}\) of water from \(20^{\circ} \mathrm{C}\) to \(100^{\circ} \mathrm{C}\) if all the energy of the photons is converted to heat.