Question

Microwave ovens convert radiation to energy. A microwave oven uses radiation with a wavelength of \(12.5 \mathrm{~cm} .\) Assuming that all the energy from the radiation is converted to heat without loss, how many moles of photons are required to raise the temperature of a cup of water \((350.0 \mathrm{~g},\) specific heat \(\left.=4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\) from \(23.0^{\circ} \mathrm{C}\) to \(99.0^{\circ} \mathrm{C} ?\)

Short answer

Answer: 1,140 moles of microwave radiation photons are required.

Step-by-step solution

Determine the required energy to heat the water

To find the energy required to raise the temperature of the water, we can use the formula: \(Q = mc\Delta T\) Where \(Q\) is the energy required, \(m\) is the mass of the water, \(c\) is the specific heat of the water, and \(\Delta T\) is the change in temperature. Given the mass of the water \(m = 350.0 \mathrm{~g}\), the specific heat \(c = 4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\), and the temperature change \(\Delta T = 99.0^{\circ} \mathrm{C} - 23.0^{\circ} \mathrm{C} = 76.0^{\circ} \mathrm{C}\), we can calculate the energy required: \(Q = (350.0\,\mathrm{g})(4.18\,\mathrm{J/g\,^{\circ}C})(76.0\,^{\circ}\mathrm{C}) = 109444\,\mathrm{J}\)

Convert wavelength to energy per photon

To find the energy per photon, we can use Planck's equation: \(E = \dfrac{hc}{\lambda}\) Where \(E\) is the energy per photon, \(h\) is Planck's constant \((6.626 \times 10^{-34} \mathrm{Js})\), \(c\) is the speed of light \((3 \times 10^8\,\mathrm{m/s})\), and \(\lambda\) is the wavelength of the radiation. Given the wavelength \(\lambda = 12.5\,\mathrm{cm} = 0.125\,\mathrm{m}\), we can calculate the energy per photon: \(E = \dfrac{(6.626 \times 10^{-34}\,\mathrm{Js})(3 \times 10^8\,\mathrm{m/s})}{0.125\,\mathrm{m}} = 1.5936 \times 10^{-22}\,\mathrm{J}\)

Calculate the number of photons required

To find the number of photons needed to provide the required energy, we can use the equation: \(N = \dfrac{Q}{E}\) Where \(N\) is the number of photons, \(Q\) is the energy required, and \(E\) is the energy per photon. With the given values \(Q = 109444\,\mathrm{J}\) and \(E = 1.5936 \times 10^{-22}\,\mathrm{J}\), we can calculate the number of photons: \(N = \dfrac{109444\,\mathrm{J}}{1.5936 \times 10^{-22}\,\mathrm{J}} = 6.866 \times 10^{26}\)

Convert the number of photons to moles of photons

To calculate the number of moles of photons, we can use Avogadro's number: Moles \(= \dfrac{N}{N_A}\) Where Moles are the moles of photons, \(N\) is the number of photons, and \(N_A\) is Avogadro's number \((6.022 \times 10^{23}\,\mathrm{mol^{-1}})\). Given the number of photons \(N = 6.866 \times 10^{26}\), we can calculate the moles of photons: Moles \(= \dfrac{6.866 \times 10^{26}}{6.022 \times 10^{23}\,\mathrm{mol^{-1}}} = 1140\,\mathrm{mol}\) In conclusion, 1,140 moles of microwave radiation photons are required to raise the temperature of a cup of water from \(23.0^{\circ} \mathrm{C}\) to \(99.0^{\circ} \mathrm{C}\).