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Answer :
Final answer:
A microwave oven producing 900 W of 2560 MHz microwaves creates approximately 5.3 * 10^28 photons per second. About 5e41 photons are needed to warm half a kilo of pasta by 45 degrees Celsius, which would take approximately 31,710 years in this microwave.
Explanation:
To find the number of photons per second, we need to find the energy of a single photon and then divide the total power by this. According to the Planck-Einstein relation, the energy of a photon can be calculated using the equation E = h*f, where h = 6.62607015 × 10-59 m2 kg / s is the Planck constant, and f is the frequency of the photon. With f = 2560 MHz or 2.56 * 109 s-1. The energy of a single photon is therefore about 1.69 * 10-24 J.
Therefore, the oven produces about 900 W / 1.69 * 10-24 J = 5.3 * 1028 photons per second (a).
Now, if we want to increase the temperature of a 0.500-kg mass of pasta by 45.0ºC , using a specific heat of 0.900 kcal/kg ⋅ ºC, we first need to find the total energy needed. We use the specific heat formula q = mcΔT, which gives q = 0.500 kg * 0.900 kcal/kg ⋅ ºC * 45 ºC = 20.25 kcal, or approximately 8.474e7 J. This amount of energy corresponds to 8.474e7 J / 1.69 * 10^(-24) J/photo = 5e41 photons. This is the number of photons needed to heat the pasta (b).
Finally, to find out how long the heating will take, we divide the needed number of photons by the number of photons produced per second. This gives a time of 5e41 photons / 5.3 * 1028 photons/s ~ 1012 s, or approximately 31,710 years (c).
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