A microwave oven heats by radiating food with microwave radiation which is absorbed by the food and converted to heat. Suppose an oven's radiation wavelength is 12.5 cm. A container with 0.250 liter of water was placed in the oven and the temperature of the water rose from 20 degrees celsium to 100 degrees celsium. How many photons of this microwave radiation were required. Assume that all energy from the radiation were used to raise the temperature of the water
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To calculate the number of photons required to heat the water, we can use the equation:
E = h * f
where E is the energy of a photon, h is Planck's constant (6.626 x 10^-34 J·s), and f is the frequency of the microwave radiation.
We know that the wavelength of the microwave radiation is 12.5 cm. The speed of light, c, can be used to determine the frequency, f, using the equation:
c = λ * f
Rearranging the equation, we have:
f = c / λ
The speed of light in a vacuum is approximately 3 x 10^8 m/s. We need to convert the wavelength from cm to m by dividing it by 100:
λ = 12.5 cm / 100 = 0.125 m
Now we can calculate the frequency:
f = 3 x 10^8 m/s / 0.125 m = 2.4 x 10^9 Hz
Now that we have the frequency, we can calculate the energy of one photon using the equation:
E = h * f
Plugging in the values:
E = (6.626 x 10^-34 J·s) * (2.4 x 10^9 Hz) = 15.90 x 10^-25 J
To find the total energy required to heat the water, we can use the specific heat capacity of water, which is 4.18 J/(g·°C). We have 0.250 liter (or 250 grams) of water, and the temperature change is from 20°C to 100°C:
ΔE = m * c * ΔT
ΔE = (250 g) * (4.18 J/g·°C) * (100°C - 20°C) = 83,600 J
Finally, we can determine the number of photons required by dividing the total energy required by the energy of one photon:
Number of photons = ΔE / E
Number of photons = 83,600 J / 15.90 x 10^-25 J ≈ 5.24 x 10^27 photons
Therefore, approximately 5.24 x 10^27 photons of microwave radiation were required to heat the water.