I am having a problem with this problem from my homework. I need some help on this problem.
Microwave ovens work by irradiating food with microwave radiation, which is absorbed and converted into heat. Assume radiation with wavelength=15.0cm is used and that all the energy is converted to heat. How many photons are necessary to raise the temperature of a 350mL quantity of water from 20.0 degree C to 95.0 degree C?
(a)
The energy of a photon is given by:
E = hf
h = Planck's constant (look it up)
f = frequency = speed of light / wavelength
f = (3.00x10^8 m/s) / (0.15m) = _____?
E = ______?
(b)
∆T = T2 - T1 = ____?
Heat needed = (4.18 J/g.C)(grams H2O)(∆T)
Heat = _________?
(c)
Divide the heat calculated in part (b) by the energy per photon calculated in part (a)
To solve this problem, we need to calculate the number of photons required to raise the temperature of the water.
Step 1: Determine the energy of one photon.
The energy of a photon can be calculated using the equation E = hc/λ, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength.
First, we need to convert the wavelength from centimeters to meters:
15.0 cm = 0.15 m.
Now, we can calculate the energy of one photon:
E = (6.626 x 10^-34 J·s) * (3.00 x 10^8 m/s) / 0.15 m = 1.325 x 10^-22 J.
Step 2: Calculate the change in energy required to raise the temperature.
The change in energy required to raise the temperature of the water can be calculated using the equation ΔE = mcΔT, where ΔE is the change in energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
The mass of the water is 350 mL, which is equivalent to 350 grams since the density of water is 1 g/mL.
The specific heat capacity of water is approximately 4.18 J/g°C.
The change in temperature is ΔT = 95.0 °C - 20.0 °C = 75.0 °C.
Now, we can calculate the change in energy:
ΔE = (350 g) * (4.18 J/g°C) * (75.0 °C) = 1.4515 x 10^5 J.
Step 3: Calculate the number of photons.
The number of photons can be calculated by dividing the change in energy by the energy of one photon:
Number of photons = ΔE / E = (1.4515 x 10^5 J) / (1.325 x 10^-22 J) ≈ 1.09 x 10^27.
Therefore, approximately 1.09 x 10^27 photons are necessary to raise the temperature of the water from 20.0 °C to 95.0 °C.