Microwave ovens heat food by creating microwave electromagnetic radiation that is absorbed by water molecules in the food. Any material that does not have water in it will not absorb the radiation and will not get hot. Metals reflect the microwaves from their surfaces and disrupt the operation of the oven. Some Australians heat water in microwave ovens to make tea. What is the minimum number of microwave photons with a wavelength of 3.28 mm that will have to be absorbed by a 112.00 gram sample of water to heat it from 25°C to 100°C? (It requires about 315 joules of energy to heat 1 gram of water 75°C)
How much heat do we need to raise T 112 g sample of water from 25 to 100? That's
mass H2O x specific heat H2O x delta T = 112 x 4.184 x 75 = 35, 146 J.
E 1 photon = hc/wavelenth
E 1 photon = 6.626E-34*3E8^2/3.28E-3
E/photon x #photons = 3,146 J
And solve for # photons.
To find the minimum number of microwave photons required to heat a sample of water, we need to calculate the energy required to heat the water and then convert it into the number of microwave photons.
First, let's calculate the energy required to heat the water from 25°C to 100°C using the given information.
The specific heat capacity of water is approximately 4.18 J/g°C. Since we want to heat 112.00 grams of water by 75°C, the energy required can be calculated as follows:
Q = mass × specific heat capacity × temperature change
Q = 112.00 g × 4.18 J/g°C × 75°C
Q = 354,120 Joules
So, we need 354,120 Joules of energy to heat the water sample.
Next, we need to convert this energy into the number of microwave photons. To do this, we can use the energy of a single photon equation:
E = h × c / λ
Where:
E is the energy of a single photon
h is Planck's constant (approximately 6.626 x 10^-34 J·s)
c is the speed of light (approximately 3.0 x 10^8 m/s)
λ is the wavelength of the microwave photons (3.28 mm, or 3.28 x 10^-3 m)
Now, we can calculate the energy of a single photon:
E = (6.626 x 10^-34 J·s) × (3.0 x 10^8 m/s) / (3.28 x 10^-3 m)
E ≈ 6.0428 x 10^-20 Joules
Finally, we divide the total energy required to heat the water by the energy of a single photon to find the minimum number of microwave photons:
Number of photons = Total energy / Energy of a single photon
Number of photons = 354,120 Joules / 6.0428 x 10^-20 Joules
Calculating this, we get:
Number of photons ≈ 5.853 x 10^25 photons
Therefore, the minimum number of microwave photons with a wavelength of 3.28 mm that would need to be absorbed by the 112.00 gram sample of water to heat it from 25°C to 100°C is approximately 5.853 x 10^25 photons.
To calculate the minimum number of microwave photons required to heat the water, we need to consider the energy required to heat the water from 25°C to 100°C.
The equation to calculate the energy required to heat a substance is:
Q = m * c * ΔT
Where:
Q is the energy in Joules
m is the mass of the water in grams
c is the specific heat capacity of water, approximately 4.184 J/g°C
ΔT is the change in temperature in degrees Celsius
Given:
m = 112.00 grams (mass of water)
ΔT = 75°C (change in temperature)
Using the equation, we can calculate the energy required to heat the water:
Q = 112.00 g * 4.184 J/g°C * 75°C
Q = 35,238.6 J
Since 1 photon of microwave radiation carries an energy proportional to its wavelength, we can use this energy to calculate the number of photons required.
The energy of a photon is given by:
E = h * c / λ
Where:
E is the energy of a photon in Joules
h is Planck's constant, approximately 6.626 x 10^(-34) J*s
c is the speed of light, approximately 3.00 x 10^8 m/s
λ is the wavelength in meters
Given:
λ = 3.28 mm = 3.28 x 10^(-3) m
Using the equation, we can calculate the energy of a single photon:
E = (6.626 x 10^(-34) J*s * 3.00 x 10^8 m/s) / (3.28 x 10^(-3) m)
E ≈ 6.049 x 10^(-22) J
Now we can calculate the minimum number of photons required:
Number of photons = Energy required / Energy of a single photon
Number of photons = 35,238.6 J / 6.049 x 10^(-22) J
Number of photons ≈ 5.82 x 10^24 photons
Therefore, the minimum number of microwave photons required to heat the 112.00 gram sample of water from 25°C to 100°C is approximately 5.82 x 10^24 photons.