Microwaves are used to heat food in microwave ovens. The microwave radiation is absorbed by moisture in the food. This heats the water, and as the water becomes hot, so does the food. How many photons having a wavelength of 3.00 mm would have to be absorbed by 2.54 g of water to raise its temperature by 1.00 °C?
How much heat (energy) do you need? That's q = [mass H2O x specific heat H2O x (Tfinal-Tinitial)
Solve for q in joules.
How much heat do you have per photon?
That's E(in joules) = hc/wavelength.
Solve for E
Then E per photon x # photons = q
Solve for # photons.
To calculate the number of photons absorbed by water, you'll need to use the equation:
N = (E / E_photon)
where:
N is the number of photons
E is the energy required to raise the temperature
E_photon is the energy of one photon.
First, let's find the energy required to raise the temperature of the water by 1.00 °C. The specific heat capacity of water is approximately 4.18 J/g°C.
E = (m * c * ΔT)
where:
m is the mass of water (2.54 g)
c is the specific heat capacity of water (4.18 J/g°C)
ΔT is the temperature change (1.00 °C)
E = (2.54 g) * (4.18 J/g°C) * (1.00 °C)
E = 10.5992 J
The energy of one photon (E_photon) can be calculated using Planck's equation:
E_photon = (hc / λ)
where:
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)
λ is the wavelength of the photon (3.00 mm = 3.00 x 10^-3 m)
E_photon = (6.626 x 10^-34 J*s) * (3.00 x 10^8 m/s) / (3.00 x 10^-3 m)
E_photon = 6.626 x 10^-25 J
Now, you can calculate the number of photons:
N = (E / E_photon)
N = (10.5992 J) / (6.626 x 10^-25 J)
N ≈ 1.60 x 10^24 photons
Therefore, approximately 1.60 x 10^24 photons with a wavelength of 3.00 mm would have to be absorbed by 2.54 g of water to raise its temperature by 1.00 °C.
To calculate the number of photons required to raise the temperature of water, we need to use the equation:
E = n * h * c / λ
Where:
E is the energy of one photon
n is the number of photons
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)
λ is the wavelength of light in meters
First, we need to convert the wavelength from millimeters to meters. In this case, 3.00 mm is equal to 0.003 m.
Next, we can calculate the energy of one photon using the equation:
E = h * c / λ
E = (6.626 x 10^-34 J s) * (3.00 x 10^8 m/s) / 0.003 m
E ≈ 6.625 x 10^-22 J
Now we can calculate the number of photons using the equation:
E = n * (6.625 x 10^-22 J)
To raise the temperature of water, we need to calculate the heat energy required. The specific heat capacity of water is approximately 4.18 J/g°C.
The heat energy required can be calculated by using the equation:
Q = m * c * ΔT
Where:
Q is the heat energy
m is the mass of water in grams
c is the specific heat capacity of water
ΔT is the temperature change in °C
Given:
Mass of water (m) = 2.54 g
Temperature change (ΔT) = 1.00 °C
Specific heat capacity of water (c) = 4.18 J/g°C
Q = (2.54 g) * (4.18 J/g°C) * (1.00 °C)
Q ≈ 10.58 J
Finally, we can calculate the number of photons (n) using the equation:
10.58 J = n * (6.625 x 10^-22 J)
n ≈ 1.598 x 10^21 photons
Therefore, approximately 1.598 x 10^21 photons with a wavelength of 3.00 mm would need to be absorbed by 2.54 g of water to raise its temperature by 1.00 °C.