Microwave ovens use microwave radiation to heat food. The microwaves are absorbed by moisture in the food, which is transferred to other components of the food. As the water becomes hotter, so does the food. Suppose that the microwave radiation has a wavelength of 11.2 cm
How many photons are required to heat 200 mL of coffee from 24C to 63C
q to heat 200 mL H2O = mass water x specific heat water x delta T water. = 200 g x 4.184 x (63-24) = xx joules.
E in a 11.2 cm photon.
E = hc/wavelength.
E = h is Planck's constant. c is speed of light in m/s and wavelength is in 11.2 cm convert to meters. This will be the energy of 1 photon of the 11.2 cm wavelength. Then E joules/photon x # photons = xx joules. Solve for # photons.
To calculate the number of photons required to heat 200 mL of coffee from 24°C to 63°C, we need to follow these steps:
Step 1: Calculate the energy required to heat the coffee.
The energy required to heat a substance can be calculated using the formula:
Q = m * C * ΔT
Where:
Q - energy (in joules)
m - mass of the substance (in grams)
C - specific heat capacity of the substance (in J/g°C)
ΔT - change in temperature (in °C)
Since the mass of the coffee is given as 200 mL, we need to convert it to grams by assuming the density of water (which is approximately the same as coffee) is 1 g/mL. So the mass (m) of the coffee can be calculated as:
m = V * ρ
Where:
V - volume of coffee (in mL)
ρ - density of water (in g/mL)
Substituting the values:
m = 200 mL * 1 g/mL
m = 200 g
The specific heat capacity (C) of water is approximately 4.18 J/g°C. So we can substitute the values into the formula to calculate the energy (Q) required to heat the coffee:
Q = 200 g * 4.18 J/g°C * (63°C - 24°C)
Step 2: Calculate the energy per photon.
The energy of a single photon of electromagnetic radiation can be calculated using the formula:
E = hc / λ
Where:
E - energy of a photon (in joules)
h - Planck's constant (approximately 6.63 x 10^-34 J*s)
c - speed of light (approximately 2.998 x 10^8 m/s)
λ - wavelength of radiation (in meters)
Since the wavelength is given as 11.2 cm, we need to convert it to meters by dividing by 100:
λ = 11.2 cm / 100
λ = 0.112 m
Substituting the values into the formula:
E = (6.63 x 10^-34 J*s * 2.998 x 10^8 m/s) / 0.112 m
Step 3: Calculate the number of photons.
To find the number of photons required to provide the calculated energy, we divide the energy required (Q) by the energy per photon (E):
Number of photons = Q / E
Now you can plug in the calculated values and perform the calculation to find the required number of photons.
To calculate the number of photons required to heat the coffee, we'll need to use the equation:
E = nhν
where:
E is the energy of the photons,
n is the number of photons,
h is Planck's constant (6.626 x 10^-34 J·s),
ν is the frequency of the photons.
First, we need to determine the frequency (ν) of the microwaves. The speed of light (c) is given by c = νλ, where λ is the wavelength. Rearranging this equation, we have ν = c/λ.
The speed of light is approximately 3 x 10^8 m/s (or 300,000,000 m/s). We need to convert the wavelength from centimeters to meters, so 11.2 cm is equal to 0.112 m.
ν = (3 x 10^8 m/s) / (0.112 m)
ν = 2.68 x 10^9 Hz
Now that we have the frequency, we can calculate the energy of each photon using the equation E = hν.
E = (6.626 x 10^-34 J·s) x (2.68 x 10^9 Hz)
E = 1.77 x 10^-24 J
Next, we can calculate the total energy required to heat the coffee. The equation for energy is Q = mcΔT, where Q is the total energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Since the specific heat capacity of coffee (c) is approximately 4.18 J/g·°C, and the density of water is approximately 1 g/mL, we can calculate the mass (m) of 200 mL of coffee.
m = 200 mL x 1 g/mL
m = 200 g
Now we can find the total energy (Q):
Q = (200 g) x (4.18 J/g·°C) x (63°C - 24°C)
Q = 16,648 J
Finally, we can determine the number of photons (n) required by rearranging the equation n = Q / E:
n = 16,648 J / (1.77 x 10^-24 J)
n ≈ 9.4 x 10^27 photons
Therefore, approximately 9.4 x 10^27 photons are required to heat 200 mL of coffee from 24°C to 63°C using microwave radiation with a wavelength of 11.2 cm.