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 200 mL of water was placed in the oven and heated. How many photons are required to heat the water in the container from 20.0 degrees C to 95.0 degrees C?
To determine the number of photons required to heat the water, we need to calculate the energy required to raise the temperature of 200 mL of water from 20.0°C to 95.0°C using the specific heat capacity of water.
Step 1: Calculate the mass of the water.
Density of water = 1 g/mL
Volume of water = 200 mL
Mass of water = Density x Volume = 1 g/mL x 200 mL = 200 g
Step 2: Calculate the energy required to heat the water.
Specific heat capacity of water = 4.184 J/g°C (approximate)
Change in temperature = Final temperature - Initial temperature
Change in temperature = 95.0°C - 20.0°C = 75.0°C
Energy required = Mass x Specific heat capacity x Change in temperature
Energy required = 200 g x 4.184 J/g°C x 75.0°C = 627,600 J
Step 3: Calculate the energy per photon.
Energy of a photon = Planck's constant x speed of light / wavelength
Planck's constant (h) ≈ 6.626 x 10^-34 J·s
Speed of light (c) ≈ 3.0 x 10^8 m/s
Convert wavelength to meters: 12.5 cm = 0.125 m
Energy per photon = (6.626 x 10^-34 J·s x 3.0 x 10^8 m/s) / 0.125 m
Energy per photon ≈ 1.5952 x 10^-16 J
Step 4: Calculate the number of photons.
Number of photons = Energy required / Energy per photon
Number of photons = 627,600 J / 1.5952 x 10^-16 J
Number of photons ≈ 3.9342 x 10^21 photons
Therefore, it would require approximately 3.9 x 10^21 photons to heat the water in the container from 20.0 degrees C to 95.0 degrees C.
To calculate the number of photons required to heat the water in the container, we first need to understand the relationship between energy and wavelength of photons.
The energy of a photon can be calculated using the equation:
E = hc/λ
Where:
E is the energy of the 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 photon.
Since the wavelength of the microwave radiation is given as 12.5 cm (which can be converted to meters as 0.125 m), we can substitute the values into the equation to determine the energy of a single photon:
E = (6.626 x 10^-34 J s x 3.0 x 10^8 m/s) / 0.125 m
E = 5.301 x 10^-25 J
Next, we need to calculate the energy required to heat the water. The specific heat capacity of water is approximately 4.18 J/g°C. However, we need to convert the volume of water from milliliters to grams before calculating the energy.
To convert volume (in mL) to mass (in grams) for water, we use the equation:
m = V x density
Where:
m is the mass of water (in grams),
V is the volume of water (in milliliters),
density is the density of water (approximately 1 g/mL).
Substituting the given volume of 200 mL into the equation, we can calculate the mass of water:
m = 200 mL x 1 g/mL
m = 200 g
Now, we can calculate the energy required to heat the water using the equation:
Q = m x c x ΔT
Where:
Q is the energy required (in joules),
m is the mass of water (in grams),
c is the specific heat capacity of water (in J/g°C),
ΔT is the change in temperature (in °C).
Substituting the values into the equation, we can calculate the energy required to heat the water from 20.0°C to 95.0°C:
Q = 200 g x 4.18 J/g°C x (95.0°C - 20.0°C)
Q = 147,400 J
To find the number of photons required, we can divide the total energy required by the energy of a single photon:
Number of photons = Q / E
Number of photons = 147,400 J / 5.301 x 10^-25 J
Performing the calculation, we find the number of photons required to heat the water in the container.