Suppose that the microwave radiation has a wavelength of 12 cm . How many photons are required to heat 295 mL of coffee from 25.0 ∘C to 62.0 ∘C? Assume that the coffee has the same density, 0.997 g/mL , and specific heat capacity, 4.184 J/(g⋅K) , as water over this temperature range.
To determine the number of photons required to heat the coffee, we need to follow these steps:
Step 1: Calculate the mass of the coffee.
Step 2: Calculate the heat energy required to raise the coffee's temperature.
Step 3: Convert the heat energy into the number of photons.
Let's go through each step in detail.
Step 1: Calculate the mass of the coffee.
To find the mass of the coffee, we can use its volume and density.
Given:
Density of coffee = 0.997 g/mL
Volume of coffee = 295 mL
Mass of coffee = Density × Volume
Mass of coffee = 0.997 g/mL × 295 mL
Mass of coffee = 293.815 g (rounded to three decimal places)
Step 2: Calculate the heat energy required to raise the coffee's temperature.
The heat energy can be calculated using the equation:
Q = mcΔT
Where:
Q = Heat energy (Joules)
m = Mass of coffee (grams)
c = Specific heat capacity of water (4.184 J/(g⋅K))
ΔT = Change in temperature (62.0 °C - 25.0 °C = 37.0 °C)
Q = (293.815 g) × (4.184 J/(g⋅K)) × (37.0 °C)
Q ≈ 48,864.52 J (rounded to two decimal places)
Step 3: Convert the heat energy into the number of photons.
The energy of a photon is given by the equation:
E = hc/λ
Where:
E = Energy of a photon (Joules)
h = Planck's constant (6.626 × 10^-34 J⋅s)
c = Speed of light (3 × 10^8 m/s)
λ = Wavelength of microwave radiation (12 cm = 0.12 m)
E = (6.626 × 10^-34 J⋅s) × (3 × 10^8 m/s) / (0.12 m)
E ≈ 1.6555 × 10^-18 J (rounded to four decimal places)
To find the number of photons, we divide the total heat energy (Q) by the energy of a single photon (E):
Number of photons = Q / E
Number of photons ≈ 48,864.52 J / 1.6555 × 10^-18 J
Number of photons ≈ 2.95 × 10^25 (rounded to two significant figures)
Therefore, approximately 2.95 × 10^25 photons are required to heat 295 mL of coffee from 25.0 °C to 62.0 °C.
To determine the number of photons required to heat the coffee, we need to follow a few steps:
Step 1: Calculate the mass of the coffee.
The mass of the coffee can be calculated using the volume and density of the coffee:
Mass = Volume × Density
Given:
Volume of coffee = 295 mL
Density of coffee = 0.997 g/mL
Mass of the coffee = 295 mL × 0.997 g/mL
Step 2: Calculate the temperature change of the coffee.
The temperature change can be calculated by subtracting the initial temperature from the final temperature:
Temperature change = Final temperature - Initial temperature
Given:
Initial temperature = 25.0 °C
Final temperature = 62.0 °C
Temperature change = 62.0 °C - 25.0 °C
Step 3: Calculate the energy required to heat the coffee.
The energy required can be calculated using the mass of the coffee, specific heat capacity, and temperature change:
Energy = Mass × Specific heat capacity × Temperature change
Given:
Specific heat capacity of water = 4.184 J/(g⋅K)
Energy = Mass × Specific heat capacity × Temperature change
Step 4: Calculate the energy of one photon.
The energy of one photon can be calculated using the equation:
Energy of one photon = Planck's constant × speed of light / wavelength
Given:
Wavelength of microwave radiation = 12 cm
Convert wavelength from centimeters to meters:
Wavelength = 12 cm × 0.01 m/cm
Now, we can calculate the energy of one photon.
Step 5: Calculate the number of photons required.
The number of photons required can be calculated by dividing the energy required to heat the coffee by the energy of one photon:
Number of photons required = Energy required / Energy of one photon
Now, we can calculate the number of photons required to heat the coffee.
numberphotons*Energy/photon=masscoffie*c*changetemp
= 295*.997*4.18*(62-25) joules
Now the energy/photon= h lambda/speedlight=6.6e-34 m2 kg / s *.12m/3e8m/s
= 6.6e-34*.12/3e8 joules
so finally, number jphotons= (295*.997*4.18*(62-25))/(6.6e-34*.12/3e8)
= 1.72e47 check that.