Suppose that the microwave radiation has a wavelength of 12.4 cm. How many photons are required to heat 255 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.
How much heat is needed to heat the coffee?
q = mass water x specific heat water x (Tfinal-Tintial)
How much energy in 1 photon?
E = hc/wavelength.
Now set up a proportion.
(1 photon/E) = (#photons/q)
solve for # photons.
To calculate the number of photons required to heat the coffee, we need to determine the energy required to raise its temperature.
Step 1: Calculate the mass of the coffee
Given that the density of the coffee is 0.997 g/mL and the volume is 255 mL, we can calculate the mass:
mass = density x volume
mass = 0.997 g/mL x 255 mL
Step 2: Calculate the change in temperature
The change in temperature is the final temperature minus the initial temperature:
change in temperature = final temperature - initial temperature
change in temperature = 62.0 °C - 25.0 °C
Step 3: Calculate the energy required to heat the coffee
The energy required to heat a substance can be calculated using the formula:
energy = mass x specific heat capacity x change in temperature
energy = mass x 4.184 J/(g/ K) x change in temperature
Step 4: Convert the energy from joules to photons
The energy of a single photon can be calculated using the formula:
energy of a photon = Planck's constant x speed of light / wavelength
energy of a photon = 6.626 x 10^-34 J s x 2.998 x 10^8 m/s / wavelength
Step 5: Calculate the number of photons
The number of photons can be calculated by dividing the energy required to heat the coffee by the energy of a single photon:
number of photons = energy required / energy of a photon
Now, let's calculate the number of photons:
Step 1: mass = 0.997 g/mL x 255 mL
mass = 256.335 g
Step 2: change in temperature = 62.0 °C - 25.0 °C
change in temperature = 37.0 °C
Step 3: energy = mass x 4.184 J/(g/ K) x change in temperature
energy = 256.335 g x 4.184 J/(g/ K) x 37.0 °C
Step 4: energy of a photon = 6.626 x 10^-34 J s x 2.998 x 10^8 m/s / wavelength
wavelength = 0.124 m (converted from 12.4 cm)
energy of a photon = 6.626 x 10^-34 J s x 2.998 x 10^8 m/s / 0.124 m
Step 5: number of photons = energy required / energy of a photon
number of photons = (256.335 g x 4.184 J/(g/ K) x 37.0 °C) / (6.626 x 10^-34 J s x 2.998 x 10^8 m/s / 0.124 m)
Performing the calculation above will give you the number of photons required to heat the coffee from 25.0 °C to 62.0 °C.
To determine the number of photons required to heat the coffee, we need to calculate the amount of energy needed to raise the temperature of the coffee from 25.0 °C to 62.0 °C.
Step 1: Calculate the mass of the coffee.
Given that the density of the coffee is 0.997 g/mL and the volume is 255 mL, we can calculate the mass using the formula:
Mass = Density × Volume
Mass = 0.997 g/mL × 255 mL
Step 2: Calculate the specific heat capacity of the coffee.
Given that the specific heat capacity of the coffee is 4.184 J/(g/K), we can use this value to calculate the amount of energy required to raise the temperature of the coffee.
Step 3: Calculate the change in temperature.
The change in temperature is the final temperature minus the initial temperature.
Change in temperature = 62.0 °C - 25.0 °C
Step 4: Calculate the energy required.
The energy required can be calculated using the formula:
Energy = Mass × Specific Heat Capacity × Change in Temperature
Step 5: Calculate the number of photons.
The energy of a single photon is given by Planck's equation:
Energy of a photon = Planck's constant × Speed of light / Wavelength
To get the total number of photons, divide the energy required by the energy of a single photon.
Number of photons = Energy required / Energy of a photon
Now let's calculate:
Step 1: Mass = 0.997 g/mL × 255 mL = 254.935 g
Step 2: Specific Heat Capacity = 4.184 J/(g/K)
Step 3: Change in temperature = 62.0 °C - 25.0 °C = 37.0 °C
Step 4: Energy = 254.935 g × 4.184 J/(g/K) × 37.0 °C
Step 5: Energy of a photon = Planck's constant × Speed of light / Wavelength (convert 12.4 cm into meters by dividing by 100)
Finally, divide the Energy required by the Energy of a photon to get the number of photons.