Suppose that the microwave radiation has a wavelength of 12 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.
Well, let's first calculate the mass of the coffee. Since the density is 0.997 g/mL and the volume is 255 mL, the mass can be calculated as:
mass = density * volume = 0.997 g/mL * 255 mL = 254.235 g
Now let's calculate the heat required to increase the temperature of the coffee. The specific heat capacity is 4.184 J/(g⋅K), and the change in temperature is 62.0 °C - 25.0 °C = 37.0 °C. Therefore, the heat required can be calculated as:
heat = mass * specific heat capacity * change in temperature = 254.235 g * 4.184 J/(g⋅K) * 37.0 °C = 38850.445 J
Now, let's calculate the energy of one photon with a wavelength of 12 cm. The formula for the energy of a photon is given by:
energy = Planck's constant * speed of light / wavelength
Planck's constant is approximately 6.626 x 10^-34 J⋅s, and the speed of light is approximately 3.00 x 10^8 m/s. However, we need to convert the wavelength from cm to m, so the wavelength is 12 cm * 0.01 m/cm = 0.12 m. Therefore, the energy of one photon can be calculated as:
energy = 6.626 x 10^-34 J⋅s * 3.00 x 10^8 m/s / 0.12 m = 1.6565 x 10^-18 J
Finally, let's calculate the number of photons required to heat the coffee. Since the total energy required is 38850.445 J and the energy of one photon is 1.6565 x 10^-18 J, the number of photons can be calculated as:
number of photons = total energy / energy of one photon = 38850.445 J / 1.6565 x 10^-18 J ≈ 2.3495 x 10^21 photons
So, approximately 2.35 x 10^21 photons would be required to heat the coffee. That's a lot of photons! Maybe we should call the coffee "Photon Brew" instead.
To determine the number of photons required to heat the coffee, we need to calculate the energy required to raise its temperature using the given specific heat capacity and the mass of the coffee.
Step 1: Calculate the mass of coffee.
Given: volume of coffee = 255 mL
density of coffee = 0.997 g/mL
Using the formula: Mass = Volume × Density
Mass = 255 mL × 0.997 g/mL
Mass = 254.835 g (rounded to three decimal places)
Step 2: Calculate the change in temperature.
Given: initial temperature = 25.0 °C
final temperature = 62.0 °C
Change in temperature = final temperature - initial temperature
Change in temperature = 62.0 °C - 25.0 °C
Change in temperature = 37.0 °C
Step 3: Calculate the energy required to raise the coffee's temperature.
Given: specific heat capacity of coffee = 4.184 J/(g·K)
mass of coffee = 254.835 g
change in temperature = 37.0 °C
Using the formula: Energy = Mass × Specific Heat Capacity × Change in Temperature
Energy = 254.835 g × 4.184 J/(g·K) × 37.0 °C
Energy = 40026.577 J (rounded to three decimal places)
Step 4: Calculate the energy per photon.
Given: wavelength of microwave radiation = 12 cm
The energy of a photon can be calculated using the formula: Energy = (Planck's constant × speed of light) / wavelength
Planck's constant (h) = 6.626 × 10^-34 J·s (rounded to three decimal places)
Speed of light (c) = 3.00 × 10^8 m/s (rounded to three significant figures)
Wavelength (λ) = 12 cm = 0.12 m
Energy per photon = (6.626 x 10^-34 J·s × 3.00 x 10^8 m/s) / 0.12 m
Energy per photon = 1.6565 x 10^-18 J (rounded to four decimal places)
Step 5: Calculate the number of photons.
Given: Energy required to raise the coffee's temperature = 40026.577 J
Energy per photon = 1.6565 x 10^-18 J
Number of photons = Energy required / Energy per photon
Number of photons = 40026.577 J / 1.6565 x 10^-18 J
Number of photons = 2.414 x 10^22 photons (rounded to three decimal places)
Therefore, approximately 2.414 x 10^22 photons are required to heat 255 mL of coffee from 25.0 °C to 62.0 °C.
To calculate the number of photons required to heat the coffee, we need to follow these steps:
1. Calculate the mass of the coffee.
2. Calculate the energy required to heat the coffee.
3. Convert the energy to the number of photons.
Let's go through each step in detail:
1. Calculate the mass of the coffee:
Given the volume of the coffee (255 mL) and its density (0.997 g/mL), we can use the formula:
mass = volume x density
mass = 255 mL x 0.997 g/mL
2. Calculate the energy required to heat the coffee:
We will use the specific heat capacity formula:
energy = mass x specific heat capacity x temperature change
The temperature change is the difference between the final temperature (62.0 °C) and the initial temperature (25.0 °C).
3. Convert the energy to the number of photons:
To do this, we need to know the energy per photon. The energy of a photon is given by:
energy = Planck's constant x speed of light / wavelength
Plank's constant (h) is approximately 6.626 x 10^-34 J·s, and the speed of light (c) is approximately 3.00 x 10^8 m/s.
We need to convert the wavelength from centimeters to meters by dividing it by 100.
Once we have the energy per photon, we can divide the total energy required to heat the coffee by the energy per photon to find the number of photons.
Now, substitute the values into the formulas and calculate:
1. Calculate the mass of the coffee:
mass = 255 mL x 0.997 g/mL
2. Calculate the energy required to heat the coffee:
energy = mass x specific heat capacity x temperature change
3. Convert the energy to the number of photons:
- Convert the wavelength from cm to meters: wavelength = 12 cm / 100
- Calculate the energy per photon: energy per photon = (Planck's constant x speed of light) / wavelength
- Calculate the number of photons: number of photons = energy required / energy per photon
Using these steps, you can calculate the number of photons required to heat the coffee from 25.0 °C to 62.0 °C.