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.
I am not sure how to find the answer
NVM got it
Energy/photon = hc/wavelength.
Remember to change wavelength meters and use c in m/s.
heat required = mass x density x specific heat x delta T. Everything is here to calculate heat required.
Knowing heat/photon and the total energy required you can calculate the number of photons required.
To find the number of photons required to heat the coffee, we need to understand the relationship between the energy of a photon and its wavelength.
The energy of a single photon is given by the equation:
E = hf
where E is the energy of the photon, h is Planck's constant (approximately 6.626 x 10^-34 J s), and f is the frequency of the radiation.
Given the wavelength of the microwave radiation (12.4 cm), we can calculate the frequency using the equation:
c = λf
where c is the speed of light (approximately 3.00 x 10^8 m/s).
Converting the wavelength to meters:
12.4 cm = 0.124 m
Rearranging the equation to solve for frequency:
f = c / λ = (3.00 x 10^8 m/s) / (0.124 m) = 2.42 x 10^9 Hz
Now that we have the frequency, we can calculate the energy of a single photon using the equation E = hf, where h is Planck's constant:
E = (6.626 x 10^-34 J s) x (2.42 x 10^9 Hz) = 1.60 x 10^-24 J
Next, we need to calculate the energy required to heat the coffee from 25.0°C to 62.0°C. The formula for calculating the energy needed to heat a substance is given by:
q = mcΔT
where q is the energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
We know the density of the coffee is 0.997 g/mL, and the volume is 255 mL. Therefore, the mass of the coffee is:
m = density x volume = (0.997 g/mL) x (255 mL) = 254 g
Using the specific heat capacity of water for the coffee, which is 4.184 J/(g/K), we can calculate the energy required:
q = (254 g) x (4.184 J/gK) x (62.0°C - 25.0°C) = 34,442.8 J
Finally, we can calculate the number of photons required by dividing the energy needed to heat the coffee by the energy of a single photon:
Number of photons = (34,442.8 J) / (1.60 x 10^-24 J) ≈ 2.15 x 10^28 photons
Therefore, approximately 2.15 x 10^28 photons are required to heat 255 mL of coffee from 25.0°C to 62.0°C using microwave radiation with a wavelength of 12.4 cm.