Warm objects emit electromagnetic radiation in the infra-red region. Heat lamps employ this principle to generate infra-red radiation. Water absorbs infra-red radiation with wavelengths near 2.80um. Suppose this radiation is absorbed by the water and converted to heat. A 1.00 L sample of water absorbs infra-red radiation, and its temperature increases from 20C to 30C. How many photons of this radiation are used to heat the water?
q=heat absorbed by water = mass x specific heat water x delta T. Assuming the density of water is 1.00 g/L, the mass of 1.0 L is 1,000 grams.
q = 1000 x 4.184 J/g*K x 10.
delta E = hc/wavelength.
Plug in h, c, and wavelength to calculate delta E. That is the energy per photon. Then set up a proportion to determine the number of photons required to reach q.
Post your work if you get stuck.
Thanks for the speedy response!
Here is my work:
q = 1000 x 4.184 x 10
q = 41840
detalE = hc/wavelenght
wavelenght = 2.8 x 10^-6 m
deltaE = [(6.626 x 10^-34)(3.0 x 10^8)]/2.8 x 10^-6
delta E = 7.09 x 10^-20 m per photon
Proportion:
1 photon = 7.099 x 10^-20m
q = 41840
Therefore, 5.894 x 10^-16
Is this correct? Thanks again for you help!
To determine the number of photons of infrared radiation used to heat the water, we can use the formula:
n = E / (hf)
Where:
n is the number of photons
E is the energy absorbed by the water
h is Planck's constant (6.626 x 10^-34 J.s)
f is the frequency of the radiation
First, we need to calculate the energy absorbed by the water:
E = mcΔT
Where:
m is the mass of the water
c is the specific heat capacity of water (4.18 J/g°C)
ΔT is the change in temperature
Given that the volume of water is 1.00 L, and the density of water is approximately 1 g/mL, we can assume the mass of the water is 1.00 kg (1000 g).
ΔT = T final - T initial = 30°C - 20°C = 10°C
Now we can calculate the energy absorbed:
E = (1000 g) * (4.18 J/g°C) * (10°C) = 41800 J
Next, we need to determine the frequency of the radiation using the speed of light formula:
c = λf
Where:
c is the speed of light (3.00 x 10^8 m/s)
λ is the wavelength of the radiation
Convert the given wavelength of 2.8 μm to meters:
λ = 2.8 μm = 2.8 x 10^-6 m
Solve for the frequency:
f = c / λ = (3.00 x 10^8 m/s) / (2.8 x 10^-6 m) = 1.07 x 10^14 Hz
Finally, we can calculate the number of photons:
n = E / (hf) = (41800 J) / [(6.626 x 10^-34 J.s) * (1.07 x 10^14 Hz)]
Calculating this equation will give us the number of photons of infrared radiation used to heat the water.
To determine the number of photons of infrared radiation that are used to heat the water, we need to consider the energy per photon and the total amount of energy required to raise the temperature of the water.
First, let's calculate the energy per photon using the formula:
E = hc/λ
Where:
E is the energy per photon,
h is Planck's constant (6.626 x 10^-34 J·s),
c is the speed of light (3.00 x 10^8 m/s), and
λ is the wavelength of the infrared radiation (2.80 μm = 2.80 x 10^-6 m).
Plugging in the values, we get:
E = (6.626 x 10^-34 J·s) * (3.00 x 10^8 m/s) / (2.80 x 10^-6 m)
E ≈ 7.12 x 10^-20 J
Now, let's calculate the total energy required to raise the temperature of the water. We can use the specific heat capacity of water, which is approximately 4.18 J/g·°C.
The sample of water has a volume of 1.00 L, which is equivalent to 1000 grams (since 1 mL of water is roughly 1 gram). The temperature increase is from 20°C to 30°C, which is a change of 10°C.
The total energy required to raise the temperature of the water can be calculated using the formula:
E_total = mass * specific heat capacity * temperature change
Plugging in the values, we get:
E_total = (1000 g) * (4.18 J/g·°C) * (10°C)
E_total = 4.18 x 10^4 J
Now, to find the number of photons, we divide the total energy (E_total) by the energy per photon (E):
Number of photons = E_total / E
Number of photons ≈ (4.18 x 10^4 J) / (7.12 x 10^-20 J)
Number of photons ≈ 5.86 x 10^23 photons
Therefore, approximately 5.86 x 10^23 photons of infrared radiation are used to heat the 1.00 L sample of water from 20°C to 30°C.