At noon on a clear day, sunlight reaches the earth\'s surface at Madison, Wisconsin, with an average power of approximately 1.00 kJ·s–1·m–2. If the sunlight consists of photons with an average wavelength of 510.0 nm, how many photons strike a 2.60 cm2 area per second?
To find out how many photons strike the given area per second, we need to calculate the number of photons using the given power of sunlight.
1. Begin by calculating the energy (E) of a single photon using Planck's equation:
E = hc/λ,
where h is Planck's constant (6.626 × 10^(-34) J·s), c is the speed of light (3.0 × 10^8 m/s), and λ is the average wavelength of the photons (510.0 nm).
Convert the wavelength to meters:
λ = 510.0 nm = 510.0 × 10^(-9) m.
Calculate the energy of a single photon:
E = (6.626 × 10^(-34) J·s × 3.0 × 10^8 m/s) / (510.0 × 10^(-9) m).
E ≈ 3.892 × 10^(-19) J.
2. Next, determine the power per photon, which is the given average power of sunlight per unit area:
Power per photon (P) = 1.00 kJ·s^(-1)·m^(-2).
Convert kJ to J by multiplying by 1000:
P = 1.00 kJ·s^(-1)·m^(-2) × 1000 J/kJ.
P = 1.00 × 10^3 J·s^(-1)·m^(-2).
3. Now calculate the number of photons striking the given area per second:
Number of photons (N) = Power per unit area / Energy of a single photon.
N = (1.00 × 10^3 J·s^(-1)·m^(-2)) / (3.892 × 10^(-19) J).
N ≈ 2.57 × 10^21 photons.
4. Finally, determine the number of photons striking the given area per second:
Number of photons per second = Number of photons / Area.
Area = 2.60 cm^2.
Convert cm^2 to m^2 by multiplying by (1 m / 100 cm)^2:
Area = 2.60 cm^2 × (1 m / 100 cm)^2.
Area = 2.60 × 10^(-4) m^2.
Number of photons striking the area per second:
Number of photons per second = (2.57 × 10^21 photons) / (2.60 × 10^(-4) m^2).
Number of photons per second ≈ 9.89 × 10^24 photons.
Therefore, approximately 9.89 × 10^24 photons strike a 2.60 cm^2 area per second under the given conditions.