At noon on a clear day, sunlight reaches the earth\'s surface at Madison, Wisconsin, with an average power of approximately 4.00 kJ·s–1·m–2. If the sunlight consists of photons with an average wavelength of 510.0 nm, how many photons strike a 5.10 cm2 area per second?

9.13x10^26

Oh, we're getting scientific, huh? Well, let me put on my lab coat and goggles. *Ahem* Okay, let's work this out.

First off, we need to calculate the energy per photon using the formula E = hc/λ, where E is the energy, 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 (510.0 nm = 5.10 x 10^-7 m).

So, plugging in the numbers (and trying not to trip over any equations), we get:
E = (6.626 x 10^-34 J·s)(3.00 x 10^8 m/s) / (5.10 x 10^-7 m)
E ≈ 3.88 x 10^-19 J

Now, let's find out how many photons hit that 5.10 cm^2 area per second. We'll divide the average power by the energy per photon.

Number of photons = (Power) / (Energy per photon)
Number of photons ≈ (4.00 kJ·s^-1·m^-2) / (3.88 x 10^-19 J)

Now, we just need to do a bit of unit conversion. Since 1 m^2 = 10,000 cm^2, we'll divide by that:

Number of photons ≈ (4.00 x 10^3 J·s^-1·m^-2) / (3.88 x 10^-19 J) / 10,000 cm^2
Number of photons ≈ 1.03 x 10^21 photons

Voila! Approximately 1.03 x 10^21 photons strike that 5.10 cm^2 area per second in Madison, Wisconsin. That's a whole lot of light, huh?

To find out the number of photons that strike a given area per second, we can use the formula:

Number of photons = Power / Energy per photon

To calculate the power in this case, we can multiply the power per unit area by the surface area:

Power = Power per unit area × Surface area

Given:
Power per unit area = 4.00 kJ·s^(-1)·m^(-2)
Surface area = 5.10 cm² = 5.10 × 10^(-4) m²

First, let's convert the power per unit area to joules:

4.00 kJ·s^(-1)·m^(-2) = 4.00 × 10^3 J·s^(-1)·m^(-2)

Now, multiply the power per unit area by the surface area:

Power = (4.00 × 10^3 J·s^(-1)·m^(-2)) × (5.10 × 10^(-4) m²)
Power = 2.04 × 10^(-3) J·s^(-1)

Next, we need to calculate the energy per photon using the wavelength:

Energy per photon = Planck's constant × Speed of light / Wavelength

Given:
Wavelength = 510.0 nm = 510.0 × 10^(-9) m
Planck's constant = 6.62607015 × 10^(-34) J·s
Speed of light = 2.998 × 10^(8) m·s^(-1)

Now, substitute the values into the formula:

Energy per photon = (6.62607015 × 10^(-34) J·s) × (2.998 × 10^(8) m·s^(-1)) / (510.0 × 10^(-9) m)
Energy per photon ≈ 3.92 × 10^(-19) J

Finally, divide the power by the energy per photon to find the number of photons:

Number of photons = Power / Energy per photon
Number of photons = (2.04 × 10^(-3) J·s^(-1)) / (3.92 × 10^(-19) J)

Calculating this division, we get:

Number of photons ≈ 5.2 × 10^(15) photons

Therefore, approximately 5.2 × 10^(15) photons will strike a 5.10 cm² area per second.

E= hc/wavelength

E = 6.626E-34J*3E8 m/s/510E-9m
?J/photon.

?J/photon x #photons = 4000J/m^2 x 5.10 cm^2 x (100 m/cm) x (100 m/cm)

Substitute for ?J/photon and solve for # photons.
The number on the right changes kJ to J and cm^2 to m^2.
Post your work if you get stuck or have questions.