How many moles of photons with a wavelength of 660 nm are needed to provide 6 kJ of energy?

Answer 1

The energy carried by a photon is given by

E = hf

Where h is Planck's constant (6.626x10^-34 Joule-seconds) and f is the frequency of the photon in Hertz (Hz).

We are given the wavelength of the photon in the question in nanometers (nm). First, we need to convert this to (SI) units, because our equations only work with SI units. Then, we will calculate the frequency of the photon from its wavelength. Once we know the frequency of the photon we're interested in, we simply use the equation above to find the energy carried by one of them. Then we divide 6 kJ by that amount of energy, and the quotient will be the number of photons needed to carry 6 kJ. Finally, when we know the number of photons we need, we divide by the number of photons in a mole to get the number of moles.

The SI unit of length is the meter (m). 1nanometer (nm) is 10^-9 meters.

660 nm = 660 *10^-9 m = 6.6*10^-7 m.

Now we will calculate this photon's frequency from its wavelength. These are related by the equation

c = fL

where c is the speed of light (3*10^8 m/s), f is the frequency of the photon and L is the wavelength of the photon.

c = fL

(3*10^8 m/s) = f * (6.6*10^-7 m)

solving for f, we have

f = (3*10^8 m/s) / (6.6*10^-7 m) = 4.54*10^15 s^-1

Note that the unit of seconds (s) raised to the -1power is defined as 1 Hertz (Hz).

f = 4.54*10^15 Hz

Now we will use the top equation to solve for the energy carried by one photon having this frequency.

E = hf

E = (6.626*10^-34 Js) * (4.54*10^15 Hz)

E = 1.369*10^-17 J

This is how much energy is carried by one photon of wavelength 660 nm (which will also have a frequency of 4.54*10^15 Hz).

How many of these do we need to provide 6 kJ? This is solved by simple division. Keeping in mind that 1 kJ = 1000 J, we have

Number of photons * Energy per photon = 6 kJ

Number of photons * (1.369*10^-17 J/photon) = 6 kJ

Number of photons * (1.369*10^-17 J/photon) = 6000 J

Number of photons = 6000 J / (1.369*10^-17 J/photon)

Number of photons = 4.382*10^20 photons

This is how many photons (at this frequency) are needed to provide 6 kJ. How many moles of photons is this?

Number of photons / number of photons in a mole = number of moles

Recall that a mole of something is defined as 6.02*10^23of it. The same way a dozen eggs is defined as 12 eggs, a mole of eggs is 6.02*10^23 eggs. Equivalently, a mole of photons is 6.02*10^23 photons. So

Number of photons / (6.02*10^23 photons per mole) = number of moles

(4.382*10^20 photons) / (6.02*10^23 photons per mole) = number of moles

7.279*10^-4 moles = number of moles

Forgive me if my arithmetic is off, as I don't have a good calculator handy. However, I believe this is the correct method to use.