Question

A 100-watt light bulb radiates energy at a rate of 100 J/s. (The watt, a unit of power or energy over time, is defined as 1 J/s J/s.) If all of the light emitted has a wavelength of 525nm 525nm, how many photons are emitted per second? Express your answer to three significant figures.

Answer 1

To find the number of photons emitted per second by a 100-watt light bulb, calculate the energy of one photon using its wavelength and then divide the bulb's power by the energy per photon. E = (6.626 x 10-34 J·s) (3.00 x 108 m/s) / (525 x 10-9 m)

Step-by-step explanation:

Calculating the Number of Photons Emitted

To figure out how many photons are emitted by a 100-watt light bulb per second, we first need to calculate the energy of a single photon using the wavelength provided. The energy (E) of a photon can be found using the formula:

E = (hc) / λ

Where h is Planck’s constant (6.626 x 10-34 J·s), c is the speed of light (3.00 x 108 m/s), and λ (lambda) is the wavelength of the light (525 nm for this question).

First, convert the wavelength from nanometers to meters (525 nm = 525 x 10-9 m). Then, using the energy equation:

E = (6.626 x 10-34 J·s) (3.00 x 108 m/s) / (525 x 10-9 m)

Calculate the energy per photon and then use the given power of the bulb (100 J/s) to determine the total number of photons emitted per second by dividing the power by the energy per photon. Remember, the bulb emits 100 J/s, and each photon carries a discrete amount of energy, so by dividing the total energy per second by the energy per photon, you get the number of photons emitted per second.

Answer 2

Approximately
`2.64* 10^(20)` (assumption: wavelength
`525\; \rm nm` is measured in vacuum, where the speed of light is approximately
`3.0* 10^(8)\; \rm m \cdot s^(-1)`.)

Step-by-step explanation:

Convert the unit of wavelength to meters:

`\displaystyle \lambda = 525\; \rm nm = 525 \; \rm nm* (10^(-9)\; \rm m)/(1\; \rm nm) = 5.25 * 10^(-7)\; \rm m`.

Assume that the wavelength
`525\; \rm nm` is measured in vacuum, where the speed of light is approximately
`2.99792* 10^(8)\; \rm m \cdot s^(-1)`. Calculate the frequency of this light from its wavelength:

`\displaystyle f = (c)/(\lambda) \approx (2.99792* 10^(8)\; \rm m \cdot s^(-1))/(5.25 * 10^(-7)\; \rm m) \approx 5.71429* 10^(14)\; \rm s^(-1)`.

The Planck's Constant can help find the energy of a photon given its frequency. Look up this constant to more than three significant figures:

`h \approx 6.62607* 10^(-34)\; \rm J \cdot s^(-1)`.

Calculate the energy of one such photon:

`\begin{aligned} E &= h \cdot f\\ &\approx 6.62607* 10^(-34)\; \rm J \cdot s^(-1) * 5.71023* 10^(14)\; \rm s \\ &\approx 3.78370* 10^(-19)\; \rm J \end{aligned}`.

Calculate the number of these photons that
`100\; \rm J` of energy can produce under the assumption of
`100\%` conversion:

`\displaystyle (100\; \rm J)/(3.78370* 10^(-19)\; \rm J) \approx 2.64* 10^(20)`.

(Rounded to three significant figures.)