Question

Question #d40ff

Answer 1

`1.72 * 10^(18)"photons"`

Explanation:

The first thing to do here is to figure out the energy of a single photon of wavelength equal to `"693 nm"` by using the Planck - Einstein relation, which looks like this

`color(blue)(bar(ul(|color(white)(a/a)color(black)(E = h * nu)color(white)(a/a)|)))`

Here

`E` - the energy of the photon
`h` - Planck's constant, equal to `6.626 * 10^(-34)"J s"`
`nu` - the frequency of the photon

As you can see, you need to use the wavelength of the wave to find its frequency. As you know, wavelength and frequency are inversely proportional to each other as described by the equation

`color(blue)(bar(ul(|color(white)(a/a)color(black)(lamda * nu = c)color(white)(a/a)|)))`

Here

`lamda` - the wavelength of the photon
`c` - the speed of light in a vacuum, usually given as `3 * 10^8"m s"^(-1)`

To find the frequency of the photon, rearrange the equation as

`nu = c/(lamda)`

Make sure to convert the wavelength from nanometers to meters when plugging it into the above equation

`nu = (3 * 10^8color(red)(cancel(color(black)("m"))) "s"^(-1))/(693 * 10^(-9)color(red)(cancel(color(black)("m")))) = 4.329 * 10^(14)"s"^(-1)`

Use this value to find the energy of a single photon

`E = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 4.329 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))`

`E = 2.868 * 10^(-19)"J"`

Now, you know that the laser produced a pulse of energy `"0.494 J"`, which means that at this frequency it produces

`0.494 color(red)(cancel(color(black)("J"))) * "1 photon"/(2.868 * 10^(-19)color(red)(cancel(color(black)("J")))) = color(green)(bar(ul(|color(white)(a/a)color(black)(1.72 * 10^(18)color(white)(.)"photons")color(white)(a/a)|)))`

The answer is rounded to three sig figs.