Which has higher energy, infrared radiation with λ=1.0 ×10^-6 m or an X ray with λ=3.0 ×10^-9 m

step 1 There are two sets of two radiations, I guess we would say. So first we're asked, which has a hi

Which has higher energy, infrared radiation with λ=1.0...

Key Concepts

Frequency-Wavelength Relationship

The relationship between frequency and wavelength is given by the equation c = ?v, where c is the speed of light, ? is the wavelength, and v is the frequency. This inverse relationship implies that for electromagnetic radiation traveling at the speed of light, a shorter wavelength corresponds to a higher frequency and, according to the photon energy formula, greater energy.

Photon Energy

Photon energy is determined by the equation E = hv, where E is the energy of the photon, h is Planck's constant, and v is the frequency of the electromagnetic radiation. This concept is fundamental in quantum physics as it shows that the energy of a photon is directly proportional to its frequency, meaning that radiation with a higher frequency carries more energy.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all forms of electromagnetic radiation, ranging from radio waves to gamma rays. Different regions of the spectrum, such as infrared and X-rays, have distinct wavelengths and frequencies. Understanding where a particular type of radiation falls on the spectrum helps in determining its energy, behavior in different media, and potential applications, such as imaging techniques or communication technologies.

Transcript

00:01 There are two sets of two radiations, i guess we would say.

00:10 So first we're asked, which has a higher energy, infrared radiation with a wavelength of 1 .0 times 10 to the negative 6 meters or an x -ray with a wavelength of 3 .0 times 10 to the negative 9 meters.

00:27 And so just intuitively, we should know that x -ray radiation, so radiation in the x -ray portion of the spectrum is going to be higher energy than radiation in the infrared spectrum.

00:45 We could also just look at these two wavelength values.

00:53 And so anything times 10 to the negative 9 meters is going to have a smaller wavelength than light with a wavelength given as 10 to the negative 6 meters.

01:09 And so since frequency more directly corresponds with energy and as wavelength decreases, frequency increases, we would expect the x -ray radiation to have the higher energy.

01:28 But to actually show this, we need to do some math.

01:32 And so i've written out at the top of the screen here, what we're going to need.

01:36 So this is first the equation that relates energy to wavelength and frequency.

01:45 So e equals hv, h being planck's constant, which i've also written out, 6 .62 times 10 to negative 34.

01:54 Joule seconds.

01:57 And we can also express this equation as energy equals hc over wavelength, over lambda.

02:06 And so that would be planks constant times the speed of light, which is the third thing that i've written out, 3 .0 times 10 to the 8 meters per second.

02:16 And so that is what we're going to use to solve this first part.

02:20 So we'll say e equals hc over lambda.

02:29 And then the pieces of information that we're given for the first part are the two different lambdas.

02:35 And so we'll have planks constant times the speed of light.

03:08 And so first we'll put this over the wavelength that we're given for the infrared radiation, so 1 .0 times 10 to the negative 6 meters.

03:24 And when we do this math, we end up with 2 .0 times 10 to the negative 19 joules.

03:36 Okay, so that's the energy of the infrared radiation here from the first part.

03:44 We'll just do the exact same thing over again for the x -ray radiation.

04:05 So, of course, this time, the value for lambda is going to be 3 .0 times 10 to the negative 9 meters.

04:25 And when we do this math, we end up with 6 .6 times 10 to the negative 17 joules.

04:38 Okay? and so the x -ray radiation is going to be higher energy.

04:44 And we knew that intuitively, right? as i talked about at the beginning of this problem...

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