At the atomic-scale, the electron volt and nanometer are well-suited units for energy and distan

At the atomic-scale, the electron volt and nanometer are...

Key Concepts

Photon Energy Formula

This concept involves the fundamental relation E = hc/?, which shows that the energy of a photon is inversely proportional to its wavelength. It is a core principle in quantum mechanics and electromagnetic theory, illustrating how light’s quantized energy depends on its wavelength.

Planck's Constant

Planck's constant (h) is a fundamental quantity in quantum mechanics that sets the scale for the sizes of energy quanta. In the photon energy formula, it serves to connect the energy of electromagnetic radiation with its frequency, encapsulating the idea of quantization of energy.

Speed of Light

The speed of light (c) is a universal constant that determines the speed at which electromagnetic waves propagate through a vacuum. It is crucial in the photon energy formula because it links the frequency and wavelength of light, thus directly influencing the energy of a photon.

Unit Conversion in Atomic-Scale Physics

In atomic-scale calculations, units such as electron volts (eV) for energy and nanometers (nm) for distance are used to simplify computations and provide more intuitive scales for atomic and molecular phenomena. Converting fundamental constants into these units, as seen in the derivation of the 1240 eV·nm constant, allows for direct and convenient applications in quantum physics problems.

Transcript

00:05 Okay, so in this problem, we want to prove that the energy of a photon can be written as 1 ,240 electron volts times nanometers divided by the wavelength of the photon, which i'm just going to write as w, which is in the units nanometers.

00:24 So first things first, we want to start with the equation that we might be most familiar with, which is e equals hc over wavelength.

00:33 In this instance, h is planx constant, which is equal to 6 .62 times 10 to the power of, which i'm writing as e, 10 to the power of negative 34.

00:50 And the units are meters squared times kilograms divided by seconds.

00:58 These units are very, very important to remember, especially for a problem like this, because this is all about units.

01:04 And c is the speed of light, which is 3 times 10 to the power of 8 meters per second.

01:13 And then the wavelength is a wavelength that we don't know yet for this first part of the problem, so we're not going to worry about that yet.

01:19 So what we want to do is we want to understand how we can convert the units that we have into the units that we want.

01:26 So we end up wanting things in electron volts and nanometers.

01:31 We start off with meters, kilograms, and seconds.

01:36 So when we do e equals hc over w, we are going to have the following units.

01:43 E equals.

01:44 So if we multiply the units for planck's constant and the speed of light, we end up getting meters cubed times kilograms divided by seconds squared.

01:57 And all of this is going to be divided by the wavelength, which is in nanometers and we're not going to touch this okay we're not going to worry about the denominator we're only going to worry about the the units in the numerator because the units in the denominator are already where we want them right so we don't need to do anything to the to the sorry the denominator is already where we want it we don't need to make any changes to it okay so we have e equals meters cubed times kilograms divided by second squared.

02:40 So we can rewrite this slowly to get electron volts.

02:48 So what we want to start off with is we can rewrite this as e equals, e, which is in joules.

02:56 That's generally the unit of energy that we want.

03:01 We have meters times meters squared.

03:03 So i'm just rewriting that meters cubed as meters times meters squared times meters squared times kilograms.

03:10 Divided by seconds squared.

03:13 So we know that meters per second squared is acceleration, right? meters per second squared is acceleration.

03:29 And we know that kilograms is mass.

03:32 So if we do mass times acceleration, and again, this is just understanding units.

03:38 We don't actually, we're not actually trying to find an acceleration or a mass.

03:42 It's literally just units.

03:45 So if we do mass times acceleration, we get force, which is in the units of newton's or n, capital n.

03:56 Right? so if we go to a new page, then we know that force is equal to neutens.

04:12 So we can rewrite this e is in joules, is equal to newton's times meters.

04:31 Times meters times meters, right? so we have meters square.

04:36 And then divided by the wavelength, which is in nanometers, which we're not going to touch.

04:42 So we have newton's times meters times meters.

04:45 So if you remember, a newton meter can be rewritten like this.

04:53 So if we have the equation, work equals force times distance.

04:56 Again, we're not trying to find the work.

04:57 We're not finding force.

04:58 We're not finding distance.

04:59 This is just about units.

05:01 So we know that work is in joules and force is in neutons and distance is in meters.

05:08 So one jewel is equal to one newton meter.

05:13 So we can replace this newton meter with jewels.

05:18 So we end up getting e, which is in joules, is equal to a newton meter, which is a jewel, times meters divided by the wavelength.

05:30 All right? so now what we want to do is we want to convert joules to electron volts.

05:36 We understand how to do that.

05:38 That's much simpler...

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