What frequency of light would be required to excite an electro...

Question

What frequency of light would be required to excite an electron if the HOMO–LUMO energy gap was 33.9 kcal/mol (142 kJ/mol)?

Accepted Answer

Hi, everyone. Let's look at our next problem. It says the value of a Homo luo energy gap is 53.8 k cows per mole or kilojoules per mole to excite an electron across this energy gap. What frequency of light would be required? Choice? A 3.39 times 10 to the 35th Hertz B 3.39 times 10 to the 38th Hertz C 5.64 times 10 to the 11th Hertz or D 5.64 times 10 to the 14th Hertz. So I'm given the amount of energy that a mole of electrons needs to jump the gap to be excited from the highest occupied molecular orbital to the lowest unoccupied molecular orbital. And I'm being asked what frequency of light is required to cause one electron to be excited by this amount. So I need to relate the energy of light that's needed with the frequency of light. We recall that um light is emitted in photons. So I need the frequency of the photon that will be needed to have enough energy to accomplish this task. So what equation do I have that relates the energy of a photon with its frequency. And that's Plank's equation which says that or Plank's law, excuse me, which says that the energy of a photon is equal to planks constant H times new or the frequency. That's that Greek letter new that looks like A V is often written as a V. Now, I'm looking for the frequency so I can rearrange that equation to solve for frequency. And then we'll get that frequency equals the energy divided by Plank's constant. And I'll just note right now as we're going forward, that Plank's constant is 6.6 to 6 times 10 to the negative 34 Jules seconds. I'd like to do that because I know I'm going to use it. And that helps me keep track of what units I'm going to want. And when I solve an equation like this, I know that if I'm solving for frequency, you know, frequency is hurt or uh one over seconds or seconds to the minus one. So that's what I'm aiming for. Everything to cancel out except for seconds on the bottom of my fraction. So I need to find out what ener the energy is of that photon. So the energy of the gap I'll say, and I have this uh energy in two different units worth, I have a K cows per mole and kilojoules per mole. But I know I have this planks constant in jewels. So the more useful units will be kilojoules per mole. So this would be the energy that you need to excite a mole of electrons over this gap. But I just want to excite one electron. So we're going to need to factor that into our equation. So the energy of the photon that we're going to be looking for, I need to convert it just to add the energy for a single electron. And then I want to go from kilojoules to Jews so that I'll be in the shape to cancel all my units out. So I'm gonna set up my conversion factors here. So I'll start with 225 kilojoules in the numerator and then moles in the denominator multiply that by my conversion factor kilos to jewels. So that will be 1000 jewels on the numerator of the next expression and one kilojoule in the denominator. And then finally, I'm going to want to go from a mole of electrons to a single electron. So what's my conversion from a mole to a single entity? And that's Avogadro's number. I have moles in the denominator as I'm multiplying multiplying out and converting things. So I want to put, want to put one mole in the numerator of the next expression and then Avogadro's number in the denominator six point oh times 10 to the 23rd. So to convert it all out, I have again, my terms that are being multiplied 225 kilojoules per mole multiplied by 1000 Jews per one kilojoule multiplied by one mole per six point oh times 10 to the 23rd. So that will give me my energy in useful units that I can use. And that calculates out to 3.74 times 10 to the negative 19th jules. So this would be the energy needed to excite one electron across that Homo Limo gap. Now, I just need to find out what frequency of light I'll need to make a photon of that amount of energy. So again, frequency will equal this energy 3.74 times 10 to the negative 19th Jules. And we're going to divide that by planks constant. So 6.6 to 6 times 10 to the negative 34th jewel seconds, that's when equal. And I see that the jewels are going to cancel out here and my value will be my number and numerical answer will be 5. times 10 to the 14th. And I'm left with units of seconds to the minus one. Just have seconds in the denominator. That's the last thing I'm left with and that will equal, I'm going to erase seconds to the minus one and that will equal the unit of Hertz, which is how my answers are given. So 5.64 times 10 to the 14th Hertz. So that is my calculated frequency for my photon of light to excite this electron across the Homo lima energy gap. And indeed, I see that choice D is 5.64 times 10 of the 14th Hertz. See you in the next video.

What frequency of light would be required to excite an electron if the HOMO–LUMO energy gap was 33.9 kcal/mol (142 kJ/mol)?

Key Concepts

HOMO-LUMO Gap

Energy and Frequency Relationship

Conversion of Energy Units

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