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A lithium atom has mass 1.17 $ imes$ 10$^-$$^2$$^6$ kg, and a hydrogen atom has mass 1.67 $ imes$ 10$^-$$^2$$^7$ kg. The equilibrium separation between the two nucl...

Question

A lithium atom has mass 1.17 $ imes$ 10$^-$$^2$$^6$ kg, and a hydrogen atom has mass 1.67 $ imes$ 10$^-$$^2$$^7$ kg. The equilibrium separation between the two nuclei in the LiH molecule is 0.159 nm. (a) What is the difference in energy between the $l$ = 3 and $l$ = 4 rotational levels? (b) What is the wavelength of the photon emitted in a transition from the $l$ = 4 to the $l$ = 3 level?

A lithium atom has mass 1.17 $\times$ 10$^-$$^2$$^6$ kg, and a hydrogen atom has mass 1.67 $\times$ 10$^-$$^2$$^7$ kg. The equilibrium separation between the two nuclei in the LiH molecule is 0.159 nm. (a) What is the difference in energy between the $l$ = 3 and $l$ = 4 rotational levels? (b) What is the wavelength of the photon emitted in a transition from the $l$ = 4 to the $l$ = 3 level?



Answers

A lithium atom has mass 1.17 $\times$ 10$^-$$^2$$^6$ kg, and a hydrogen atom has mass 1.67 $\times$ 10$^-$$^2$$^7$ kg. The equilibrium separation between the two nuclei in the LiH molecule is 0.159 nm. (a) What is the difference in energy between the $l$ = 3 and $l$ = 4 rotational levels? (b) What is the wavelength of the photon emitted in a transition from the $l$ = 4 to the $l$ = 3 level?

So is this question would have a hypothetical molecule, so it doesn't necessarily exist. We have nitrogen and hydrogen and when you try to bounce them together, so we want them to make a rotational. Never transition from our equals. 32 l equals one, and it gives off a photo of wavelengths 1.78 centimeter when transits. So it won't to work out what is a separation between the two. Adams is a small Tigger. What is separation? And we know that the mass of hydrogen and H is one point 67 times. Change is a negative. 27. Hugh Grant is the best off. Nitrogen is 2.33 times 10 to the negative 26 kilograms. So give us that. We would like to know it was the distance between them. So in order to do that, we have to again use our energy. For Miller, it's the energy for each level is times air plus one H square over two. So the energy transition from L equals 32 equals One meal to eat is where equals three. It's three times for three times for what L, because when it's what brought us to each square over too high. So this is equal to three times four is 12. 12. Minus two is 10 then. So it is five times each square over at so two castles is Ensign Steel. Toe e also equals two. Did you see over Linda? We can establish this equations. That agency over Lambda equals five times each square over I. So from this, we are here to find out what Iess. So sorry. I gives i e i equals five times. Uh, I it's over. See, that, uh, is five age named are over. See? Okay. And so if we calculate this Oh, is a book. It is more often that we use H bar instead of age, so we can also use five times. H bar is one. Quit over five times 10 is addictive. 34 Just second over to pipe times one point 70 Donald, Meter C is three times 10 to 8, you know, per second. So this is full point Night 81 times 10 to Z negative. 52 kilogram meter square is in. We know that I equals the reduced mess. Times are not square, so we can find the separation are not if we do are not square equals. Uh, I'm a square equals I over our m r s o are not equals I over And our square root is there's a reduced mass is equal to massive nitrogen times massive hydrogen over massive nitrogen plus massive hydrogen. Okay, so, uh, now if we start you this is one putting or everything. We will get four point 981 times 10 to the negative. 52 over. Well, put 67 times. 10 to the negative. 26 27 times two points. 33 times 10 to the negative. 26. Then we Here we have some plus together 1.67 10 to times 10 to the negative. 27 clause two points. There's three times. Chances are negative. 26. Okay. And if we can't charism, we will get that this equals five point 65 times 10 years. An active searching meter. So, uh, if you actually compare this to a normal, more curious e, that's the separation is much, much smaller than the diameter of a typical Adam. So, um, this is not very realistic. So this hypothetical moniker actually shouldn't exist

So in this question where Cubans is hydrogen Io night, what a cute and became a bunch of informations about it And is this question We want to calculate books. It's vibrational modes and its traditional boat. So in part, everyone to first start calculating its rotational boat home. We want to find the moment of inertia of it. So you want to defend the moment of inertia is the moment of inertia is keeping by reduced mass Times are square. So we have to first calculates the reduced mass off this molecule. So, um, the reduced mess in our equals massive hydrogen times, massive eye all night divided by a massive regimen plus massive tonight. So this is 1.67 times 10 to the next 27 kin o gram times, um, tonight is 2.11 times 10 to 0 negative. 25 kicker over 1.67 times to interdicted 27 plus 2.11 times 10 to the collective 25 kilo okra. And we see this gives us 1.657 times 10 to z factor of 27 Clower. And with that information, we can check It is a moment of pinochle, which is just a reduced mask. Times a separation. Which is there? A 0.1690 meter. This gives us 4.24 times 10 to the negative 47 kilogram meter script. So now let's record party. Party asked us to look at the energy involved in several different transitions. Um, so for number one, we have the transition from an equals one. Our week was one to an equal. Cyril, How equal time are too. We have an equals one l It was too to an equal Cerro ill equals one. So both rotational and vibrational energy under went, um, lower transition. Finally, we have n equals two hour equals two, two in equals fund. Yeah, he cause three. Okay, so this truck, it is the 1st 1 The 1st 1 iss a simpler case in this case, we only have, um uh, operational energy. Is that ghost on by one level. So the vibrational energy is, um, given by operational energy. E n is in plus one, Huh? Each bar k prime over in. Ah, here. We do not know what caped crime over M s, but this is K prime over m is just omega. So we should It might be more helpful to read it as a TSH borrow Mika and we do not know what Oh, make eyes but we know what frequency is in frequency. Which bar? Oh my God. So omega equals to pi times frequency and each bar equals h over to pipe. So we really it is easier to your rights. This equation iss b n equals in plus 1/2 age. So this gives us a vibrational energy off all those systems. But and now let's trick it. Number one, number one, we have some vibrational energy decrease by one level. So the change of detail to eat is, um 12 two It's he okay? Yeah. Sorry. A quick mistake. Number one, we actually both levels decrease by one esos among decrease pies. Vibrational energy. Duffy is ah, age bar, times f. So this is how much of a vibrational energy decreases is. Then let's go catch the rotational energy. Rotational energy is given by out. I'm still lost one inch bar square over to us and we already calculated I means the previous, uh, question in part a So now we can just use a result. So when decreases again, we also need to add all times l plus one which are over too high. When is that? Ellie Goes one. This number is equal to two attempts EJ far over to I when our equals zero e l equals zero. So this is a total amount of energy without changes. Is, um each bar over? It just grew large square over just I plus h f and friendly. We want to find the way plants associated with the photons. I was admitted in this process and the way plants is given by Linda equals. Did you see over Deputy? Okay, so let's get a new page so we can write what clearly l equals one in because 12 hour equals zero in equal. Cyril, we have change of rotational energy is H half change. Um, change a vibrational energy is Jeff. Change of rotational energy is H bar square over I. So the total change of energy is an added together ends and weapons fault is HC over Delta E equals H C over inch bar square over I plus h f. So this is an equation for is this question. Um And if we if we're given each for an h A constant, um, I we found the previous question and it By putting those numbers, we will get weightless off 4.3 sterile microbe meter. So this is ah, first question. The second question we have and equals one. Yeah. Equals to two and equal zero. Jail equals one. So in this case, most things are the same. We still have an equals one to n equals zero. So the vibrational energy is the same. It's still itch if, but the rotational injury is different in l equals two. We have two times three times each bar square over to I. Well, l equals one. We have one times two each bar square over to us. So if we subtract them, we'll see that now a different a difference in rotational energy is to be two times each far over instead of just each march over. So the wave lands will be in a bit smaller. It will be 4.28 micrometer. So, finally, four parts three. Now we have, um, vibrational energy goes down by one, but rotational energy increase too increased by one. So now is named, uh, did you see over Delta E rotational energy from 2 to 3. So it was actually, um, it was actually a negative value. So when l equals two, we have two times three equals six times. Which bar? Square over to at and when it's three, we have 12 times each bar over to I. So we have negative three each bar square over to I. Oh, really? And then, well, it's a change of rotational off aberration. Energy is doing, Chef. So this one when we calculated, um, it is a very small energy difference, actually because, say castle each other and the final witnesses, 4.4 micro meter.

In a set of experiments on a hypothetical one electron Adam, you measure the wavelengths of the photons emitted from transitions ending in the ground level, as shown by these arrows. In this figure, you can also observe that it takes 17.50 Let Sean volts toe ionized this, Adam. The first thing we want to know is what is the energy of the atom in each of the energy in each of the levels shown in the figure that is any is equal to one all the way through and is equal to five. And then we want to find out that if an electron makes the transition from the N equals for energy level to the N equals two energy level, as shown by this blue line, what wavelength of light it would admit it. So we know in this transition from that this higher level down to this lower level that is going to admit I'm a photon of some wave like, and we just want to find out what that wave lights going to be. So we want to start with part A, and we want to calculate the energy of the Adam in each of these levels because then part B, if we can just find the difference between the energy levels, the energy difference between these energy levels, we can then use planks relation to find what that wavelengths should be. So to find these energy levels, let's first look back to the hydrogen atom. Since this may give us an idea as Teo, what the energy levels of our hypothetical Adam are going to be so the energy levels of the hydrogen atom is given by negative 13.60 electron volts divided by and squared where Anne is a positive integer greater or equal to what So some important information were given about a hypothetical Adam. That might help is this ionization energy of 17.50 like John Bulls. So we're called that ionization Asian corresponds to a transition from the ground state and is equal to one to a state where the electron is in an infinitely large orbit radius, which is labelled by an is equal to infinity and then the ionization energy. So this is the energy that needs to be put into the atom energy we need to give to the atom in orderto strip you. Let's run away from it is given by Infinity Highness are sorry infinity minus the one. So just by looking at this relationship with this equation for the energy levels of the hydrogen atom, we see that as end goes to infinity, then one ends one over and squared goes to zero, so infinity must be zero. So our ionization energy, then zero minus e one. And again, we can just plug in this relationship into this equation for anyone. And we don't want to forget this important minus sign. We get negative Negative 13.60 A trans bolts divided by one squared, which is just equal to positive 13.60 electron volts. So the ionization energy of our hydrogen atom ene, is just 13.60 They're chan volts, which has already appears in our equation. So if we wanted, we can think about just substituting this and and we wantto just let e I equals 13 point B 13.60 electron volts. Then we can rewrite the energy levels for a hydrogen atom as just negative ionization energy over and squared. And now we can move on to our hydrogen like Adam, so h like Adam. And we guess that the energy levels of this Adam are going to be negative. The ionization energy over and squared, which is just negative. 17.50 electron volts over and squared. Cool. So we can use this relationship this equation to go ahead and just calculate Hee Won, which is negative 17.50 electron volts over one squared, which is just negative 17.50 Like crumbles, Andy, too is equal to negative. 17.50 electron volts divided by two squared, which is just, uh, just 17 negative for point three 75 electron volts. Awesome. And I'll leave the rest of these equation at these energy levels e 334 and five for you to calculate in a similar way. So now we have our the energy levels we wanted. E n is equal to negative 17 0.50 electron volts divided by n squared. We can actually check to make sure that this equation is true since we're given these wavelengths for these transitions. So let's go ahead and check that. So this were given. Let's focus on this. Ah, equals to tow n equals one transition, and we're told that the wave like that submitted for this transition is 94.5 for nano meters, so we can actually try to calculate the wavelength by ourselves using this equation and make sure that we get this correct 94.5 for nanometers. So to do that, we recall that when we we want to look at the emission of a photon when we go from and is equal to two towe n is equal to one. So let's first find the energy of that vote on the energy of the photon. When we goto and eagles to toe and equals, one is equal to minus a final minus initial. Now you might be wondering why this minus sinus here in the last part, when we were looking at going from a lower energy to a higher energy, this was a positive because we were adding energy into the system in order to make that transition happen. But now we're actually removing energy from the system to go from a higher energy to a lower energy. So the electron is the photon is being admitted out of the Madam so it's being taken away from the atom. So we have this minus sign so we can also even rewrite this just as the initial minus the final, which is e to minus hee won. So we have the energy of our photon is negative 17.50 electron volts. And I'm gonna just factor that out since it's common to both e to anyone. One over two squared minus one over one squared, which is going to be so we have negative 17.5 times, one over two squared, minus one over one squared. And that is 13.125 electron volts. So we have the energy now of our There we go. So we have our energy of this transition from an eagles to tow n equals one. So the energy of the photo on that submitted is 13 0.125 electron volts. But now how do we compare this to the wave like that? Were given, which is a wavelength. But we just have energy. We recall that the energy of a photon is given by H times. See, so that's plaints. Constant times the speed of light divided by its way of light. So solving for its wavelength we have Landa is equal to h time See over energy Now I want o point something out when If you wanted to look up the equation for the value for H and look up the value for sea on DH, just plug these numbers into a calculator. You'll actually get the wrong answer here because we have our energy in electron volts and planks. Constant has energy in jewels. So Teo, get around that It's actually pretty easy to Trent to go from electron volts to Jules and that is on election vault. So one electron volt, we just We can get that to Jules by just multiplying the electric charge, which is, uh, so the electron charge, which is one 0.6 here, I'll rewrite this somewhere where we have more room. So one electron charge or one electron like jumbo TV is the electron charge 1.60 two times 10 to the negative 19 times. Uh, cool arms times one vote. And we noticed that the combination of the unit's Coolum, which is electric charge and volts, was elect, which is electric potential, actually gives us Jules So we have 1.602 times 10 to the negative. 19. Jules. So now we use this value of energy. So we have e is equal to 13 0.1225 times 1.602 times 10 to the negative. 19. Jules. Um and we can just type that into a calculator. 13.125 times 1.60 two time sensitive. You're 19 is equal to two point one. 03 times 10 two times 103 times 10 to the minus 18. Jules. And now we can plug this in to this equation here. Since now we have this in jewels and get our get our answer. So let's rewrite this. We have Landa is equal to H C over E and we have e is equal to 2.103 times 10 to the negative 18 Jules and we know that H is six point 6 to 6 times 10 from negative 34 Jule seconds and C is equal to 2.998 times 10 to the eight meters for a second. So we just type this all into a calculator 6.626 times 10 to the negative, 34 times, 2.998 times Sense of eight. Divided by 2.103 times sent to the negative 18 and we get Landa equals nine point for 45 times 10 to the negative eight meters which is 94 point for five nano meters. So clearly there was some rounding errors in there somewhere. But that's close enough and that's close enough to confirm that we have the correct equation for our energy states. Awesome. So now we have just one more thing to do and that is we want to find out how, what the wavelength of light will be and we go from any single four down two n is equal to two. And this is actually going to be exactly the same calculation as we did from n equals 2 to 1. Um, just with some different values for n so going a little bit more quickly this time energy of our photon is equal to e four minus e too, which is negative. 17.50 electron volts times one over four squared minus one over two squared which gives us 3.281 electron volts, which, if we multiply just by the elementary charge to give us Jules, gives us five point two. I've six times 10 to the negative 19 Jules, and then we can easily find the way find as we did before with Lambda, is equal to a plant's constant times the speed of light divided by this energy in jewels, which gives US 6.626 times 10 to the negative. 34 times two point 998 time sense. Of the eight divided by 5.256 times 10 to the negative, 19 is three 0.7 79 times 10 to the negative seven meters, which gives us 300 77 0.9 nana meters and there we have it, and we can use the same equation to calculate any of these transitions.

All right. So in hydrogen, the energy associate id with the quantum number and is given by negative 13.6 TV over and squared. And in part a, we were asked how much energy is needed to, um, ionized an electron from the n equals force. They So when the ionizing electron weaken, basically think of it as being removed infinitely far from the hydrogen atom or practically infinite, in which case the energy of the electron is a zero. So basically the energy that we have to put in eyes just equal to the energy that the electron initially had, which for an an equals for quantum stay is given by 13.6 TV over four squared, and this is 0.85 BB. Now, if we have a transition between Orbital's two and four, then we need to take a difference of energies using the same expression. So the energy associate ID with the N equals to orbital is, um, 13.6 TV over one over two squared, and we take the difference with the N equals for orbital and the energy supplied, um, to the photon is given by the difference of these energies. Basically, the change in energy in the orbital's is compensated for by the mission or absorption of the photo, and this comes out to 2.55 If we want to know the wavelength, then we can set the sequel to each C over Lambda with H expressed in electron volts and solving for land gives 487 nano meters, I think.


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