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Suppose that ground-state electron (fictitious) single- electron atom has an average potential energy of -20.2 eV Find the average potential energy in joules the a...

Question

Suppose that ground-state electron (fictitious) single- electron atom has an average potential energy of -20.2 eV Find the average potential energy in joules the average potential between the electron and the nucleus in volts, and the average distance between the electron and the nucleus (you may assume that the nucleus has a charge of +e):

Suppose that ground-state electron (fictitious) single- electron atom has an average potential energy of -20.2 eV Find the average potential energy in joules the average potential between the electron and the nucleus in volts, and the average distance between the electron and the nucleus (you may assume that the nucleus has a charge of +e):



Answers

The mean or average value of the potential energy of the electron in a hydrogen atom can be found from $U_{\mathrm{av}}=\int_{0}^{\infty} U(r) P(r) d r .$ Find $U_{\mathrm{av}}$ in the $1 s$ state and compare with the potential energy computed with the Bohr model when $n=1$

Demeanour average value of the potential energy of the electron in a hydrogen atom can be found by the integral from zero infinity you of RP of RDR. Find the average potential energy in the one s. State and compare it With the potential energy computed with the Bohr model went and equals one. Okay, so This is equal to the integral from 0 to infinity of negative E squared Over four pi absalon on it for over a not cubed. Are it is the negative to our over a night D. R. So this is equal to negative E squared Over four Pi absalon. Not for over A not cute one over two. Over a not squared, which gives us negative E squared over four pi. Absolutely not a nine.

This question. We need to find the average potential and carry energies off the electron in the ground States off hydrogen. Okay, so solution. Okay, First, we recognize that, uh, the average kinetic energy, plus the average potential energy is going to be Ah, the ground state energy. Okay, which is equal to Ah, minus mhm. Uh huh. Que square over to a no. Right. And then, um, the potential energy in the hydrogen atom system is equal to minus k e squared over r. And then we also need the ground state way function. Sy 100 r Is this equal to ah one over roots for pie one over. They not to the tree house King let me check. Ah, for the exact with your functions. And then we have, uh to our to times e to the minus are over in on. Okay, so, uh, yeah, you can also see that That's Ah. Square it for over here so you can simplify to one of the root pie one over in order to the tree. Half e to the minus are over, you know. Okay, So the first we find our average potential energy. So this is equal to Ah, by 100 R. What square Ah, be off our yeah times, um, or pie are square. We are Okay, so we substitute border ah functions. Okay, so our months are more square of the wave function will be one of the pie and then one of rain are cute into the minus two are over in our and then you are is minus k e squared over R there we have four pi Our square We are. Okay, so the pie cancels one of the are cancels. Okay, then we just pull out the constants minus K square over in Cuba. And then you also have a four, and and the integral from zero to infinity R E to the minus two are over. A Not you are. Uh huh. Then you just, uh, integrate and you get, uh, you're not square or four. Okay, So the four councils, uh, you know, square councils. So you are left with minus okay. A square over, you know? Hey, then the So this is the average value. The average ah, potential energy. Okay, then, uh, you can find a average kinetic energy. This is equal to uh, promised the energy. My Nesta uh, pretend average potential energy. Okay, so the ground state energy is minus K E square over to a no. Uh, and then class que es square over. A note is equal to e squared over two NR. Okay, so this is the average potential energy. Canada energy. Sorry. Okay, so that's all.

Supplies. By the terms of the hydrogen. That means the stabilized with the nucleus of hydrogen, every Egyptian's of 5.2 and 2011. So we know that at the nucleus of hydrogen atom there's only one proton type, the distance of art that is given what we knew the electric for themselves I think. But potentially deadly sins other it in three D. It is ego that will kick you upon our rights. Okay. It's nothing about 9 to 10 to the power of nine into the stars. The 1000 proton anything about 1.619. And I found distance. The distance which we have managed that 5.2911 5 Point Tuna and returned to develop -11. So it comes out to be tend to devour of China. So it comes out to be 10. Right? It will be 144 lb 5 29. Right? Uh So now what will be the potency? Um so it will be one product for Do you have my 5.29? It will be 27 27.2211. Okay. So potentially at that resistance will be 27. About 2211 hold. Okay, let me check it again. So 1 44 divided by 5.29 Yeah. Yeah. In the next scenario will to find a potential energy electron is hit at that area side. I don't eat electronic, hit this radio. So basically what we do the uh, energy. So when this will do nothing, but just potential times of the sergeant. Electronic BQ. Thank you. So that and if you converted into electronic also potential energy, that will be I think about you in to be the th minus 1.7 minutes 19. Okay. And to hold well defined in the electronic hold. In place of 1.61211 point six interested governments. And you just place one It will become 27.2211 electron hold right now in the next five. But you need to do is that the kinetic energy of electron. And you mean to be involved in a similar officer, basically, we know that this video obviously negative because uh, electron have negative giants. Now we need to find a frantic energy. It is evolving in the circular movement and energy. So see we're gonna use the conservation of energy. We know that the entire distance they have zero taking out your opponents energy. Okay. And when it reached to the radio, it has a great importance in the idea. So initially that this would be called the final difficult. There's no external work done over here. That means if there is something for negative for themselves. So it should have some equal an equal and a public science of kinetic energy so that this would be, you know, so that kind of technology that will do nothing. But yeah, Good. 1 2211 electron volts. Right? This was a C and D. How much energy is required to organize the hydrogen atom? Basically, we know that If we need to recognize that means we need to give the energy that is positive with the early 70s so that it should be the energy becomes, you know, so in order that required is nothing but just 27.2 electron world. Mm And, uh, now this was a team. Okay, thank you.

Okay, So the electric potential, um, is given by vehicles constant k time charge Q over. Babies are so council cares. Nine times 10 of the nine in s I units charge is the elementary charge 1.6 times tend to the negative 19 cool ums, radius and sub or radius. That's 5.3 times 10 to the negative. 11 meters, giving you, um, potential of 27 volts. Um, R b um, we know that energy will be equal to charge times. Voltage. Eso charge is minus 1.6 tons. 10 of the negative 19. Cool ums. Electron. It's the negative. Protons charge Voltage is 27 volts. Now the answer is minus 4.3 times tender. That, uh, negative 18 jewels. You can convert this to electron volts by dividing by 1.6 cents. 10 of the negative 19. Jules, for electron faults. This gives you, um, no surprises here, but this gives u minus 27 electron volts. So note that this is just E. This is elementary charge. So 27 times eased. These have involves times E is just 27 electron volts. So you didn't need to do all of this. You could just note that fact


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