How much energy does the electron have initially in the n=4 ex...

Question

How much energy does the electron have initially in the n=4 excited state

Accepted Answer

Welcome back everyone. We need to find the energy and jewels of an electron in the hydrogen atom. Given each of the following scenarios, we have our energy level being the fourth energy level and then for be our energy level being the eighth energy level determined the wavelength in meters of the radiation emitted when the electron moves from B to A. So we have a two part example. Beginning with part one, we're going to need to recall our boar energy formula in which what we have is our energy of our electron. At the given level end is equal to negative one times are Rydberg constant, which is multiplied by the atomic number of hydrogen Z squared, which is divided by our energy level squared and specifically our energy level of our electron. So going into our first example for A we're told that the energy level is the fourth energy level. So we have E sub four set equal to negative one times are Rydberg constant, which we should recall is 2.18 times 10 to the negative 18th power jewels. Multiplying this by then the following quotient where in our numerator we have our atomic number of hydrogen which we see on our periodic table as the value one. This is squared and divided by our energy level which is given for a as the fourth energy level squared in the denominator. And so this will simplify in our next line. 2 -2.18. So let's write that out without parentheses negative 2.18 times 10 to the negative 18th power jewels, which is then multiplied by the result of our quotient being 0.625. So taking the product between these two terms on the right hand side, we have a value of negative 1.3625 times 10 to the 19th power jewels. And sorry, that should be a negative 19th power in the exponents. So we can round this to a value of about negative 1.36 times 10 to the negative 19th power jewels as 366. And this would be our first answer as the energy of our electron at the fourth energy level in a hydrogen atom. Then moving on to our energy level for B, which is given as N equals eight. We have e sub eight for our board energy formula equal to negative one times are Rydberg constant. Again 2.18 times 10 to the negative 18th power jewels, multiplied by the quotient. In our numerator, we have one squared for the atomic number of hydrogen squared. And then in our denominator we have our eighth energy level squared. And so simplifying this, we have to be equal to negative 2.18 times 10 to the negative 18th power jewels, multiplied by the result of our quotient which is the value 0.15625. Taking the product here we get a result of negative 3.40625 times 10 to the negative 20th power. Which we can round to about 36 figs as negative 3.41 times 10 to the negative 20th power jewels. And this would be our second answer so far for the energy of our electron in the hydrogen atom at the eighth energy level. Now we need to focus on part two of our prompt which is to find the wavelength of our radiation emitted when our energy sorry, when our electron moves from the And equals for energy level to the n equals eight energy level. Or actually that's backwards. So when our electron moves from B which is N equals eight to N equals four which we have for a given in the prompt in order to figure this radiation value out. We have to go through two steps. Step one is going to be finding the energy of a photon of an electron. And so we're going to need to take the energy difference from the two levels that we calculated above. So we have the energy level of our final energy level minus the initial energy level. And so according to our prompt, our final energy level is going to be B. Which is rather a Sorry because we're going from B which is eight to A. Which is four. And so this would be our final energy level and then equals eight would be our initial. So we have E four for final minus E eight for initial. So plugging in our values from our energy values calculated above. We have our difference in energy equal to for our final energy level being the fourth energy level of value of negative 1. times 10 to the negative 19th power jewels. Subtracted from our energy level of an electron at the eighth energy level. Our initial energy level being negative 3.40625 times 10 to the negative 20th power jewels. And so taking the difference here in our calculators, we get a value of negative 1. times 10 To the negative 19th power jewels. And so now we're going to use their results and go to step two of funding the wavelength radiation that is emitted when our electron travels from the fourth energy level to the eighth energy level. And sorry, I keep saying that backwards from the eighth energy level to the fourth energy level. Step two of finding this result is then recalling our Plank Einstein equation in which we want to recall the formula where lambda or wavelength is equal to Planck's constant H. Multiplied by the speed of light C divided by delta E. The difference in energy or the energy of our photon of a single photon specifically. And we want to recognize that for delta E. We want to use its its exact magnitude or its absolute value. So we want to use a positive value. And so going into our formula, we would find that our wavelength of radiation emitted is equal to Planck's constant recall that that is a value of 6.626 times 10 to the negative 34th power jewel's time seconds multiplied by the speed of light 3. times 10 to the eighth power with units of meters per second and then dividing by our denominator, which is our value delta E. For the energy of a single photon. Which above we found the value 1.021875 times 10 to the negative 19th power jewels. And again, this is the absolute value. So it's a positive integer here in the denominator. So now canceling out our units notice we can get rid of jewels with jewels in the denominator seconds with inverse seconds. And that leaves us with meters as our final unit for wavelength of our radiation. Which is what we want according to the prompt. And what we'll find is that we have a value from the result of this quotient equal 1.945-8. Sorry, that's 5248 times 10 to the negative six power meters, which we can round 2366 as 1.95 times 10 to the negative sixth power meters. And this would be our third and final answer to complete this prompt as the wavelength of radiation emitted when our electron transfers from the eighth energy level to the fourth energy level. And so our final answers will correspond to choice B as the correct answer. So I hope that this made sense and let us know if you have any questions.

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How much energy does the electron have initially in the n=4 excited state

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