Calculate the wavelength and energy in kilojoules necessary to co...

Question

Calculate the wavelength and energy in kilojoules necessary to completely remove an electron from the second shell (m = 2) of a hydrogen atom (R∞ = 1.097 * 10-2 nm-1).

Accepted Answer

Welcome back everyone. What is the wavelength in nanometers needed to remove an electron in the fourth shell and equals four of a hydrogen atom. Where the Rydberg constant to infinity is equal to 1.97 times 10 to the negative second power inverse nanometers. And how much energy is this in killed joules? Per mole that is needed. So according to the prompt, we have an electron in the fourth shell. So our principle quantum number is N equals four as given in the prompt. And we would say that because this is where we are removing our electron from. This would be our initial energy level. So N. I. Is equal to four, meaning that our final energy level based on our principal quantum number and our understanding of its definition, which means that it should only include positive integers ranging to infinity. We would have a final energy level of infinity. And so that is why we have our rights for constant in terms of infinity Equal to 1.97 times 10 to the negative second power inverse nanometers as given in the prompt. And in our textbooks, our first step is to calculate our wavelength. So we want to recall our formula where are inverse wavelength is related to our Rydberg constant. Multiplied by the difference between Our atomic number for a hydrogen atom being one divided by our initial energy level squared subtracted from one over our final energy level squared. And so plugging in what we know into our formula to find our wavelength first will have won over our wavelength equal to our rights for constant. In terms of infinity being 1.97 times 10 to the negative second power inverse nanometers multiplied by the difference between one over our initial energy level. Given in the prompt as four squared, subtracted from one over our final energy level, which will say is infinity squared. So simplifying this will have won over our wavelength lambda equal to the product between our quotient and its difference where we would have our rights for constant in the numerator as 1.97 times 10 to the negative second power, inverse nanometers divided by 16 when we take four squared here. And so simplifying this quotient, we would have won over our wavelength equal to 6.8563 times 10 to the negative fourth power. And recall that when we have diagonals in algebra we can just switch places. And so we would say that our wavelength is going to equal and let's actually follow how things are written. So we would have won over 6.8563 times to the negative fourth power. And we still have our units of inverse nanometers. So let's carry that over. This is equal to our wavelength lambda. And so simplifying this quotient, we would say that our wavelength we'll say our quotient which equals a value of 1459 nanometers are no longer in the inverse because it's not in the denominator is equal to r value for wavelength. So this would be our first answer as the wavelength in nanometers required to remove this electron from our fourth shell of the hydrogen atom. And now we need to figure out how much energy is needed to remove this electron in terms of killer jewels. So we next want to recall our second formula where the energy of a photon is related to plank's constant H. Multiplied by the speed of light. C. Divided by our wavelength lambda in the denominator. And so plugging in what we know, we have our planks constant 6.626 times to the negative 34th power units of joules times seconds being multiplied by the speed of light. 2.998 times 10 to the eighth power meters per second being divided by our wavelength. That we just calculated as 1459 nanometers. And again keep in mind from the prompt. We need energy to be in units of joules per mole. And so we're going to focus on just converting from nanometers in the denominator two m so that we can cancel it out in the numerator. So we called it our prefix nano tells us that we have 10 to the negative nine power of our base unit meters, canceling out nanometers. And also canceling out meters as well as seconds were left with jewels as our final unit first. So this quotient when we simplify will lead us to a result of, we would get 1.36153 times 10 to the negative 19 power jewels. And now we want to convert from our jewels amount of energy to kill a jules Permal. So we have and sorry, let's write energy better. We have 1.36 times 10 to the negative 19 powered jewels, having jewels in our denominator and placing kill jewels in the numerator, recall that our prefix kilo tells us that we have 10 to the third power of our base unit jewels. And so now canceling out jewels, we can go and incorporate our units of moles utilizing avocados number. Where we would recall that one mole has an equivalence of six point oh 22 times 10 to the 23rd power for a Vos number. And now we can cancel out or not cancel out. But now we are left with units of kilograms per mole as our final unit for energy. And this is going to give us a result of 81 point 99 kg jewels Permal. As our second final answer for the amount of energy needed to remove this electron from the fourth shell for a hydrogen atom Equal to 81.99 kg per mole. So our two answers and our first answer was our wavelength here. So let's highlight that in yellow. Our two answers for wavelength and our energy required to remove this electron from the fourth shell are our final answers. I hope everything that I reviewed was clear. If you have any questions, please leave them down below and I will see everyone in the next practice video.

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