(c) The hydrogen atom now has a single electron in the 3d subshel...

Question

(c) The hydrogen atom now has a single electron in the 3d subshell. What is the energy in kJ>mol required to remove this electron?

Accepted Answer

welcome back everyone. We're told that the electron in an energetically excited hydrogen atom is located in the five F sub shell. A photon is released as the electron descends to the four D sub shell. How much energy is required to remove the electron in a four D. Or in the four D. Sub shell. So from our designation in the prompt four D. Where we focus on the four will tell us our principle quantum number represented by the term N. Where we can say that N is equal to four. And from this principle quantum number, we should recognize that since this electron is defending descending to this level, the fourth energy level is going to be our initial energy level. So we would say and I is equal to four, meaning that our final energy level. Based on our understanding of our principle quantum number should be positive integers ranging from one all the way up to infinity. So we would say that our final energy level is equal to infinity. And so for our rights burg constant for our electrons, we would recall that from our notes. This is equal to a value of 1.97 times 10 to the negative second power inverse nanometers. So because we need to figure out the energy required to remove the electron. Our next step is to recall our formula where energy is related to Plank's constant represented by H multiplied by the speed of light C. And divided by our wavelength represented by lambda. Where because we understand that Plank's constant and the speed of light are standard values from our notes and textbook. We need to calculate the wavelength by next recalling our second formula where our inverse wavelength is equal to our rights were constant. Multiplied by the difference between one over our initial energy level of our electron squared, subtracted from one over the final energy level of our electron squared. And so what we're going to do is first figure out what this wavelength would be. So simplifying this formula and plugging in what we know, we have one over our wavelength equal to our rights were constant as 1.97 times 10 to the negative second power inverse nanometers being multiplied by One over our initial energy level, which we agreed is four squared where one represents our atomic number of our hydrogen atom as given in the prompt subtracted from one over our final energy level, which we agreed would be infinity squared. And so simplifying this, we would have our inverse wavelength equal to once we take the product of these two terms. And the difference between our quotients, we'll just be able to simplify so that we have 1.97 times 10 to the negative second power inverse nanometers divided by 16. And so now we're going to simplify so that we have one over lambda equal to the value of this quotient on the right hand side being six point 8563 times 10 to the negative fourth power inverse nanometers. And so now we would recall that when we have diagonals in algebra we can just switch places and sorry, we can switch places with lambda here. So we would have actually one over 6.8563 times 10 to the negative 4th power inverse nanometers. And this is set equal to our wavelength. And we would see that the value of this quotient is 1458.52 nanometers. Since we're no longer in the denominator equal to our wavelength. Now that we have this value for lambda, We're gonna go into again our formula for energy to calculate the energy required to remove this electron and we're going to have planks constant, which we should recall is 6.626 times 10 to the negative 34th power with units of joules times seconds being multiplied by the speed of light. From our textbook as 2.99. Eight times 10 to the eighth Power units of meters per second which is then divided by our wavelength which you just found to be 1458.52 nanometers. However, we're going to need to convert this to meters. So we're going to multiply by converting from nanometers two m. Where our prefix nano tells us that we have 10 to the negative nine power meters. And because from the prompt we need to give energy and units of joules Permal. We're going to multiply by another conversion factor in our denominator. To cancel out jewels where we would recall that we have 10 to the third power jewels equivalent to one kg jule. And sorry jules should be here. And we also want to expand our denominator so that we can introduce avocados number where we would recall that we have for one mole an equivalent of six point oh 22 times 10 to the 23rd. Power being avocados number in our numerator. And so canceling out our units. We can get rid of jewels with jewels in the denominator. We can get rid of meters with meters in the denominator here, we can cancel out nanometers as well as seconds with seconds in the denominator. And we're left with kilo jewels in the numerator. Per mole in our denominator as our final units. Which is what we want for our final answer for the energy required to remove our our electron in the four deception. And so this is going to yield a result of And sorry, Just to be clear, we should actually do the math for our conversion from Tequila joules per mole separately so that we don't end up with the incorrect answer. So all of our units, as we've written out, will cancel out accordingly. And this is going to yield an energy where we would only have the unit jewels left that we haven't canceled yet. So we would have an energy equal to one 0. 1.36198 times 10 to the negative 19th power jewels. And now we're going to take this energy value and say that we have our 1.3698 times 10 to the negative 19 power jewels, which we want to convert from jules into kilo jewels. And then we want to introduce our units of moles. So we're going to recall that One mole has an equivalent of avocados number being 6.02, 2 times 10 to the 23rd power as avocados number. And so again for jules, tequila jules, we're going to recall that our prefix kilo tells us that we have 10 to the third power of our base unit jewels, allowing us to cancel out jewels and leaving us with killer joules per mole as our final unit. And so now we're going to carefully type this next step into our calculators to get our energy required to remove our electron from the four D sub shell equal to 82. killer joules per mole. And so we can round this to about just 82 point oh kilo jewels Permal as the amount of energy required to remove our electron in the four D sub shell. So it's highlighted in yellow is our final answer. To complete this example. I hope everything 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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