Order the following transitions in the hydrogen atom from
smalles...

Question

Order the following transitions in the hydrogen atom from
smallest to largest frequency of light absorbed: n = 3 to
n = 7, n = 4 to n = 8, n = 2 to n = 5, and n = 1 to n = 3.

Accepted Answer

hey everyone in this example we need to rank the given transitions below from largest to smallest frequency. So we should recall a few different formulas. The first being that the energy of a photon is equal to negative one times are Rydberg constant, which is then multiplied by the difference between one over our final energy level squared minus one over our initial energy level squared. We also want to call the formula where we would calculate the energy of a photon from our frequency multiplied by or sorry, Plank's constant. H multiplied by our frequency represented by this simple here. And so we can simplify this and say that our frequency is equal to our energy of our photon divided by plank's constant. And so we can substitute this in our first formula so that we can say that frequency is equal to -1 times our rights burg constant, which is then multiplied by one over our final energy level squared -1 over our initial energy level squared. So this would be a numerator where we're going to divide this entire numerator by plank's constant. H. So beginning with our first transition according to the prompt, we have the initial energy level being eight and the final energy level being five. So solving for our frequency, what we would have is negative one times are Rydberg constant, which we would recall is 2.178 times 10 to the negative 18th power and units of jewels times seconds. Or sorry, we just have units of jewels for our rights burg constant. This is then multiplied by the difference between one over our final energy level. Which according to our prompt for Part one is going to be the fifth energy level because that's where we end up. So it says so it should be 1/5 squared minus one over our initial energy level for part one, which is going to be eight squared according to part one of our prompt. So this is then going to be divided by our planks constant in our denominator which we recall is 6.626 times 10 to the negative 34th power in units of jewels times seconds. So we should be careful with our parentheses in our calculators and make sure that we're including the following parentheses as far as our units, we're going to be able to cancel our units of jewels leaving us with inverse seconds as our unit for frequency, which is the correct unit for frequency and what we should get here is a value equal to 8.1 to 1 times 10 to the negative 13th power inverse seconds as our frequency value. So this would be our first answer for part one or rather our first calculation for part one. Now we're going to consider the second transition In part two. So according to part two, we're going to have the same formula we're applying. However, now our initial energy level is the first energy level and the final energy level is the fourth energy level. So what we're going to plug in is negative one times our rights for constant 2.178 times 10 to the negative 18th power which is then multiplied by one over our initial energy level which is now one squared or rather our final energy level. Sorry. So according to part two, that's going to be four squared. So 1/4 squared minus one over our initial energy level, which in part two is one squared. So this is our numerator. And then for our denominator we're going to just plug in our planks constant, which we recall is 6.626 times 10 to the negative 34th power in units of jewels times seconds. And we should have had jewels appear in our numerator for our rights for constant. So yet again jules will cancel out. We're left with inverse seconds As our final unit for frequency. And what we're going to get for a value here is a frequency equal to 3.08 times 10 to the 15th power inverse seconds. So this would be our second calculated value here. And we're not going to box anything in yet. So moving on to part three were given in the prompt. The initial energy level being at the third energy level and the final energy level being at the sixth energy level. So we're going to calculate for frequency based on that transition. And what we would have is again the negative value of our rights for constant 2.178 times 10 to the negative 18th power jewels. Multiplied by what we should have Now for our difference is one over our initial energy level. For part three is given in the prompt as three squared. We're sorry our final energy level for part three when we look above is given as six. So it would be 1/6 squared minus one over our initial energy level. Given for part three as three squared. This is our numerator. And then in the denominator we plug in planks constant which again is 6.626 times 10 to the negative 34th power jewels times seconds were left with inverse seconds of the final unit. After we cancel out jules. And what we're going to get is a frequency equal to a value of 2.739 times 10 to the 14th power inverse seconds. And so this would be our third frequency. And then finally for part four we're calculating its frequency based on it's given transition in the prompt where we have an initial energy level of six and a final energy level of seven. So what we should have is again negative 2.178 times 10 to the negative 18 power jewels for our rights for constant, multiplied by one over our initial energy level for part four which is our final energy level for part four, which is going to be this seven here. So 1/7 squared minus our initial energy level for part four given as six squared this is our numerator. And then in our denominator we would plug in planks constant 6.626 times 10 to the negative 34th power jewels times seconds again we canceled our units were left with inverse seconds for frequency which is the proper unit. And what we're going to get is a value of two .422, 4 times 10 to the 30. And sorry, nothing is showing here. So it's two points 4224 times 10 to the 13th power inverse seconds as our frequency. And so comparing all of these frequency values. To answer this question, We would say that our highest frequency value is going to be the frequency that yielded the highest exponents. So that would be the 15th power here for the transition in part two. And then so we would begin our our ranking by saying that transition in part two is greater than the next highest exponents which is our transition which was in part three which had exponent of 14. So we would say transition level two Is greater than transition given in part three which is then greater than the transition from just to make a correction here. This should not have a negative sign, that should be a positive 13. So we're going to compare the transition in part one to the transition we calculated for part four. So which frequency would be higher? That would be this first frequency for the first transition which is greater than the frequency generated from the fourth transition here. So we would say that for ranking our first transition. So our transition in the part one is greater than our transition in part four. And this would be our final answer for our ranking of largest frequency to smallest frequency based on our given transitions and parts one through four. So what's boxed in is our final answer? I hope that everything I went through is clear. If you have any questions, please leave them down below and I will see everyone in the next practice video.

Order the following transitions in the hydrogen atom from smallest to largest frequency of light absorbed: n = 3 to n = 7, n = 4 to n = 8, n = 2 to n = 5, and n = 1 to n = 3.

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