How Many Lines?

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Every hydrogen atom has just one electron, okay? Now take a sample of hydrogen, send the electrons up to level five, and let them fall back to the ground state. One electron per atom, so just one line in the spectrum, right? No, no! The spectrum shows many lines. So how many, exactly? And here is the twist: one single atom can never give you all of them. Now, this is a common one-line question in JEE, and the trap often hides in one word of the question. So, let's begin.

0:38A sample: many atoms, many routes

So, first, what does a sample mean? Many, many atoms, not just one, okay? On screen, n two, the n with a small two below, is the upper level where the electrons start. And n one, with a small one below, is the lower level where they stop. All routes means every atom comes down in its own way: some jump straight down, some stop on the way. Clear? Let's start with the smallest case. The electrons start in level three and fall to level one. How many different lines can the sample give you? Three, yes! Let's list them. The arrow means falls to. Three to two, two to one, and three straight down to one. Did you say two? Then you missed the straight jump from three to one, right? One atom takes one route, another atom takes another, and the sample shows all three lines together. Now see the pattern. Every line joins two levels, an upper one and a lower one, okay? So the real question is simply this: how many pairs of levels can I pick? That's it, count the pairs!

1:44Count the pairs: the formula

So, how many levels are there from n one up to n two? It is n two minus n one, plus one. Why plus one? Because both end levels count, the top one and the bottom one, right? Now, from these levels, pick any two: that gives one line. The number of pairs is n two minus n one, times n two minus n one plus one, all over two. Why divide by two? Because otherwise the same pair gets counted twice, once in each order. Write this down. This is the formula JEE expects you to know. Got it? And the most common case: electrons falling all the way to the ground state. Put n one equal to one, and call the starting level just n. The formula becomes n times n minus one, over two. But careful! This short form works only for the ground state. Remember that; it will come back to trick you. Okay?

2:41One atom: at most one photon per step

Alright, now the twist from our hook. Take just one hydrogen atom, with its one electron up in a high level. Can this one atom give you every line that the whole sample gives? No, never! One atom has one electron, so it comes down by just one route. Each step down gives one photon. So the most photons come when it steps down one level at a time: at most n two minus n one photons, okay? If it skips levels, it takes fewer steps, so fewer photons. That is why we say at most. Let's take a course example. One single hydrogen atom is excited to n equals four, and comes back to the ground state. One step at a time: four to three, three to two, two to one. Three steps, so at most three photons. Simple as that. So, back to our hook. One atom from level five: at most five minus one, that is four photons. But the sample has many atoms, each on its own route, so together they give many more lines. How many? Example one will tell us. And remember, this is the JEE trap: always look for one word in the question. Sample, or single atom? Got it?

3:56Absorption from the ground state: only Lyman

Now, let's flip the question: absorption. Take hydrogen at room temperature. All its atoms sit in the ground state, n equals one: the bottom line of the ladder in the picture, okay? Now shine white light on it. White light has all wavelengths in it, every single one. So tell me. Every wavelength is there in the light. So the Lyman, Balmer and Paschen series, all of them get absorbed, right? No, no, careful! To absorb, the electron jumps up from the level it is already in, the occupied level. And every electron is sitting in n equals one. Look at the orange Balmer arrows in the picture: every one of them touches level two. To absorb a Balmer line, the electron must start from level two. But nobody is in level two, right? So no Balmer jump. Same story for the green Paschen arrows on level three. So only the jumps that start at n equals one count: the blue Lyman arrows in the picture, just run them upwards in your mind. Only the Lyman series appears. Exactly! And the arrows here point down because the picture shows emission, clear?

5:04Example 1: lines from n = 5 to the ground state

Now, remember our puzzle from the start? Let's finish it. How many spectral lines can appear when electrons in a sample of hydrogen atoms fall from n equals five to the ground state? Pause the video, and try it first. Done? Okay, let's check. First, the key word: sample. So many atoms, and all routes. And they fall to the ground state, so we can use the short formula: n times n minus one, over two. Here n is five. Five times four, over two: ten lines! There you go, puzzle solved. Each atom has only one electron, but different atoms take different routes, so the sample shows ten lines. And one single atom from level five? It can never give all ten, as we saw. Yes, that was the twist!

5:57Example 2: lines from n = 6 to n = 2, how many Balmer

Now a slightly harder one, JEE level. Read it with me. Electrons in a sample of hydrogen fall from n equals six to n equals two, by all possible routes. How many lines can appear, and how many of them are Balmer lines? Pause and try. Hint: look at where the electrons stop. Alright, the number of lines first. Electrons start at six, so six times five, over two: fifteen, right? No, no, careful! This is the JEE trap. That short formula is only for the ground state. Here the electrons stop at n equals two; they never go below it. So n two is six, n one is two, and n two minus n one is four. Put it in: four times five, over two. Ten lines. Or the topper's way: levels two to six, that is five levels, and any two of them give a line. Five choose two, the tall brackets on screen: ten again. Clear? Now the second part. Which of these ten lines are the Balmer lines? Yes, the ones that end at n equals two! Six to two, five to two, four to two, and three to two. Four Balmer lines. And the other six end at n equals three, four or five: the Paschen, Brackett and Pfund lines. So the answer: ten lines, and four of them are Balmer lines. See how fast that was, once you dodge the trap?

7:33Quick revision: line counts for a sample and one atom

So, quick revision. For a sample, count the pairs of levels: n two minus n one, times n two minus n one plus one, over two. To the ground state, n times n minus one, over two. For a single atom, at most n two minus n one photons, one per step. And ground-state atoms absorb only the Lyman lines. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!