Other Orbital Quantities and Limitations

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Inside a hydrogen atom, one electron goes round and round the nucleus. The electron carries charge, right? And charge moving round in a loop is an electric current! So can one single electron really make a current? And if yes, how big? So tiny that it hardly counts? Hmm, think about it. And one more guess: how many rounds does it make in one second? Interesting, isn't it? Today we will find out both. And look, these orbit quantities and the limitations of Bohr's model are asked in JEE too. So, let's begin.

0:44Time period: build it from r and v

So, first, the time period. Capital T with a small n below it is the time for one full round in orbit number n. Time is distance over speed, right? One round is the circumference, two pi r n, where r n is the radius. And v n, at the bottom, is the speed. So T n equals two pi r n over v n. Okay? Now recall Bohr's results from the first video of this chapter. The sign like a stretched alpha means proportional to. r n is proportional to n squared over Z, where capital Z is the nuclear charge, and v n is proportional to Z over n. And here is a topper's trick: don't mug up one more formula; build it. Two pi is just a number, so T goes as r divided by v. So tell me: how does T depend on n and Z? Yes, exactly! Dividing by Z over n means multiplying by its flip, n over Z, okay? So n squared over Z, times n over Z: n cubed over Z squared. Higher orbit, a much longer round; more charge, a much shorter one. Neat, isn't it? Next, f, the frequency of revolution: the number of rounds in one second. If one round takes time T, then one second holds one over T rounds. So f equals one over T n, just the flip: Z squared over n cubed. Got it?

2:16Hydrogen, first orbit: the real numbers

Alright, now real numbers: hydrogen, capital H, first orbit, n equals one. T one, with a small one below, is approximately one point five two times ten to the power minus sixteen second; small s for second, and the wavy equals sign means approximately. Really tiny! So, our second guess from the start. How many rounds in one full second? Just flip it! f equals one over T one: approximately six point six times ten to the power fifteen hertz. Capital H, small z, hertz means per second. That many rounds, every single second! Way more than you guessed, right?

2:58The orbiting electron is a current

Now, back to our first puzzle. The electron carries charge, and goes round and round, crazy fast. So tell me, is this really a current? Yes, it is! Pick any point on the orbit. In each round, the electron passes that point once, carrying its charge, small e, the size of the electron's charge. And current means charge passing per second. The electron passes f times every second. So the current, capital I, equals e times f. As simple as that! And f is one over T, which is v n over two pi r n. So I equals e v n over two pi r n. Clear? Example two puts real numbers into this.

3:42Limitations of Bohr's model

Okay, Bohr's model works beautifully for hydrogen, and for the helium ion, H e plus, also. So it should work for the helium atom too, right? No, it does not! The helium atom has two electrons, and they repel each other. Bohr's model has no place for this. H e plus has just one electron. So the model works only for one-electron species: hydrogen, H e plus, and L i two plus. Got it? So that is the first big failure. For atoms with many electrons, it cannot explain the spectra. Even helium, with just two electrons, is already too much for it. With me? Next, the fine structure of the spectral lines: Bohr's model cannot explain that either, okay? In a magnetic field, the lines split: the Zeeman effect. In an electric field, they split too: the Stark effect. Bohr's model explains neither. Don't mix them up: magnetic is Zeeman, electric is Stark. Nor chemical bonding: how atoms join to make molecules. On that, Bohr's model has nothing to say, right? And the deepest problem. Bohr gives the electron an exact orbit and an exact speed. That goes against the uncertainty principle, which you will meet soon in this unit. But for one-electron species, Bohr's formulas still give quick, sure marks, okay?

5:11Example 1: ratio of time periods for H and He+

Now, example one. Find the ratio of the periods of the electron in the first orbits of hydrogen and H e plus. Pause the video, and try it first. Done? Okay, let's check. T is proportional to n cubed over Z squared. First orbits, so n is one for both, and T is proportional to one over Z squared. Hydrogen has Z equal to one, and the helium ion has Z equal to two. So the ratio of the periods is one is to two, right? No, no, careful! Two mistakes in one line. Z is squared, and it sits at the bottom. T with a small H below it, for hydrogen, over T with H e plus below it, equals one over one squared, divided by one over two squared. Flip it: two squared over one squared, four. So T H is to T H e plus equals four is to one. Hydrogen's electron takes four times longer per round. Makes sense, right? More charge, a faster electron in a smaller orbit.

6:19Example 2: current in the first Bohr orbit of hydrogen

Now a harder one, JEE level. Find the current due to the electron circulating in the first Bohr orbit of hydrogen. Yes, our puzzle! Pause the video and try it. Hint: find f first. Done? Okay, let's check. In the first orbit of hydrogen, the speed v one, the small one for the first orbit, is two point one eight times ten to the power six metres per second, written m s to the power minus one. And the radius r one is zero point five two nine angstrom: the capital A with a small ring on top. Now, f is v over two pi r. So we simply put in zero point five two nine, right? No, no, careful! That is a common trap. Speed is in metres per second, so the radius must be in metres too. r one is zero point five two nine times ten to the power minus ten metre, small m for metre. So f equals v one over two pi r one: two point one eight times ten to the power six, over two pi times zero point five two nine times ten to the power minus ten. That is six point five six times ten to the power fifteen, per second. It matches the frequency we saw earlier. Perfect! Now I equals e f. The electron's charge is one point six times ten to the power minus nineteen coulomb, capital C, times six point five six times ten to the power fifteen, per second. That gives approximately one point zero five times ten to the power minus three ampere, capital A for ampere. Ten to the power minus three ampere is a milliampere, m A on screen. So, puzzle solved: one single electron makes a current of one point zero five milliampere! Not so tiny after all, is it?

8:16Quick revision: time period, current and limitations

So, quick revision. T n is two pi r n over v n, proportional to n cubed over Z squared; the frequency is the flip, Z squared over n cubed. The orbiting electron is a current: I equals e f, or e v n over two pi r n. And Bohr's model works only for one-electron species; it fails for the Zeeman effect, the Stark effect and bonding. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!