Atomic Mass and Isotopes

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. In your periodic table, the atomic mass of chlorine is about thirty five point five. Now think. An atom is made of protons and neutrons, and you cannot have half a proton, or half a neutron, right? So how can one chlorine atom weigh thirty five point five? Hmm, is half a neutron hiding somewhere? Today we will solve this mystery. And look, this is not a small topic. In JEE, the whole mole concept begins right here, with atomic mass. So, let's start.

0:37One atomic mass unit: the carbon-12 ruler

So, first question. An atom is far too tiny for any balance. So how would you weigh one single atom? You can't! No balance in the world can do it. So chemists did something clever. They chose one atom as a ruler, and measured every other atom against it. That ruler is an atom of carbon twelve. One unified atomic mass unit, written as small u, is exactly one twelfth of the mass of one carbon twelve atom. On screen, the small m means mass, and the little twelve on top of C tells you which carbon we mean, okay? Now, how big is one u in grams? About one point six six zero five, times ten to the power minus twenty four grams. Look, that is a decimal point, then twenty three zeros, and only then the digits. Tiny, right? And it is the same as one over N A grams, where N A, the N with a small A below it, is the Avogadro number. This unit has one more name: the dalton, written D a, okay? So if a book says dalton, don't panic. Same unit, new name. Now, the atomic mass of an element is simply the mass of one of its atoms, measured in units of u, okay? So when we say oxygen is sixteen u, we mean one oxygen atom weighs sixteen of these tiny carbon twelve units. And when we write it as a relative atomic mass, it is a pure number, with no unit at all. Just sixteen.

2:05Average atomic mass: weight by abundance

Clear? Now, the real reason behind our puzzle. Most elements in nature are not one kind of atom. They are mixtures of isotopes: atoms of the same element, but with different masses. Alright, so which mass goes into the periodic table? Should we just add the masses and divide by the number of isotopes? Should every isotope count equally? No! If most of the atoms are of one kind, the average must lie close to that kind, right? So the common isotope must count more. So, we take a weighted average. On screen, A bar, where the bar on top means average, equals the sum of isotope mass times abundance, divided by the sum of the abundances. This big Greek letter, sigma, just means add them all up, okay? Simple, isn't it? And one small tip. If the abundances are given in percent, they add up to a hundred, so the bottom is just a hundred. That's it: multiply, add, and divide by a hundred.

3:09Two isotopes: find the abundance

Now, the JEE favourite: just two isotopes. Let a fraction x of the atoms have mass A one. Then what fraction is left for the other isotope, with mass A two? Yes, one minus x, because all the fractions together make one. So, on screen: fraction x has mass A one, and fraction one minus x has mass A two. All clear? So, A bar equals x times A one, plus one minus x, times A two. The same weighted average, just written for two isotopes. With me? But JEE often turns it around. They give you the average, and ask for the abundance. So how do we find x? Open the bracket, and collect all the x terms on one side. That's all. Do the algebra once, and you get this: x equals A two minus A bar, divided by A two minus A one. Write this down. It saves time in the exam. And here is a topper's check. Look at the top of that formula: it is the distance of the average from A two. The closer the average sits to A one, the bigger this distance, and the bigger x becomes. So the average always lies nearer the more abundant isotope. Five second check, every time.

4:28JEE trap: mass number is not atomic mass

Now, a classic JEE trap. Mass number is the count of protons plus neutrons. We count particles, so it is always whole, okay? So if an atom has mass number thirty five, its mass is exactly thirty five u, right? Hmm, careful. Students mix up these two all the time. The atomic mass of an element is usually not a whole number. And there are two reasons. Reason one: it is an average over several isotopes. See, this is our chlorine puzzle! No half neutron anywhere, just an average. Reason two: inside the nucleus, nuclear binding makes each isotope's mass slightly different from a whole number. Carbon twelve is the exception: exactly twelve u, by definition. Remember: mass number, always whole; atomic mass, usually not. Got it?

5:21Example 1: average atomic mass of chlorine

Alright, now let's finish our puzzle. Chlorine is seventy five percent chlorine thirty five, and twenty five percent chlorine thirty seven. Find its average atomic mass. Pause the video, and try it first. Done? Okay, let's check. The two masses are thirty five and thirty seven, and exactly halfway between them is thirty six. So the answer is thirty six, right? No, no, careful! That is the simple average, and we already said every isotope cannot count equally, remember? Three out of every four chlorine atoms are chlorine thirty five, so it must count more. Weighted average: thirty five times seventy five, plus thirty seven times twenty five, all divided by a hundred. Thirty five times seventy five is two thousand six hundred and twenty five. Thirty seven times twenty five is nine hundred and twenty five. Together, three thousand five hundred and fifty. Divide by a hundred, and the answer is thirty five point five u, okay? And there it is, mystery solved! No atom has half a neutron. Thirty five point five is just an average, and it sits nearer thirty five, because chlorine thirty five is more common. As simple as that.

6:38Example 2: percentage of Cu-63 in copper

Now a harder one, JEE level. Copper has atomic mass sixty three point five five. It is made of copper sixty three, with mass sixty two point nine three, and copper sixty five, with mass sixty four point nine three. Find the percentage of copper sixty three. Pause and try. Hint: use the two isotope formula. Done? Okay, let's check. First, which masses do we use: sixty three and sixty five, or the masses given? The given masses, always! See, copper sixty three weighs sixty two point nine three, not sixty three. That is reason two from our trap, nuclear binding, at work. So, A one is sixty two point nine three, for copper sixty three, the isotope we want. Then A two is sixty four point nine three, and A bar is sixty three point five five. Now put them in. On top, sixty four point nine three minus sixty three point five five. At the bottom, sixty four point nine three minus sixty two point nine three. The top is one point three eight, and the bottom is exactly two. One point three eight divided by two: zero point six nine. So copper sixty three is sixty nine percent. Now the topper's check. The average, sixty three point five five, sits nearer sixty two point nine three, so copper sixty three must be the more common one. More than half. Absolutely correct!

8:09Quick revision: atomic mass unit and isotope averages

So, quick revision. One u is one twelfth of the mass of one carbon twelve atom. The atomic mass of an element is a weighted average of its isotope masses, and for two isotopes, x equals A two minus A bar, over A two minus A one. And mass number is always whole; atomic mass usually is not. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!