Counting Particles and Average Atomic Mass

Transcript

0:00Introduction

Hello everyone! So, one small puzzle before we start. Open any periodic table and look at chlorine. Its atomic mass is thirty five point five. Thirty five point five? But the mass number counts protons plus neutrons, and nobody has ever seen half a neutron, right? So where does that point five come from? Today we will find out, and we will even work out the exact mix of atoms inside chlorine. We will also learn to count protons, neutrons and electrons in any sample. Both ideas come up often in JEE numericals. So, let's begin.

0:38Counting particles in a sample

Alright, first, counting. Every counting question starts with moles. Small n, the number of moles, equals the mass of the sample divided by its molar mass, okay? You already know this line from the mole concept. Now, how many molecules? Multiply n by N A, the capital N with a small A below it. That is Avogadro's constant: six point zero two times ten to the power twenty three, per mole. The mol with a minus one on top is read as per mole. Clear? Now think. Suppose the question asks for electrons, not molecules. You already have the number of molecules. So what is the next step? Yes, exactly! Multiply by the number of electrons in one molecule. Same for protons, and same for neutrons. So the count is n times N A, times the number per molecule, the part in brackets. Three quick steps, that's it. Got it?

1:39Counting inside one molecule

Now, how do we count inside one molecule? Protons first. Each atom brings its atomic number, capital Z, so just add the Z values of all the atoms, okay? And if the molecule carries no charge, the electrons balance the protons. So electrons equal protons. Really simple. Neutrons need one more step. For each atom, the neutrons are A minus Z: the mass number minus the atomic number, from our last video. Then add them up for all the atoms. With me? Let's try one. Carbon dioxide, C O two: the small two below the O means two oxygen atoms, okay? Carbon has Z equal to six, and oxygen has Z equal to eight. How many protons in one molecule? Six plus eight plus eight: twenty two. Correct! Don't forget the second oxygen. Always count every atom in the formula, okay? Now neutrons. Take water, H two O, made of hydrogen one and oxygen sixteen. The small number on top of each symbol is its mass number, okay? Hydrogen one has mass number one and atomic number one. So how many neutrons does it have? Zero! One minus one. Ordinary hydrogen has no neutron at all, and students often miss this. Oxygen sixteen: sixteen minus eight, eight neutrons. So zero plus zero plus eight: just eight neutrons in each water molecule.

3:11Average atomic mass

Clear? Now the second idea: average atomic mass. Most elements are a mixture of isotopes: same element, different mass numbers, okay? So the atomic mass we use is an average. On screen, you see capital A with a bar on top. The bar means average. It equals sigma of x i times A i. The big Greek letter sigma means add up, over all the isotopes. The small i below each letter is just a label: isotope one, isotope two, and so on. So capital A sub i is the mass of that isotope, and x sub i is its fractional abundance: the fraction of all the atoms that are of this isotope. Each mass times its share, then add. Right? Now, in questions the abundance is usually given as a percentage. But x sub i is a fraction. So what do you put in the formula? Divide by a hundred, exactly! The percentage over a hundred. And every atom belongs to some isotope, so all the x i values add up to one. Remember this line. Example two needs it. Now, back to our puzzle. Each isotope has a mass number that is a whole number. So why is the average not a whole number? Yes! Because the element is a mixture. Atomic masses are not whole numbers mainly because elements are mixtures of isotopes. Average a mix of different masses, and the result usually lands in between, not on a whole number. So no chlorine atom has half a neutron! In example two, we will find the exact mix.

4:47Example 1: number of electrons in 18 mg of water

Now, example one. How many electrons are in eighteen milligrams of water? The unit, written m g, means milligrams, okay? Pause the video, and try it first. Done? Okay, let's check. First, moles. Water has a molar mass of eighteen grams per mole, and we have eighteen. So exactly one mole, right? No, no, careful! Read the unit again. Milligrams, not grams. Eighteen milligrams is zero point zero one eight grams. Divide by eighteen, and n is ten to the power minus three moles, just one thousandth of a mole. That unit is the trap in this question. Next, electrons in one water molecule. Each hydrogen brings one, and oxygen brings eight. One plus one plus eight: ten. Neutral molecule, so electrons equal protons, right? Now our three steps: moles, times N A, times ten per molecule. Ten to the power minus three, times six point zero two times ten to the power twenty three, times ten. Now the powers of ten: minus three, plus twenty three, plus one, gives twenty one. So, six point zero two times ten to the power twenty one electrons. That's it!

6:10Example 2: abundance of chlorine-35 from 35.5 u

Now example two, and this is our puzzle from the start. Chlorine, with atomic mass thirty five point five u, is a mixture of chlorine thirty five and chlorine thirty seven. The small u is the atomic mass unit, and the numbers on top are mass numbers, okay? Find the abundance of chlorine thirty five. Pause and try. Hint: call one fraction x. Done? Okay, let's check. Here is a topper's trick. Only two isotopes, so we need only one unknown. Let the fraction of chlorine thirty five be x. Then the rest, one minus x, is chlorine thirty seven, because the fractions add up to one. Clear? Now the average atomic mass formula: each mass times its fraction, then add. Thirty five x, plus thirty seven times bracket one minus x, equals thirty five point five. Now you solve it. What is x? Open the bracket: thirty five x, plus thirty seven, minus thirty seven x. So thirty seven minus two x equals thirty five point five. The double arrow means so: two x is one point five, and x is zero point seven five, okay? So the answer is zero point seven five percent, right? No, no, wait! Here x is a fraction, not a percentage. Multiply by a hundred: seventy five percent. So chlorine thirty five makes up seventy five percent of all chlorine atoms. Always check. Zero point seven five times thirty five is twenty six point two five. The rest, zero point two five, times thirty seven is nine point two five. Add them: thirty five point five. Perfect! Puzzle solved! Three quarters of the atoms have mass thirty five, one quarter have thirty seven, and the average comes out as thirty five point five. No half neutrons anywhere.

8:06Quick revision: counting particles, average atomic mass

So, quick revision. Particles: n times N A, times the number per molecule. Neutrons: A minus Z for each atom, and ordinary hydrogen one has none. Average atomic mass: add each isotope's mass times its fraction, and with two isotopes, use x and one minus x. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!