Solid State part 9

追加: 2026-05-12

[00:00:00]now coming to the packing fraction or the packing efficiency all right write it down what is packing efficiency or packing fraction the percentage of the total space filled by the particles is called packing efficiency the percentage of the total space filled with the particles is called packing efficiency particles means atoms molecules are ions suppose if you are having a lattice structure so how much space is being occupied by the atoms or the molecules of the ions all right which is expressed in percentage that how much let us say seventy percent thirty percent forty percent so that is known as packing efficiency so the remaining what it will be suppose seventy percent space is occupied then what will be the remaining whites empty spaces because seventy percent space is occupied by the atoms or the molecules of the iron so remaining thirty percent is your voids voids means what empty space or vacant space remember it whites v o i d s voids means vacant space or the empty space we can say that clear there is a to be us okay or we can say the fraction of the total space filled by the atoms or by the particles or the molecules or the ions is called as packing fraction the fraction of the total space filled means that is suppose total space let us say the solid is cubical okay the solid is cubicle and the side of the cube is a the edge of the cube so what is the volume of the cube a cube so out of a cube how much percent is being filled by the atom because that is the total space availability right or in terms of fraction how much fraction of the space is is being occupied with atoms molecules or ions that is called as packing fraction i hope you got it what is packing fraction now based upon the type of solid or the type of lattice already we have read it we are having scc simple cubic cell we are having bcc body centered cubic we are having fcc face centered cubic i said you already that one is obsolete that is ecc and centered cubic is absolutely no more use this particular one we can say that clear you know it already so first we are going to start from simple cubic that is scc from there we are going to start first let us see what is the packing fraction or the packing efficiency remember it first we always find the fraction then you multiply into 100 percent that will give you efficiency very simple first you will always calculate in terms of fraction then when you multiply into 100 that you get the efficiency that is packing efficiency so now suppose in case of scc look at the diagram so what we find is that scc comes under the primitive cells and dcc fcc comes under the non-primitive cells we can say that what is the difference between that you have seen in the av in case of primitive cells the atoms of the corners touch each other the atoms are the corners they touch each other see here they are touching each other the atoms are the corners they touch each other here you can see the atoms they are touching each other at the corners let us say the four corners are there these are four corners the atoms of the corners you can see very clearly it is touching each other so this space which you can see it here what is this whites or empty space this one is the voids or empty space all right now what is the words or empty space in this way so now we are going to find that in a particular cubicle solid how much percent of the space is occupied by scc if the atoms are arranged in scc in simple cubic cell then how much percent space is being occupied with the atoms molecules and ions and the remaining from 100 if you subtract you will get the voids the empty spaces i hope you understood so in case of scc that is simple cubic cell again i repeat it is a primitive cell the atoms are the corners they are touching each other you can see the atoms at the corner let us say this is the center of the unit cell see they are touching each other now this uh the radius of the atom is r all right radius of the atom and this is the center of one units at the center of another unit cell so as you can see i'm drawing a unit cell a particular unit cell i got it now this is your a a is the edge of the unit cell this is a this is a this is a this is because this is nothing but ah what is that square in terms of two dimension i can say this square and when i arrange in three dimensional this square only becomes a cube product lengthwise bread twice height wise when we expand it then it becomes a cube you know it so here i can say a is equal to two r isn't it this a and this 2i this is the radius of the atom which is nothing but a is equal to 2 alright now first is your volume of the unit cell this one is the unit cell volume of the unit circle so what is the volume of the unit cell it is a cube a is your 2 r so 2 r whole cube so it is nothing but 8 r cube we got it now number of atoms per unit cell number of atoms per unit cell is nothing but is that is that how much it is one we have seen it how it is one one right into eight there are eight corners are there so what is the contribution to each corner one by eight one by eight into number of corners how much you got it one so number of atoms per unit cell in f in case of scc this is simple cubic is what one so here you can see four items similarly four four more corners are there this is four corners you can see contribution to each corner is how much one by eight here four atoms are there similarly four more items are there all right so each item will the contribution will be one by eight all right so in this way now what is the [Music] volume of that is each atom this atom volume of each other now atom is spherical so what is the volume 4 by 3 pi r cube we can say that the volume is four by three pi r cube here four by three power q if you know it let me write this one little bit here otherwise you will get confused later on a is equal to two r fine now next is what is the total volume of atoms in the unit cell four by three pi r cube into one because number of unit cell is one number of atoms unit cell is one so total volume of atoms occupied uh what you can say by the atom per unit so this four by three five cube into one into number of units that is four by three so packing fraction p f we can say is what that is nothing but the volume of the atoms occupied so volume of the atom support is four by three irq on the total volume total volume i hope you understood after that i write a sentence i am not writing it here you understand it total volume of atoms ah what you can say per unit cell in case of scc how much will be four by three pi r cube into number of atoms per unit cell number of items is one so you've got four by three pi divided by packing fraction is total volume of atoms per unit cell divided by the volume of the units the volume of the unit cell is how much 8r into 100 if i convert it into now if i keep this much only this much then what i get packing fraction i get it you cancel it for 2 so pi by 6 so packing fraction is five by six now if i have to convert into packing efficiency what i need to do into hundred percent i'll do it into hundred percent so how much it will come you calculate and see it comes to this fifty two point four percent so that means fifty two point four percent of the space is occupied by the atoms and the remaining is points all right empty spaces so this is fifty two point four percent so packing efficiency is fifty two point four percent we can say clear so this is for scc similarly we'll calculate for fcc now fcc comes under non-primitive units in non-primitive unit cell the atoms of the corners they do not touch each other that's the most important thing you have to understand it while drawing the diagram the atoms at the corners do not touch each other they are not touching each other see these are the atoms at the corners clear are they touching each other no the atoms of the corner did not touch each other let me draw this one little bit better okay fine now in fcc the atoms here at the center of each face this is one face correct this is one particular face we can say that clear so there is a center the center is center and this is the center we can see so atoms this is at the center you can see i have done it here dotted line you can see the atoms of the center so i have to find here same thing packing efficiency of the fine group so what will i do i will just join this particular sentence i'll join it all right i'll join this diagonal let me name this as a triangle a b c all right now you know this is r this is also r this is also same atom the volume of the atom is same thing here that is radius of the atom is same thing here r so this one again it is r and this one is again it is r okay of course it does not look like same but you consider it that insane because i'm poor in drawing i'm not so good in drawing so to say so i can make little bit better this one yeah a little bit better i can make it fine yeah this one so whatever it is you can see here this one that is this one is rr so hence what you get here that ac here it is come to be at 4 r and here a b is your nothing but a the edge of the unit cell and bc is also your a the edge of the unit cell all right edge of the unit cell is what i have already said you that is a here the atoms of the corner they are not touching each other mind it so here in triangle abc which is a right angle triangle you know that ac is equal to root over of a b square plus bc square you know a this one in massive studies pythagoras theorem so it is a square plus a square is equal to nothing but root 2 a square all right fine ac is equal to a b square plus b c square a whole root of so we got it now here what we get it is that that is root 2 a square or i can write this one as s 2 2 a square so that is nothing but a root 2 i can write it no problem fine now ac is 4r i've written it there you can see 4 r is equal to a root 2. so a is equal to 4 r by root why did i find a because i have to find the volume of the cube volume of the unit cell that is volume of the cube so i already said it one in the unit cell is what it is cubical and what is volume of the cube it is a cube so i need to find the value of a so 8 is here nothing but we got it this fine so hence we got it a cube is nothing but your 4 r by root 2 whole cube 4 r by root 2 now one more thing one more thing listen here that is in case of scc you got it scc a is equal to 2 are you added so fine they may also also ask you r r is equal to what a by 2 r is what radius f and if it is diameter diameter is equal to 2 r is equal to nothing but a minute this is for scc this one it is then in the numericals you will see this will be used r will be used diameter will be used diameter in the numericals in case of ionic compounds it will be the inter ionic distances the word will be there inter ionic distances listen carefully in the numericals for in case of ionic compound in case of ionic solid all right because this atoms will be the oppositely charged ions let us say sodium ions chloride ions it will be oppositely charged ions so there they will be asking you that is the inter ionic distance means the distance of the two ions what will be it is nothing but is your diameter that particular venus so inter ionic distance is something but it's your d and d will be how much it is 2r which is nothing but is equal to a in case of scc if it is scc mind it if it is scc then it will be d is equal to 2 r is equal to a i am saying you if it is a cc repeatedly i am saying all right now if it is not a cc let us say if it is fcc in scc you understood how to calculate r and d similarly in case of fcc how will you get that is r from here now 4 r is equal to a root 2 so r is equal to what i is nothing but a root 2 by 4 4 r is equal to a 2 so r will be a root 2 by 4 and what is d d is equal to 2 r 2 into 4 root a a root 2 by 4 will be 2 into a root 2 by 4 so 2 and 4 will get cancelled so a root 2 by 2 will be that is nothing but d clear are you getting what i am saying so this will be definitely you will see it is being used all right now here we got it 4 r by root 2. so you this one if you simplified 4 for the 16 for the 64 r cube by 2 root 2. we got it okay now i have to love it little bit i need to rub so here i'm doing it so here that is number of atoms per unit cell is how much in case of fcc number of atoms per unit cell is nothing but four how it is for one by 8 into 8 all right plus half into 6 number of atoms at the corner is what 1 by 8 into 8 plus how many faces are there six faces contribution to each face is what half half into six is three three plus one it is four got it so hence it is coming to be as four i hope you understood i have told in the previous av also you know that so now next is total volume of atoms total volume of atoms per unit cell so 4 into 4 by 3 4 into 4 by 3 pi r cube so packing fraction will be that is nothing but 16 by 3 pi r cube that is total volume of atoms divided by volume of the unit itself volume of the unit cell is how much a cube a cube is how much you got it 64 r cube by 2 root 2 so here it will be 2 root 2 clear understood 16 4 5 16 pi r cube by root 3 uh sorry 16 pi r cube by 3 i got it here now by total volume total volume is 64 rq by 2 root 2.2 root 2 goes up rp bar cube cancel here 16 4 64 it gets cancelled if you multiply it this you convert into packing efficiency 200 percent you do it you will get it to be as nothing but 74 percent is the packing efficiency which is the maximum because of that so fcc is the most efficient type of packing because we have 74 percent of the space occupied by the items one more is left bcc is it but it will be less you'll see that then fcc so it's the most efficient packing is considered means the atoms are most uh what you can say more tightly packed in case of fcc then bcc then scc of course this is i have not found it out and i'll find it out and say you in the same way we'll do all right and the remaining that is 26 percent easier nothing but voids we can say i hope you understood how to find the packing fraction as ccn fcc will come to bcc next you