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Hello everyone.
00:01
For this video, we'll be looking at the second -thromodynamics, and we'll, particularly in this problem, we'll be looking at macostates and micros, macro stakes and microstates.
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So this problem, in this problem we're in this situation where we have this box.
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So we have this box, and it's split in half.
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And there's a total of five atoms in this box.
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And they can either be on the left side or the right.
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Side.
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So we'll write the to describe the left side we'll write n sub capital l and for the right side or number of atoms on the right side will say that we'll have n sub capital r.
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So n sub l will be will describe the number of atoms on the left side of the box and m sub r will describe the number of atoms on the right side of the box.
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So for to have an expression for the microstates, we can express these microstates as being w and sub l is equal to w nl, comma, and r being equal to the number of atoms divided by n sub l, the number of atoms on the left side, times n sub r, the number of atoms on the left side, times n sub r, the number of atoms, the number of atoms on the left side, number of atoms on the right side.
01:49
So we know that the total number of atoms on the box or in the box is n is the left side, no number of atoms on the left side and l plus the number of atoms on the right side and r and we know this is a total of five atoms so that that was given to us that we have a total of five atoms and we can describe the distribution of the five atoms as this.
02:24
So we have n l comma and r and this will be equal to so the so the macro states distribution of the five atoms can be described as five comma zero so we can have all five atoms be on the left side and we can also say that we can have four atoms on the left side and one atom on the right side we can also say that we have three atoms on the left and two on the right and we can say we that we can also say that we can have four atoms on the left side and one atom on the right side and finally we can say we have zero atoms on the left side and five atoms on the right side.
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So these are the six possible macrosates.
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So with that in mind, so this premise the solution or, yeah, there's a solution to part a where they asked us to identify the five possible macrosates.
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So we, these are all the possibilities.
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So let me, i'll label this a and and highlight you.
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It all in red.
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So this is, so these are all the possible macrosates, right? so we can, there's only five atoms and they can either be in the right side or left side.
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And we're left with six possible, six possible macrosates in which these atoms can be arranged.
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So for part b of this problem, whereas to determine the number of micro, states for each microsstate and to do that we have to use this expression that we have to use this expression here so we'll do part b in blue.
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So for part b we're going to find the number of microstates.
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So find the number, so we're going to find the number of microstates so we're going to find the number of microstates.
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So find the number of microstates for each macro state.
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So what we're going to do is lay out all of our possible micro states.
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So we'll have our macro states.
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And we're pretty much just going to plug that into our expression.
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Our expression, which is wn sub l, is equal to...
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So we have the number of atoms on the left side, comma, number of atoms on the right side.
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And we can write that expression.
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And we know that that will pretty much be the total number of atoms divided by the times the factorial of the total number of atoms, divided by the number of atoms on the left side, factorial times and sub r total number of atoms on the right side factorial.
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So with this expression, we can pretty much write, we pretty just plug in our all these possibilities that we are all these possibilities for our for macro states that we found on this on part a, so we have these six possibilities, and we just plug in our values for the number of atoms on the left side and right side.
07:19
And this expression will then give us the number of microstates for each of these macro states.
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So for the first one, we'll have total number of atoms, factory.
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So that's 5 factorial divided by the number on the left side, which is 5.
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So 5 factorial times the number on our right side, which is 0 factorial.
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And this will leave us with 1.
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And answer of 1.
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So for our first, for our macro state in which we have 5 atoms on the left and 0 on the right, that gives us 1 macro state.
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And next we'll do 1 comma 5, so if we have, or 1 comma 4, so if we have one atom on the left side and 4 on the right side, we could then write.
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So for our number of microstates, we'll have 5 factorial divided by number on our left side, which is 1.
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So 1 factorial times 4 factorial.
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5 divided by 5 factorial divided by 1 factorial times 4 factorial will give us 5 microstates i'll give us anter 5 so and if you understand what i did so we know that a factorial a factorial is simply every number so this 5 factorial will be five times all the numbers before it.
09:23
So this will be 5 times 4 times 3 times 2 times 1.
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And in this case, since we're doing 4 factorial, so 4 factorial will be 4 times 3 times 2 times 1 and we are dividing.
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We are essentially all these cancel out and we're left with 5 as our answer for 5 factorial divided by 4 factorial.
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And so that's what we did here.
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And we'll be following that same, the same math, all of these micro state calculations.
10:05
So next we'll have two comma four, three, so two atoms on the left side of the box and three atoms on the right box.
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So for that we'll have five factorial divided by two factorial times three factorial.
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And for this one, i'll write it out for everyone.
10:26
We'll have 5 times 4 times 3 times 2 times 1 divided by 2 times 1 times 3 times 2 times 1 so one thing we can just do is say 2 times 2 is 4 and so we could cancel out the 4 and 3 times 2 is 6 and 6 divided by 3 is 2.
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Well, instead, i'll actually just cancel the threes.
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And then we can just say 5 times 2 is 10.
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And times 1, of course, it's still 10.
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And divided by 1 is still 10...