Consider the distribution of ideal gas molecules among three bulbs (A, B, and C) of equal volume. For each of the follow-ing states, determine the number of ways (W) that the state can be achieved, and use Boltzmann's equation to calculate the entropy of the state. (a) 2 molecules in bulb A (b) 2 molecules randomly distributed among bulbs A, B, and C

Video transcript

Hello. Everyone in this video, we're being told that we have four different boxes labeled 1-4 of equal volume that distributed with ideal gas molecules. So we're being asked in how many ways can the following states be obtained? So we have this here ST one estate to were asked to also determine the entropy of each state using the bolts pins equation. So the ultimate equation, let's go and write this out. Actually that is equal to us being equal to K. So our constant multiplied by the natural log of W. So what K here is is the Holtzman's constant. So K equals to 1.38 times 10 to the negative 23 units being drools per kelvin. And the W is going to be the number of ways that the states can be obtained. So to calculate for w we're gonna go ahead and use w equal to be raised to the power of N. Let's go ahead and kind of break this down. So we said W is the number of ways the states can be attained or be here is the number of boxes. And this end here that is the number of molecules. Alright, so for our first scenario here, we can go ahead and do some math. We'll do a green to the right. So for one is the three molecules in box one. So for W that equals to again be raised to part of N. So B is the number of boxes. We have one box and then and this is the number of molecules and we are being told that we have three molecules so one raised to the power of three, which of course is just equal to one now for our s using the bolton equation. So we have our cake constant, the bolts mina constant, that's 1.38 times 10 to the negative 23 joules per kelvin and retain the natural log of W. Which we just saw for above. And that's one. So of course we know that the natural log of one is just equal to zero and they multiplied by zero is just zero. Therefore s here is equal to zero. So s is our entropy. So to formalize this answer for a situation one R W is equal to one and our entropy is equal to zero. So this answer for our first part here we're continuing on to our state number two, I'll do in purple, we will go ahead and scroll down a little bit for more space so far situation to hear R. W. Or the number of ways that the states can be obtained is equal to again be raised. The power of N. So R B here is the number of boxes. So we have four total boxes because we have box one through four. There's four different boxes to raise the power of end. Here is the number of molecules. So we have again three molecules. So doing four ratio power of three which of course is just that calculate entropy here, Starkey constant. Deep Altman constant, there's 1.38 times 10 to the negative jewels per kelvin. And then for the natural log of w we already solved for this and that's 64. So now I put everything into vocabulary, see them at entropy is equal to 5.74 times 10 to the negative 23 units being jewels per kelvin. Again. We're gonna go ahead and finalize his answer. So we have a number of ways that the states can be obtained is equal to 64 And our entropy here is equal to 5.74 times 10 to the -23 jewels per kelvin. So this here is my second and final answer for this problem.