Okay, in this video I'm going to continue on with my tutorial series on thermodynamics and thermochromatism mechanics. This is video at number 42. I'm going to discuss the Gibbs free energy and chemical potential. So let's draw your attention to my website universityphysicsturors.com. So to do a small bit of revision, we saw in a previous video that the Gibbs free energy capital G is given by u plus pv minus T s. Okay, so basically what this was was we had mechanical work being done against the surroundings and because of surroundings our environment was at constant temperature we could also avail a free a heat, we'll say, the value or quantity of T s. And we found that this is at constant pressure and constant temperature. Okay, constant pressure and constant temperature. So what where do we go from here? We're able to solve that the infinitesimal change in G is going to be dg is equal to du plus vdp and it's supposed to be vdp believe it or not, vdp plus pdv minus s dt minus tds. That's just by the product root. So when we subbed in the first law we got and cancelled that the remaining term to minus s dt plus vdp plus mu dn was equal to the infinitesimal change in the Gibbs free energy. All right, so what does that mean? Well if we look at this what we're trying to do is we're trying to translate the Gibbs free energy and the chemical potential. So we see here this term is the chemical potential. So we notice that if we take the partial derivative of G so say we take we take del g del n if we take that derivative and we hold everything else fixed. So in this case we hold the temperature and the pressure fixed what we'll get is the chemical potential. So this suggests that in order to mention the chemical potential we take the we take let's say we somehow get an experiment that measures del g del n. Now in the in video number 41 I discussed extensive intensive properties. So generally an extensive property is one which would say doubles when you double the number of particles so when in doubles. Okay and we've found that an intensive property is one which does not double when you double when you double the number of particles. So difficult extensive properties we had were pressure pressure usually for example it is one which is an extensive or yeah an extensive property. But intensive properties we had things like the chemical potential and we had the temperature and so on. So each look about previous video if you want to look at this but if we look here if we look here we find that both the variables which we're holding fixed in this particular and partial derivative are intensive. So both of these do not double would say when you double the number of particles. They're both intensive, EINT, ENSI, VE. Okay they're both intensive particles. So at the moment what this partial derivative is is implying as well. If we integrate of course what that means that it's it's implying that d g is equal to mu d n or if we integrate both g capital g the Gibbs free mg is equal to mu times n. So it implies that the chemical potential is the Gibbs free energy per particle of something. So does that make sense? Well it should make sense because we're having two intensive variables. Now what that means is because the both of our intensive this leaves the extensive quantities to grow with n or grow proportionally with n which is exactly what extensive properties do. So what we're saying here is that where you add a single particle del g del n is the added chemical potential to the system. Okay so that's why we say the chemical potential is the energy added to the system when you add a single particle. That's what I've been saying all along. So the point here is because these these fixed variables are both intensive as you increase the number of particles the chemical potential remains unchanged and that's very important. Okay so just one last thing on this and it's the showing contrast. If we take a similar partial derivative if we take del f del n so this one is the the helm holds free mg we take their partial derivative with respect to the number of particles we take it and hold it temperature and volume fixed right now why the partial derivative this also should give us the chemical potential. Now the next thing we need to do is evaluate or we'll say analyze excuse me what is happening here. So we note that both of these are neither intensive or extensive because an actual fact warm is extensive and one is intensive. So the volume the volume is an extensive the volume is an extensive quantity and the temperature is an intensive quantity. Now why is this important? It's important because as we add particles to the system we are automatically going to be changing we're going to be changing the volume because the volume if you double the number of particles what the volume is going to double. So this is going to change which means the chemical potential is going to change. Okay so the point here is that we can't actually take del f del n and use mu and get mu excuse me because the the variables are totally fixed one is intensive and one is extensive. So when n increases so too does the density holding the volume fixed does the mu actually changes as n increases and this is because the fixed quantities are both intensive and extensive. So summarize then what we say is that mu is the Gibbs free energy for a particle. Okay, where we have T and P fixed. Great stuff. Okay thanks for watching please pass it on to your friends subscribe to my channel and you will also visit universityphysics.com.