Lecture 19

We will start the lecture 19, and today's topic is graphical understanding of free energy, which is known as G, and composition which is known as X, that is mole fraction of condensed solution. mixing enthalpy or mixing volume, these two will be zero in the ideal system, and entropy of mixing, as we have expressed, will be minus R into X A ln X A plus X B ln X B, and so on. So, we have already got an expression for free energy in the case of binary solution, and we have also got an expression for mixing free energy, and by combining these expressions, we have got some graphs, and we have plotted the graphs as well. Now, let us see how these graphs vary, that is, how they change, if I increase the temperature or decrease the temperature, then what will happen? Now, if I take a binary system, and if I assume that there are two elements A and B, those two element crystal structure are same, that is, if F C C is two F C C structure, that is, body centered cubic, with the B C C or B C C, F C C body centered cubic, that is, if I assume that it is a single element, then there will be F C C body centered cubic, So, let us see what we have already in our hand, what we have got from the previous lectures. delta G mix equal to X A delta G A bar M plus X B delta G B bar M, G X A G A bar plus X B G B bar, delta H mix it is nothing but 0, delta V mix equal to 0 and delta S mix equal to minus R, R is universal gas constant, X A ln X A plus X B ln X B, here X is mole fraction X B ln X B and it is a binary solution of A and B elements, I have taken this as alloy system, metallic alloy system and I have taken this as metallic alloy system, metallic alloy I have taken ok. So, I have these equations, I have seen this plot, I have seen del X mix face centred cubic f c c or it a lattice parameter which a it a lattice parameter ok. So, lattice parameter to say number of atoms a number of atoms n a, b number of atom n b are total atom which a n a plus n b equal to Avogadro number which a n 0 Avogadro number fine. So, this is a lattice parameter which a n 0 Avogadro number which a n 0 Avogadro number which a n 0 Avogadro number which a n 0 Avogadro number which a n 0 Avogadro number, ok. . of temperature on del S mix and delta G mix . . greater than 0, positive we have it, . So, finally, minus term , the considered . minus positive of it and if I take x A 1 or x B 1 or x A 0 or x B 0, means at the extreme point where there is pure substance, then I am taking del G del S mix as 0, right? will be a symmetric plot. This is del S mix plot. Now, if I take del G, delta G mix, then it is 0, plus R T x A ln x A plus x B ln x B which is nothing but minus T del S mix. it should be always negative, ok. At extreme point, means 0 and 1, this is x B, x A equal to 1, mole fraction 1, this is x B equal to 1, pure B, pure A. So, it will just be opposite. Fine, ok. So, this is equal to minus T 1 del S mix equal to del G mix at T 1 at T, del S mixture, no minus T 2 del S mix equal to del G mix at T 2 at T 1 at T 2 at T 1 at T 2 at T 1 at T T 2. of temperature on del G del S mix and del G mix effect of temperature on del G mix not del G mix rather G free energy of solution effect of temperature on G1 unmixed condition the melting point of A or melting point of B, if I compare them, melting point of B is more than melting point of A, right? for solid, it is for liquid, we have seen earlier in pure condition, right? And here, these two are equal, that means, it is equilibrium melting point, another one Tm A, ok, and heater value to G A 0 liquid and this one is G A naught solid, temperature less than let us say Tm A, I have taken a temperature here, let me show you, this temperature is here, this temperature, if it can be held, ok. Now, here, if I hold only A and hold only B, ok, that means, I have held A, first I am holding A, second I am holding B, third I am holding A, I am holding B, fourth I am So, the temperature of A is equal to the temperature of solution, right? less than T m a less than T m b before mixing at temperature T which is less than T m a solid standard is the lowest rather lowest lowest hence stable solid stable solid b stable it out solid similarly solid a stable at temperature at T at T relative position the G 1 equal to X a G a solid not plus X b G b solid not ok X a and X b temperature T 1 temperature at T 1 free energy for solid free energy of S not at T 1 G b S not at T 1 at T G G 1 equal to X a G a not S plus X b G b not S T 1 just G 1 equal to X a G a not S X b G b not S at T 2 T 2 G a S not double prime G b S not double prime temperature effect So, that is why I am referencing this free energy temperature plot in the case of pure case. In the case of pure case, I have seen how it varies. It will be a little more complicated when we assign a standard state and the best value we get to assign a standard state is 0. When we assign this 0, the plots will change a little, but the concept will remain the same. So, that is why I am using this as a standard state and the best value we get to assign a standard state is 0. It will be a little more complicated when we assign a standard state and the best value we get to assign a standard state is 0. Thank you.