More on phase diagram, chemical potential, and mixing Narayanan Kurur Department of Chemistry IIT Delhi 13 July 2013
Melting point changes with P ( ) Gα P T = V α V > 0 = G α when P Intersection point shifts to higher T because change in G sol. is less than change in G liq. when ρ sol. > ρ liq.. 2013-14 Class 9 13 July 2013 2 / 13
Melting point decreases if ρ sol. < ρ liq. Intersection point shifts to lower T because change in G sol. is higher than change in G liq. when ρ sol. < ρ liq.. 2013-14 Class 9 13 July 2013 3 / 13
System of variable composition Consider a homogeneous phase in which there are k different substances. Let n 1 be the number of moles of the substance 1 in the phase, n 2 of substance 2, etc. If n 1, n 2,..., n k are constant, the Gibbs energy depends only on S and V. However, for variable composition G = G(T, P, n 1, n 2,..., n k ), and thus the total differential of G is ( ) ( ) G G k ( ) G dg = dt + dp + dn i T P,n i P T,n i n i T,P,n j In the first two partials the subscript n i implies that the mole numbers of all species are constant. In the last term, the temperature and pressure are constant, together with all but one of the mole numbers. 2013-14 Class 9 13 July 2013 4 / 13 i=1
Definition of Chemical Potential For constant mole numbers dg = SdT + V dp is valid, and so we obtain ( ) G = S and T P,n i Let µ i be defined by ( ) G µ i = n i Thus we may write dg = V dp SdT + ( ) G T,P,n j. P T,n i k µ i dn i i=1 = V 2013-14 Class 9 13 July 2013 5 / 13
Significance of the chemical potential The quantity µ i, called the chemical potential greatly facilitates the discussion of open systems, or closed ones in which there are changes of composition. The chemical potential has an important function like temperature and pressure. A temperature difference determines the tendency of heat to pass from one body to another and a pressure difference determines the tendency towards bodily movement. Similarly, the chemical potential is the cause for a chemical reaction. 2013-14 Class 9 13 July 2013 6 / 13
Chemical Potential according to Gibbs Gibbs defined the chemical potential as If to any homogeneous mass we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered. In other words, according to Gibbs ( ) U µ i = n i S,V,n j. Would you be able to show that the Gibbs definition and our definition are the same? 2013-14 Class 9 13 July 2013 7 / 13
Temperature dependence of µ We are interested in ( µi T ) P,n = ( 2 G T n i Because G is a state function ( ) ( ) 2 G 2 G = T n i n i T and as a result ( ) 2 G n i T P,n j = P,n ) P,n P,n j ( ) ( ) G = ( S) n i T T,P,n j n i T,P,n j = S i 2013-14 Class 9 13 July 2013 8 / 13
Pressure dependence of µ By a similar analysis as with T we can show (can you?) that ( ) µi = V i P The temperature coefficient of µ is the partial molar entropy, S, and the pressure coefficient is the partial molar volume, V, where the partial molar properties are defined as ( ) X X i = n i T,n T,P,n j 2013-14 Class 9 13 July 2013 9 / 13
Pressure dependence of µ for ideal gas dµ = V dp = RT P dp. T const. Integration from a standard state, indicated by a superscript, yields µ(t ) = µ (T ) + RT ln P P By analogy the chemical potential of a species in a mixture of ideal gases is µ i = µ i + RT ln p i p i where p i is the partial pressure of the i th species. Note that the standard state is different at different temperatures. 2013-14 Class 9 13 July 2013 10 / 13,
Integration of the basic equation Enlarge the size of the system with its T, P, and the relative proportions of components remaining unchanged. The µ i, which are intensive variables like T and P, remain unchanged. Integration of k dg = V dp SdT + µ i dn i then gives G = k µ i n i. i=1 This derivation depends on the physical knowledge that the intensive variables are not affected by the size of the system, whereas the extensive properties are directly proportional to its size. 2013-14 Class 9 13 July 2013 11 / 13 i=1
Mixing of gases Consider the mixing of n A moles A with n B moles of B; both at temperature T and P. 2013-14 Class 9 13 July 2013 12 / 13
G decreases when gases are mixed G mix = G f G i G f = n A µ f A +n Bµ f B = n Aµ A +n Bµ B +n ART ln p A p +n BRT ln p B p G i = n A µ i A +n Bµ i B = n Aµ A +n Bµ B +n ART ln p p +n BRT ln p p G mix = n A RT ln p A p + n BRT ln p B p G mix = n A RT (x A ln x A + x B ln x B ) 2013-14 Class 9 13 July 2013 13 / 13