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More on , , and mixing

Narayanan Kurur

Department of Chemistry IIT Delhi

13 July 2013 More on phase diagram, chemical potential, and mixing changes with P

  ∂Gα = Vα ∂P T

V > 0 =⇒ Gα ↑ when P ↑ Intersection point shifts to higher T because change in Gsol. is less than change in Gliq. when ρsol. > ρliq..

2013-14| Class 9| 13 July 2013 2 / 13 More on phase diagram, chemical potential, and mixing Melting point decreases if ρsol. < ρliq.

Intersection point shifts to lower T because change in Gsol. is higher than change in Gliq. when ρsol. < ρliq..

2013-14| Class 9| 13 July 2013 3 / 13 More on phase diagram, chemical potential, and mixing System of variable composition Consider a homogeneous phase in which there are k different substances. Let n1 be the number of moles of the substance 1 in the phase, n2 of substance 2, etc. If n1, n2, . . . , nk are constant, the Gibbs depends only on S and V . However, for variable composition

G = G(T, P, n1, n2, . . . , nk ), and thus the total differential of G is k ∂G  ∂G  X ∂G  dG = dT + dP + dni ∂T ∂P ∂ni P,ni T,ni i=1 T,P,nj

In the first two partials the subscript ni implies that the numbers of all species are constant. In the last term, the and are constant, together with all but one of the mole numbers.

2013-14| Class 9| 13 July 2013 4 / 13 More on phase diagram, chemical potential, and mixing Definition of Chemical Potential For constant mole numbers dG = −SdT + V dP is valid, and so we obtain ∂G  ∂G  = −S and = V ∂T ∂P P,ni T,ni Let µi be defined by ∂G  µi = . ∂ni T,P,nj Thus we may write k X dG = V dP − SdT + µi dni i=1

2013-14| Class 9| 13 July 2013 5 / 13 More on phase diagram, chemical potential, and mixing Significance of the chemical potential

The quantity µi , 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 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 .

2013-14| Class 9| 13 July 2013 6 / 13 More on phase diagram, chemical potential, and mixing 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 and 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  µi = . ∂ni S,V,nj 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 More on phase diagram, chemical potential, and mixing Temperature dependence of µ We are interested in ∂µ   ∂2G  i = ∂T P,n ∂T ∂ni P,n Because G is a  ∂2G   ∂2G  = ∂T ∂ni ∂ni ∂T P,n P,nj and as a result  ∂2G   ∂ ∂G   ∂  = = (−S) = −S¯i ∂ni ∂T ∂ni ∂T ∂ni P,nj T,P,nj T,P,nj

2013-14| Class 9| 13 July 2013 8 / 13 More on phase diagram, chemical potential, and mixing Pressure dependence of µ

By a similar analysis as with T we can show (can you?) that   ∂µi = V¯i ∂P T,n 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 = ∂ni T,P,nj

2013-14| Class 9| 13 July 2013 9 / 13 More on phase diagram, chemical potential, and mixing Pressure dependence of µ for

RT dµ = V¯ dP = dP. T const. P Integration from a standard state, indicated by a superscript “◦”, yields P µ(T ) = µ◦(T ) + RT ln P ◦ By analogy the chemical potential of a species in a mixture of ideal gases is p µ = µ◦ + RT ln i , i i ◦ pi th where pi is the partial pressure of the i species. Note that the standard state is different at different .

2013-14| Class 9| 13 July 2013 10 / 13 More on phase diagram, chemical potential, and mixing Integration of the basic equation Enlarge the size of the system with its T , P , and the relative proportions of components remaining unchanged. The µi , which are intensive variables like T and P , remain unchanged. Integration of k X dG = V dP − SdT + µi dni i=1 then gives k X G = µi ni . 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 More on phase diagram, chemical potential, and mixing Mixing of gases

Consider the mixing of nA moles A with nB moles of B; both at temperature T and P .

2013-14| Class 9| 13 July 2013 12 / 13 More on phase diagram, chemical potential, and mixing G decreases when gases are mixed

∆Gmix = Gf − Gi f f ◦ ◦ pA pB Gf = nAµ +nBµ = nAµ +nBµ +nART ln +nBRT ln A B A B p◦ p◦ i i ◦ ◦ p p Gi = nAµ +nBµ = nAµ +nBµ +nART ln +nBRT ln A B A B p◦ p◦

p p ∆G = n RT ln A + n RT ln B mix A p B p

∆Gmix = nART (xA ln xA + xB ln xB)

2013-14| Class 9| 13 July 2013 13 / 13