Topic1180 Entropies: Solutions: Limiting Partial Molar Entropies A key equation relates the chemical potential and partial molar entropy of solute-j. For a given solute j in an aqueous solution, =−∂µ ∂ Sjjp() aq [ ()/ aq T] (a) In order to appreciate the importance of equation (a) we initially confine our attention to the properties of a solution whose thermodynamic properties are ideal. A given aqueous solution contains solute j at temperature T and ambient pressure (which is close to the standard pressure). µµ=+⋅⋅00 jj(aq ; id ;;) T p ( aq ;;) T p R T ln( m j / m ) (b)

=−⋅00 Using equation (a), Sjj(;;;) aq id T p S (;;) aq T p R ln(/ m j m ) (c)

0 SaqTpj ( ; ; ) is the partial molar entropy of solute j in an ideal aqueous solution having unit molality [1,2]. Therefore for both real and ideal solutions [3], →=+∞ limit ( mjj0 )S ( aq ; T ; p ) (d) In other words the limiting partial molar entropy for solute j is infinite. Interestingly if the aqueous solution contains two solutes j and k, then the following condition holds for solutions at temperature T and pressure p. →→ − =00 − lim(it mjk00 ; m )[(;;) S j aq T p S k (;;)]() aq T p S j aq S k () aq (e) Similarly →→µµ − = µµ00 − limit ( mjk00 ; m )[ j ( aq ; T ; p ) k ( aq ; T ; p )] j ( aq ) k ( aq ) (f) The partial molar entropy of solute j in a real solution is given by equation (g). = SaqTpj (;;) ∂γln( )  (g) 00−⋅ −⋅γ −⋅⋅ j S jjj(;;)aq T p R ln(/ m m ) R ln() R T   ∂T   p A given aqueous solution having thermodynamic properties which are ideal contains a solute j, molality mj. The partial molar entropy of the solvent is given by equation (h). =+⋅⋅* S11( aq ; id ;;) T p S (;;)l T p R M 1 m j (h) →=* Hence, lim(it mj 0 ) S11 ( aq ; id ;;) T p S (;;)l T p (i) When an ideal solution is diluted the partial molar entropy of the solvent approaches that of the pure solvent.

A given aqueous solution is prepared using 1 kg of solvent and mj moles of solute j at temperature T and pressure p, the latter being close to the standard pressure. The entropy of the solution is given by equation (j). ==⋅−1 +⋅ S(; aq w111 kg ) M S1 () aq mjj S () aq (j) In the event that the thermodynamic properties of the solution are ideal the entropy of the solution is given by equation (k). == Saqidw(;;1 1 kg ) (k) −1 ⋅+⋅⋅+⋅−⋅* 00 M1 [()SRMmmSaqRmm11l jjj ] [( ) ln(/ j )] Interestingly, →== limit ( mj 01 ) S ( aq ; id ; w1 kg ) ( ) −1 ⋅+⋅−⋅* 00l M1 SSaqRm1 ()l ()[00j ( ) ln(/ )] →⋅0 = But limit ( mjjj00 ) m ln( m / m ) (m) In other words the entropy for an ideal solution in the limit of infinite dilution is given by the entropy of the pure solvent. For a real solution,  ∂φ  S (aq;T;p) = S* ( ;T;p) + φ⋅ R ⋅ M ⋅ m + R ⋅ T ⋅ M ⋅ m ⋅ (n) 1 1 l 1 j 1 j  ∂   T  p Hence, = S(aq;w1 1 kg;T;p)

−  ∂φ  = M 1 ⋅[S* ( ;T;p) + φ⋅ R ⋅ M ⋅ m + R ⋅ T ⋅ M ⋅ m ⋅ ] (o) 1 1 l 1 j 1 j  ∂   T  p  ∂ ln(γ )  + m ⋅[S0 (aq;T;p) − R ⋅ ln(m ⋅ γ / m 0 ) − R ⋅ T ⋅ j  ] j j j j  ∂T    p = − = The difference [S(aq;w1 1 kg;T;p) S(aq;id; w1 1 kg;T;p)] yields the E = excess entropy, S(; aq w1 1 kg ;;) T p . Footnotes [1] For a salt solution, the standard partial molar entropy of the salt is given by the sum of standard partial molar entropies of the ions. For a 1:1 salt, 0 = 0 + 0 S j (aq) S+ (aq) S− (aq ) [2] Y. Marcus and A. Loewenschuss, Annu. Rep. Prog. Chem., Ser. C, Phys. Chem., 1984, 81, chapter 4. [3] For comments on the entropy of dilution of salt solutions see (a classic paper), H. S. Frank and A. L. Robinson, J. Chem. Phys.,1940,8,933. [4] For comments on partial molar entropies of apolar solutes in aqueous solutions see, H. S. Frank and F. Franks, J. Chem. Phys.,1968,48,4746.