Chemical Potential and Osmotic Pressure The theoretical background leading up to the derivation of the osmotic pressure relationship usually follows the sequence: (1) The Gibbs free energy change is zero for a reversible process in a closed system a t constant temperature and pressure involving pressure-volume work only. (2) For a system a t equilibrium and constant temperature and pressure, the chemical potential of a given component is the same in all phases. (3) At equilibrium the chemical potential of the solvent must be equal on both sides of a semipermeahle membrane. Many textbook authors start the osmotic pressure derivation with statement (3) and assume that (3) follows from (1) and (2) without further explanation. But several questions arise. Some authors state that the pressures of all phases are the same for a system in equilibrium, and typically uniform pressure is implied in the derivation of (2). Does constant pressure in statements (1)and (2) mean uniform pressure throughout the system? Does equilibrium in statement (2) require pressure equilibrium'! If w, itatement 131 is incorrect. Alsu we m e rarrful toensure that t1.r trmperatow and presiure o i t h r iinnl .tat(. are thesnme as the inil~nlstatewhen using stntement r l , asscriterion nfspmraneity. I)uw rtatrmmr 1,md wnaequrotly 131 hold if the solvent mows rewrrihlg acmri a membrane a ;trite uf di~frrenrpre.;surc? ~
~
.
Wall' shows that expression (I),which is the basis for statement (11, A G r y t 0 (PVwork only)
(1)
.
holds for a dosed svstem ~, with several mts a t different Dressurea ~rovidedthe nressure in each Dart remains constant. The equality applies tu reversihle processer and the mequalitv to spontnnew~sprocesses. Thus rLn+rnnt pre.rurr in *rnrrmmt (Ir can he interpreted to mean that the pressure of each suhsyiwm remains conrtnnt rather that) the rustenre ~ i u n ~ f u r m pressure throughout the system. Likewise constant pressure has the same meaning in statement 12). Consider the reversible transfer of a component from phase a t o phase P across a semipermeable membrane with the phases a t different pressures. By eqn. (1) we see that AG = 0 for the system, but the free energy of each phase changes due to the transfer of substance in or out of the phase. If the temperature and pressure of each phase remains constant, application of equ. (2) ~
~~~~
~
~
+ Zuidni
dG = VdP - S d T yields
dGr,pa = ui,dni, = uipdnip ~GT,PB
The change for the system is given by the sum dGr,p(sys) = ui,dni,
+ uipdnip = 0
and noting that dni, = -dnia, we obtain (uie - ui,)dnip = 0 Since dnia is not zero UiB
= Uio
l i t h e rompment is solvent, statement (31 is p n w d . 'l'hus with rersrd to SlGlfmpnts I I I and 121,nnd rhr r m ~ r q u m r validity of 0 1 , mita tan^ prrssure means that each phase rrmainr nt the w n r p w w r e . I t dues n
