N. J. Selley
of lfe He-lfe, Nigeria
Entropy of Mixing-
University
An Electrochemical Measurement
Recently there has been considerable interest in means of introducing chemical thermodynamics in a meaningful way. J. A. Campbell, writing in THIS JOURNAL (1) expressed regret that "too little use is made of simple statistical ideas as A S = R in (W2/Wl), ideas which students find much more useful and applicable than simple adherence to A S = q,,,/l'." The present paper describes an experiment and its preparatory theory which might be found useful for developing and reinforcing these ideas. Most standard textbooks of thermodynamics give an expression for the entropy of mixing ideal gases, and d i e r only in the notation used. For example, Fowler and Guggenheim (2)give the general equation
for the mixing of N A , N B molecules of gases A,B, a t constant temperature, and with original gases and final mixture at the same pressure. Similar equations apply to ideal solutions, and in the case of two solutes only, the expression for the mixing of Nl moles of solute 1 with N z moles of solute 2 (at constant mean concentration) is simply eqn. (2)
A derivation of this from first principles is not usually included in the discussion, but this has the advantage of providing a valuable exercise for the student (after suitable guidance). A possible route is suggested below. I t follows in part the treatment by Braunstein (3) in an article which also gives admirably clear explanations of the terms states and distinguishability. Since the enthalpy of mixing is zero for ideal gases and solutions, the free-energy of mixing is simply AGmi, = -TAS,i.
(3)
Thus eqn. (2) could be tested experimentally if there were any means of measuring the maximum reversible work obtainable from the mixing. Unfortunately, this is not practicable for most solutions (nor for gases). However the worlc of any reaction between aqueous ions is measurable by potentiometry, and there is no reason why this should not apply to the process of mixing.
between M+ and M2+ (there heing no intermediate oxidation state of M), but if an electrochemical cell were set up consisting of complementary mixtures linked by a salt bridge, with inert electrodes, there would he a potential difference. One such cell, which makes the point about work of mixing, wonld he Pt(l) 10.9 mole/l M + 0.1 m&/l 0.1 mole11Mg+ 0.9 molell M
1
Platinum electrode no. 1 wonld be the more negative due to the tendency of the reaction Mt(aq)
-
Ma+(sq)
+ e-
to occur there, where the concentration of M + is high relative to M2+,and to reverse a t electrode no. 2 where the concentration of M + is low. The cell potential reflects the tendency for the two ions to try to equalize their concentrations, since electron transfer provides a possible means of doing this. ( v = the stoichiometric coefficient of electrons, = 1 in this instance) The differential form is eqn. (5), where c = the amount of electricity passed.
By eqn. (4) the cell potential E is connected with the work done during the transfer of one mole of M Z +from half-cell 2 (where it is in excess) to half-cell 1, and one mole of M + from half-cell 1 to half-cell 2. Noa consider the sequence of events illustrated in Figure 1. The free-energy change A G for the complete mixing will equal the sum of the free-energy changes for any number of steps such as the two shown. Further, the free-energy changes are the same whether work is done or not. If the solutions are simply mixed together, no worlc is harnessed; but in the electrochemical cell described above the same mixing occurs as a result of the electrode reactions, while electrical work is produced. Under reversible conditions this work equals the free-energy change, and so the cell potential is given by eqn. (5).
The Complementary Mixtures Cell ,x?.n.,e
If salts are available with a common anion but with the cations heing the same metal in two different oxidation states, say IN+and MZ+,it would he possible to make up solutions with these salts mixed in any proportions. There would obviously be no chemical reaction 212
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Journol of Chemicol Education
Figure 1.
,OL",,ON,,
Mixing in stager.
"I.
I
As the cell runs, the passage of nl mole electrons is accompanied by, an increase of nl mole M2+and a decrease of nl mole M + in half-cell no. 1, and complementary changes in half-cell no. 2. Thus 1/2 mole electrons ( F / 2 coulomb) would suffice to bring about complete mixing in both half-cells (when the ions differ in charge by one unit), since then nl = (1 - nl). ' dn,
I n practice the mixing process would not be followed by waiting for electron-flow to occur, but by preparing a series of solutions at successive stages of mixing, for example 19:1, 9:1, 8:1, etc. through to 5:5. The cell ~otentialwould decrease as mixing- became more complete (4. Now if (and it is a big "if") the ionic solutions behave ideally, AS,,, and AG,,, can be replaced by eqns. (2) and (3). The maximum work available from the partial mixing of 1 mole of ions (nl of one sort and 1 - nl of the other) in each of the half-cells is given by eqn. (7); the factor 2 arises from the addition of the work from the two half-cells. \
,
E dn*
=
2RT[n, Inn,
+ (1 - n,) ln (1 - ndl (7)
-2RT d i n n, :. lE( = ---Inl -- + Inn,
F
d n,
+ (1 - nd d l n ( 1n,- n , ) In (1
-
- nJl
This' can be tested directly by experiment (5). The results given below for hexacyanoferrate mixtures show good agreement with theory. Also, the graphical integration to obtain AG,,, (over the experimental range of concentration ratios) gives a result within 2% of the theoretical. Suggestions for the Derivation of the Entropy of Mixing Equation
The thermodynamic probability W of N molecules of an ideal gas or solute is given by eqn. (9), where N* identical molecules are all in the ith energy state, and are indistinguishable
1
It will he noticed that, this expression for E is consistent with
the familis equations for the variation of electrode potential with concentration, in the case where v = 1, and the activity coefficients are sufficiently close that their ratio may be neglected. Writing reduction potentials:
and similarly
Therefore of the N! possible combinations of the N molecules, Nt! combinations are indistinguishable, and this applies to every energy state. Therefore md~stmguishsblecombinations distinguishable combin&ms
W = . . .total combinations
Now consider N1 molecules in every ith level being changed, by mixing, to (Nn)
