Ions in Water Solution

Before we can get into the thermodynamics of electrochemistry we have to take a look at how we deal with the chemical potentials of ions in water solution. For example, we know that soluble ionic compounds are completely ionized in water,

(1) NaCl(*aq*) →
Na^{+}(*aq*) + Cl^{−}(*aq*).

(A better way to say this is that when ionic compounds dissolve they are completely dissociated into ions. This way of stating it includes compounds like AgCl, which are not regarded as soluble ionic compounds, but do give very small concentrations of ions in water solution.)

The chemical potentials must be consistent with the formation of ions,

(2)

which also implies that

(3) .

We also know that we can (must?) write

(4)

When working with ionic solutions we would like to be a little more specific about the activities of the species in solution. It is customary to use units of molality, *m*, for ions and compounds in water solution. If the solutions were ideal we could write

(5) *a _{i}* =

and the chemical potential would be written

(6)

Two things must be said about this equation. First, it must be understood that there is an implied *m _{i}*

However, ionic solutions are far from ideal so we must correct this expression for chemical potential for the nonidealities. As usual, we will use an activity coefficient, γ
, and write the activity as

*m _{i}γ
_{i}* /

and consistent with what we have been doing, we will set

(7) .

The standard state for this equation is a hypothetical standard state. The standard state is not actually realizable. We are using the so-called Henry's law standard state in which the solution obey's Henry's law in the limit of infinite dilution. That is,

(8) γ
* _{i}* →
1 in the limit when

(There most likely is a concentration where *m _{i}γ_{
i}* = 1, but, interestingly enough that would not be the standard state. The fact that at that concentration the chemical potential would equal the chemical potential of the standard state is a coincidence.)

In what follows we will rarely, if ever, deal with more than one positive and one negative ionic species in the solution. In this case we will simplify matters by writing *m*_{+} , γ
_{+} and *m*_{−} ,γ
_{−} for the molalities and activity coefficients of the positive and negative ionic species,
respectively.

We will also refer to the ionic compound simply as the "salt." With this notation we can rewrite Equation 2 as,

(9)

or

(10) .

Equation 9 is also true when all the components are in their standard states, so we can write,

(11) .

Combining equations 10 and 11 we see that

(12) ,

from which we conclude that

(13) .

The activity coefficients, γ
_{+} and γ
_{−} can't be measured independently because solutions must be electrically neutral. In other words, you can't make a solution which has just positive or just negative ions. You can't calculate the individual activity coefficients from theory, either. However, you can measure a "geometric mean" activity coefficient and, within limits, you can calculate it from theory. (We will show how to do both of these things later.) The actual form of the geometric mean depends on the number of ions produced by the salt. Right now we will define it for NaCl and then give more examples later. For NaCl we define,

(14)

or

(15) .

So,

(16) .

But for a NaCl solution of molality, *m*, we have *m*_{+} = *m* and *m*_{−} = *m* so that

(17) .

Keep in mind that this for is for NaCl, but it is correct for any one-to-one ionic compound.

Try MgCl_{2},

(18) MgCl_{2} →
Mg^{2+} + 2 Cl^{−}

(19) ,

so

(20)

But for MgCl_{2} at molality, *m*, we know that *m*_{+} = *m* and *m*_{−} = 2*m*. Further, we define the geometric mean activity coefficient by,

(21) .

Then

(22)

Let's do one more, LaCl_{3}.

(23) LaCl_{3} →
La^{3+} + 3 Cl^{−}

(24) ,

so

(25)

But for LaCl_{3} at molality, *m*, we know that *m*_{+} = *m* and *m*_{−} = 3*m*. Further, we define the geometric mean activity coefficient by,

(26) .

Then

(27)

Other ionic compounds are done in a similar manner. After some practice you can probably figure out the expression for a_{salt} just by looking at the compound. Until then, or when in doubt, go back to the expressions for chemical potentials as we have done here. (For practice you might want to try Al_{2}(SO_{4})_{3}.) We will apply all of this by looking at the:

Relationship between K_{thermodynamic} and K_{molality}

Use an example to show this relationship;

(28) AgCl(*s*) →
Ag^{+}(*aq*) + Cl^{−}(*aq*).

(29)

Then,

(30) .

If we are just dealing with AgCl in water the concentrations of the ions are pretty small so γ_{
±}
is approximately unity. If, however, we are dealing with the common ion effect, for example, the solubility of AgCl in 0.10 m HCl, then the γ_{
±}
for the ions in this stronger solution is not equal to one, not even approximately so.

Which takes us to the:

Debye-Hückel Limiting Law (DHLL)

Theoretical calculation of γ_{
±}
.

The Debye Hückel limiting law gives the γ_{
±}
in terms of the **ionic strength**, *I*, defined as,.

(31) ,

where *z _{i}* is the charge on ion

Examples,

0.010 m HCL

(32) ,

0.10 m Na_{2}SO_{4}

(33)

We must include **all** ions in the solution. The ionic strength is a measure of the total concentration of charge in the solution. Notice that it includes contributions from both the number of ions in the solution and the charges on the individual ions. Look at Equations 33 to see how the −2 charge on the SO_{4}^{2−} ion contributes to the ionic strength.

Test yourself on LaCl_{3} .

We won't derive the Debye-Hückel Limiting Law. We will just give the result without proof. DHLL says,

(34) .

A more accurate version is,

(35) .

The *B* and *a*_{o} are parameters from the details of the theoretical model, such as the dielectric constant of water, sizes of ions, and so on.

It is not unusual to see common logarithms used instead of natural logarithms, as in,

(36)

and its other variations given above. I suspect that this is because, before the advent of electronic calculators, people used tables of common logarithms to carry out calculations involving many multiplications and divisions.

WRS

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Last updated 22 Oct 04

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