5. Effect of Temperature on Reactions

The relationship between ΔG° for a reaction and K is given by:

\[
\Delta G^\circ = -RT \ln K
\tag{443}
\]

The temperature dependence of the equilibrium constant can be found by differentiating:

\[
\frac{d\ln K}{dT}
= -\,\frac{d}{dT}\!\left(\frac{\Delta G^\circ}{RT}\right)
\tag{444}
\]

From thermodynamics we get the expression,

\[
\frac{d}{dT}\left(\frac{\Delta G^\circ}{T}\right)
= -\frac{\Delta H^\circ}{T^2}
\tag{445}
\]

Therefore,

\[
\frac{d\ln K}{dT}
= \frac{\Delta H^\circ}{RT^2}
\tag{446}
\]

Stated another way, this gives:

\[
\frac{d\ln K}{d(1/T)}
= -\frac{\Delta H^\circ}{R}
\tag{447}
\]

where, again, ΔG° and ΔH° refer to free energy and enthalpy changes, respectively, for a reaction, under standard conditions. The last expression can be integrated if the temperature dependence of ΔH° is known. For reactions involving neutral (uncharged) species ΔH° is almost constant over a modest temperature range. Then we obtain, after integration,

\[
\ln K_2
= \ln K_1
– \Delta H^\circ\left(\frac{1}{T_2} – \frac{1}{T_1}\right)
\tag{448}
\]

where K₂ is the equilibrium constant at temperature T₂, and K₁ the equilibrium constant at T₁. Temperature is on the Kelvin scale. If K and ΔH° are known at one temperature, then K can be estimate at another temperature.

However, for reactions involving ionic species ΔH° for the reaction depends strongly on temperature and the simple equation above cannot be used. This is severely limiting for reactions of interest in hydrometallurgy. Certainly we still know the qualitative effects of ΔH°, as per Le Chatelier’s principle. There are methods available to determine ΔG° as a function of temperature and hence equilibrium constants, for reactions involving ions.

There are also empirical correlations available for ΔG° as a function of temperature over limited temperature ranges. These take the form:

\[
\Delta G^\circ
= a + bT
= -RT \ln K
\tag{449}
\]

where a and b are empirical constants, determined by experiment. These are valid up to about 75°C. A few examples are listed in the Table below.

Table 15. Variation of G° with temperature for some reactions of
hydrometallurgical interest. T is in Kelvin.
Reaction Constants for G° = a + bT
a J/mol b J/mol K
H2O = H+ + OH-   55840   80.70
0.5Al2O3 s + 2.5H2O = [Al(OH)4]- + H+   62420 -69.96
Al(OH)3 s + OH- = [Al(OH)4]-   72830 1441.2
PbCl2 s = Pb+2 + 2Cl-   22980    13.85
CuO s + 2H+ = Cu+2 + H2O l -63760    73.22
CuS s + 2H+ = Cu+2 + H2S aq   74070    42.25
NiS s + 2H+ = Ni+2 + H2S aq -10700    90.17
Ni(OH)2 s = Ni+2 + 2OH-   14000   265
HSO4- = H+ + SO42- -21930    111.7

Where reactions can be added to obtain a new one, the ΔG° of the new reaction is the sum of the ΔG° values for the individual reactions. For example at 75°C,

\[
\ce{CuO(s) + 2H^+ = Cu^{2+} + H2O(l)}
\qquad
\Delta G^\circ = -38{,}269\ \text{J·mol}^{-1}
\tag{450}
\]

\[
\ce{2H2O(l) = 2H^+ + 2OH^-}
\qquad
\Delta G^\circ = 2 \times 83{,}936\ \text{J·mol}^{-1}
\tag{451}
\]

\[
\ce{CuO(s) + H2O(l) = Cu^{2+} + 2OH^-}
\tag{452}
\]

\[
\Delta G^\circ
= -38{,}269 + 2(83{,}936)
= 129{,}603\ \text{J·mol}^{-1}
\tag{453}
\]

\[
K_{sp}
= \exp\!\left(
-\frac{129{,}603\ \text{J·mol}^{-1}}
{8.314\ \text{J·mol}^{-1}\text{K}^{-1} \cdot 348.15\ \text{K}}
\right)
= 3.6 \times 10^{-20}
\tag{454}
\]

The case of bisulfate is important. Sulfuric acid is the most important acid in hydrometallurgy. Bisulfate is a weak acid, and it becomes weaker as temperature increases. At 0°C pKa₂ = 1.64, and at 75°C it is 2.54.

Precipitation of oxides and hydroxides is an important consideration in hydrometallurgy. Sometimes this is pursued as a way to recover a metal in this form, or remove an unwanted one (e.g. Fe⁺³), and sometimes care must be taken to avoid it. Take the case of Cu⁺² and CuO again. For the reaction:

\[
\ce{CuO(s) + 2H^+ = Cu^{2+} + H2O}
\tag{455}
\]

\[
K = \frac{[\ce{Cu^{2+}}]}{[\ce{H^+}]^{2}}
\tag{456}
\]

\[
\log K
= \log[\ce{Cu^{2+}}]
– 2\log[\ce{H^+}]
\tag{457}
\]

\[
\log K
= \log[\ce{Cu^{2+}}]
+ 2\,\text{pH}
\tag{458}
\]

\[
\log[\ce{Cu^{2+}}]
= \log K – 2\,\text{pH}
\tag{459}
\]

The last equation is similar in form to that for the pH of a buffer comprised of a metal oxide and soluble metal salt; the two should not be confused! The solubility of CuO as a function of pH and as a function of temperature can be determined.

(a) What is the solubility of CuO (expressed as [Cu⁺²] in M) at 25°C and pH 3.5?

\[
K = \exp\!\left[-\frac{a + bT}{RT}\right]
\tag{460}
\]

At 25°C,

\[
K
= \exp\!\left[
-\frac{-63{,}760 + 73.22 \times 298.15}
{8.314 \times 298.15}
\right]
= 2.22 \times 10^{7}
\tag{461}
\]

\[
\log[\ce{Cu^{2+}}]
= \log(2.22 \times 10^{7})
– 2(3.5)
= 0.346
\quad\Rightarrow\quad
[\ce{Cu^{2+}}] = 2.22\ \text{M}
\tag{462}
\]

(Using sulfuric acid we would obtain <2.2 M due to the solubility limit for CuSO·5H₂O, but is not important here.)

(b) How much of this Cu+2 is precipitated as CuO by raising the pH to 4.5 (decreases the H+ concentration by a factor of only 10)?

At pH 4.5, log[Cu⁺²] = -1.654  (0.0222 M). The solubility of Cu⁺² has dropped by:

\[
\frac{2.22 – 0.0222}{2.22} \times 100
= 99\%
\tag{463}
\]

(c) If a solution containing 2.22 M Cu+2 at 25°C is raised to 100°C, how much of the Cu+2 will precipitate?

(The ΔH° for the reaction is -63.76 J/mol at 25°C. Since this is negative, raising the temperature should lower the equilibrium constant, as per Le Chatelier’s principle.)

K at this temperature can be found using the a + bT relationship; this is a bit of an extrapolation, since the temperature is a bit higher than the range over which the relationship is known to be valid.

\[
K = 1.26 \times 10^{5}
\ (< 2.22 \times 10^{7})
\tag{464}
\]

\[
\log[\ce{Cu^{2+}}]\ \text{at pH 3.5}
= -1.90
\quad\Rightarrow\quad
[\ce{Cu^{2+}}] = 0.0126\ \text{M}
\tag{465}
\]

The change in concentration is -99.4%, i.e. 99.4% of the Cu⁺² precipitates. Thus the combination of temperature and pH effects can have a strong influence on solubility. Such effects become even more pronounced in autoclave leaching vessels where temperatures can be as high as 300°C.

As an illustration, iminodiacetic acid ion exchange resins contain the functional group -NH-(CH₂CO₂H)₂ bonded to a polystyrene polymer. The acetic groups can ionize (weak acid) to form a dianion group. This plus the trivalent nitrogen atom complexes to Cu⁺² quite strongly and is a viable means of recovering cupric ion from dilute aqueous solution streams. And while the resin operates best at pH ≥ 4, precipitation of Cu⁺² can limit its use to pH 4 or less.