The Effect of Temperature and Pressure on Equilibria: A Derivation of the van't Hoff Rules H. R. Kemp University College, Australian Defence Force Academy, Campbell, ACT, Australia 2600
For many years, Le Chatelier's principle has received welldeserved criticism (1-6). Simple statements of the nrincinle arc, vague and more pre~iscstatements in terms of intensive and extensive variables (31 arr difficult to remember. Consequently, it has heen strongly argued (1,5,6) that the principle should he discarded in favor of the equilihrium law and two rules of van't Hoff. This is the viewpoint taken in the present article: the equilihrium law can be used to predict the effect of adding reactants or an inert substance to a system initially in equilihrium, and the van't Hoff rules can he used to predict the effect of temperature and pressure. In this way the study of disturbed equilihria is soundly based on rules that can be deduced from the laws of thermodynam-. ics. This paper contains a concise derivation of the van't Hoff rules thnt ~ssuitahlrfur standard physical chemistry courses dealing with thermrdynamirs. Thc actual avolication of the -. rules, however, is suitable for introductory courses in general chemistry that contain little or no systematic study of thermodynamics. The van't Hoff rules for disturbed equilihria can be expressed as follows: Rule 1. For closed systems at constant pressure, initially at equilibrium, temperature and theamounts of substance on the higher enthalpy side of the equation rise and fall toaether. Rule 2. I h r closed sysrems at conatnnt ternpprature, initially at pressurp nnd the ammnts ofsulmtanre on the lower rq~il~lrium, volume side of the equation rise and fnll together. Short forms of these rules are
Then dG = -SdT
+ Vdp
(1)
and d H = TdS
+ Vdp
(2) where G is the Gihbs function, S the entropy, Vthe volume, p the pressure, and H the enthalpy of the system. These relations are two of Gibhs's four differential equations, all of which are derived from the first and second laws and various definitions (8). From eq 1a t constant pressure
Differentiating with respect to extent of reaction 6 a t constant temperature and pressure gives
Since the order of differentiation is immaterial;
In eq 4 aG/a[ is the Gihhs function of reaction a t constant temperature and pressure for equilihrium conditions and aSla[is the entropy of reaction under the same conditions. I t is confusing to symbolize these and aHIa6, as AG, AS, and AH, yet this is often done. From eq 2, at constant pressure d S = aH/T. Combining this with eq 4 gives
1. Higher temperature favors higher enthalpy 2. Higher pressure favors smaller volume.
As an example, consider ice and liquid water in equilihrium.
If the temperature is raised a t constant pressure, then ice melts because liquid water has higher enthalpy. On the other hand, if the pressure is raised a t constant temperature.' then ice melts because liquid water has smaller vofume. The two rules can he deduced from the first and second laws of thermodynamics. One such derivation, given by A. R. Bent is by way of addition of chemical potentials combined with the second derivative of the Gihhs function with respect to extent of reaction. The difficult nature of these concepts is possibly the reason why this derivationis notwell known. The derivation that follows is simpler. I t is applicable to hoth chemical reactions and phase changes.
(n,
Derivation Of the van't Hoff Rules for Disturbed Equillbrla
Consider a system in thermal and mechanical equilihrium with its surroundings. No work is done by way of electric currents, only by way of change in volume of the system. 482
Journal of Chemical Education
For a system a t a given temperature and pressure, i t follows from the first two laws (9)that the condition for advance of reaction is
for retreat
and for equilihrium
The effect on composition of altering the temperature at constant pressure is most easily predicted from a sketch of aGlac shown as a function of T. Figure 1 gives the sketches for (a) endothermic and (b) exothermic reactions. At the equilibrium temperature, aG/ a[ = 0 for hoth reactions. For dHl@ > 0, it follows from relation 5 that the slope of the curve in Figure l(a) is negative. This means that aG/a[ becomes negative when the
\
0
initial equilibrium \ temperature
/
/
\
/ /
P ' i n i t i a l equilibrium I " ,
/
I
/
pressure
initial equilibrium
\
\
pressure
P [initial equilibrium temperature
Figure 1. The change in Gibbs function of reaction with changes in temperature at constant oressure for svstems initiaiiv When aH/JL: is . in eouiiibrium. . por,live, "crease n lemperalure makes JGIdt negat ve. whereas, when dHl d l s negative dG/dt becomes posnt:ve. Hence, the effect of an increase in temperst~reis lo make lhe reacllon shill nine direction of higher snlhalpy.
temperature is increased. Hence, from relation 6 the reaction will advance, thereby increasing the enthalpy of the system. Similar reasoning applied to the exothermic reaction (Fig. l(b)) and to drops in temperature give the generalization of Rule 1. The effect of a change in pressure a t constant temperature can be deduced as follows: . From eq 1, a t constant temperature
Differentiating with respect t o ( a t constant temperature. and pressure gives
Since the order of differentiation does not matter, this can he writtren as
This gives the rate of change of the Gihbs function of reaction with pressure a t constant temperature. The significance of the relation 10 is most easily seen from Figure 2 which shows dG/at as a function of p. From eq 10, the slope of the curve is positive when aVlat >
Figure 2. The change in Gibbs function ol reaction with change in pressureat constant temperature far systems initially in equilibrium. When aV/J€ is positive, increase in pressure makes JG/a€ positive. On lhe other hand, when d v l d t is negative increase in pressure makes J W d t nqativs. The result is that increase in Dressure favors a shin in the reaction toward smaller volume.
0.Since aG/dt = 0 a t equilibrium, it must become positive when the pressure is increased. Hence, the reaction must retreat so that the volume of the system decreases. Similar reasoning applied to reaction in which aVldt < 0 and t o drops in pressure gives Rule 2. This completes the train of reasoning from the first two laws of thermodynamics t o the van't Hoff rules. Discussion The derivation of the van't Hoff rules shows that i t is the differential quantities of reaction, aHlat and aVla€, a t equilibrium that are relevant t o the effect of the temperature and pressure changes on extent of reaction rather than the integral molar values, which may have opposite signs to the differential quantities. For example, the integral molar enthalpy of solution to infinite dilution for NaOH(s) is negative at room temperature, but, for the crystalline form NaOH.H20 that is in equilibrium with the saturated solution, aHlat is positive. I t is the latter value that allows Rule 1 t o give the correct prediction that increasing the temperature increases the solubility (4). The restriction of the van't Hoff rules to closed systems is necessary even when the substances added or removed are inert. For example, consider the rise in pressure that occurs a t constant temperature and volume when an inert gas is added to a vapor in equilibrium with its liquid. Ideally, the equilibrium will not he disturbed by the rise in pressure. (The small effect of pressure on the activity of the liquid is Volume 64
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483
ignored.) On the other hand, consider the addition of the inert gas a t constant total pressure. Because of the increase in volume, more vapor is produced although the total pressure, the temperature, and the equilibrium constant are unchanged and no reactants or products are added. Relation 4 can be used to deduce the rule that the higher entropy side of the equation is favored by increase in temperature for a closed system a t constant pressure. This rule is a useful alternative to Rule 1 when the entropy change is known for the reaction. Summary
For closed systems in equilibrium, the effect of temperatureor of pressure on the compos~tioucan he found from the rules that higher temperature favors higher enthalpy and
484
Journal of Chemical Education
higher pressure favors smaller volume. These rules can be derived from the first two laws of thermodynamics and apply to both ideal and real systems. The rules are simpler to use than Le Chatelier's principle and, unlike the principle, give reliable predictions.
~n~~~~~~~ cited 1. deHeer.J. J . Chem.Educ. 1957,34337-339. 2. Bdner. George M.J.Chcm. Educ. 19M.57.111-110. a. ~ r e o t o w . ~ i c h a r dJs .ch~m.~duc. LSRO.S,.~T~-UL , , ~-~ ~4. ~ernand;z-~rini,R. J . Chem. Edue. 1982.59.55%553. 5. A I I ~ PR., T.:G ~N. H. ~E ~ U C .~chpm ~1984, M ~ ~ ~, ~C 5~ 6, . 6. Gold, Jean; Gold. Vietor. Educ Chem. 1985.82-85. 7. B ~ . R, A he SecondLow:oxford:NPW YOIL, 1965:pp271-274. 8. Pieer. Kenneth S.: Brewer, Leo. Thermodynamics, Irevision of Lewis and Randall). 2nded.; McGrsw-Hill: New York, 1 9 6 1 ; 103. ~ 9. R , r s , ~ u 3 . ~
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