jueves, 28 de enero de 2010


Wayne E. Steinmetz
Chemistry Department
Pomona College



The First Law of Thermodynamics places important restraints on the path that can be taken by a system but it does not define the path.  For example, the First Law does not rule out the possibility of warming oneself by sitting on a block of ice.  From the perspective of the First Law, this process is feasible as long as the thermal energy gained by your body equals in magnitude the thermal energy lost by the ice.  Yet we know that this process does not occur.  In fact, the reverse is the natural process and we would cool ourselves by sitting on the  ice.  Clearly something very important is missing from our thermodynamic apparatus.  It is the Second Law of Thermodynamics, the most important of the three.

In this course we shall not discuss the Third Law which makes statements about the behavior of materials at low temperature and the impossibility of reaching absolute zero in a finite number of steps.

We shall introduce the Second Law from the molecular level as this approach clearly indicates the important message.  Natural processes in nature are a consequence of blind chance and within the constraints imposed by the conservation of energy lead to an increase of disorder of the entire universe.


In order to clearly present the molecular basis for the drive to equilibrium, we shall consider a very simple model.  The results from this model can be extended to more general systems but a great deal of mathematics is required to reach this goal.  We shall be content in this discussion with the simple model and accept the claim that the extension to more complicated systems has been rigorously established.  A full discussion of the molecular basis for thermodynamics is Statistical Mechanics.

Consider two isomers, i.e. two compounds that have the same molecular formula but a different molecular structure.  Let us assume for the moment that the two isomers have the same electronic energy in the lowest or ground state.  Suppose further that the isomeric molecules are imbedded in a crystal maintained at a temperature close to 0 K.  Hence, on a reasonable time scale, the molecules do not diffuse, do not rotate, and have the lowest possible vibrational energy.

The Heisenberg Uncertainty Principle prevents us from completely suppressing vibrational motion.  However, a molecule that has the lowest possible vibrational energy possesses only one mode of vibration, irrespective of the identity of the molecule.  Locating the molecules in a crystal allows us to circumvent the Pauli Exclusion Principle.  The crystal lattice labels the molecules and the Pauli Principle only applies to indistinguishable species that cannot be labeled.

Hence, we have two molecular states corresponding to the two isomers.  Suppose that we call them H and T.  The choice of the notation is deliberate.  Our chemical problem is isomorphous with a well known problem in statistics, the outcome of flipping a fair coin.  A coin has two states, Heads (H) and Tails (T).  In the case of a fair coin, they are equally probable.  That is, 50% of the time, a fair coin when flipped will yield Heads and 50% of the time Tails.  Similarly with our molecular example, since the two isomers are isoenergetic (at least in the vicinity of 0 K), a single molecule will be present as the H and T isomers with equal probability.

Suppose that we have two molecules or two coins.  The following four permutations or arrangements are possible:

HH    HT    TH     TT

where HT indicates that the first molecule (coin) is in the H state and the second in the T state.  In the parlance of statistical mechanics, these arrangements are called microstates.  Each microstate is consistent with the total number of molecules and with conservation of energy.  The fundamental postulate of statistical mechanics states that the microstates are equally probable.

Normally in chemistry, we are not interested in which molecule is in which state but in the mole fraction of molecules in a state.  Similarly, in a gambling game, we are primarily concerned about the fraction of coins that come up heads.  To address this question, the four microstates are rearranged below in three sets of microstrates.  The sets are enclosed in braces and are called configurations:

{HH}   {HT,TH}    {TT}

Note that one of the sets, the middle one corresponding to 50% H, has twice as many microstates as the other two.  Since microstates are equally probable, the middle configuration is twice as probable as the other two.  This should come as no surprise.  In the language of the gaming table, there are two ways of getting one Head and one Tail but only one way of obtaining all Heads or all Tails and so  the fraction 50% H is the most likely outcome.  However, one would not be surprised if on two tosses of a coin to obtain Heads on both tosses.  This outcome, although less likely than the most probable outcome of 50% H, will still occur 25% of the time.  In the molecular world, this translates to the result that the most likely and also the average concentration is 50% H.  However, the other concentrations, 0% H and 100% H, would be expected to occur with a significant probability.  That is, as the system evolves in time and individual molecules randomly change state, fluctuations in concentration about the average should be observable.  This is exactly what is observed when scientists observe the properties of small numbers of molecules.  These type of measurements have only been possible with the recent improvement of detection systems.

Now, suppose that we play the game with 10 coins or 10 molecules.  This is still a very small number of molecules or a very modest investment in pennies.  However, there is a large increase in the total number of microstates, i.e. the number of arrangements of the coins or molecules.  The total number is 210 or 1024.  Note that this number is two orders of magnitude greater than the number of objects.  This increase in the number of microstates is an important feature of the problem.

The results of the calculation are shown in the figure below.  Since the number of microstates denoted as the variable W in each configuration can be a large number, the results are normalized.  That is, the vertical axis on the graph is W divided by the Wmax, the maximum value of W.  Similarly, the number of molecules in the H state is also normalized and the fraction in the H state is plotted on the horizontal axis.  This fraction is also known as the mole fraction and is often symbolized by X.  Note that one configuration, 50% heads or XH = 0.50, is clearly more likely than the other configurations.  This configuration, the one with the largest number of microstates, is called the dominant configuration.  The probability of  finding a configuration different than the dominant configuration decreases rapidly as one leaves the dominant configuration.  Note that the curve resembles the Gaussian curve used to analyze random error.  This is not an accident and is a consequence of the Central Limit Theorem, a very important result in statistics.

Figure 1. Case with 10 molecules.

The results for the case of 100 molecules, still a very small number, are shown in the next figure.  Notice that the curve has sharpened considerably.  The total number of microstates is now 2100 or 1.3 x 1030.  Most of these belong to the dominant configuration or configurations very close to the dominant configuration.  In other words, as the system of 100 molecules evolves in time and hops from microstate to microstate, it will almost invariably be found very close to the dominant configuration simply because that configuration and ones close to it are overwhelmingly more probable.  The point of equilibrium where XH = 0.50 is thus seen as the consequence of blind chance.  Furthermore, fluctuations from equilibrium resulting from the system being in a configuration measurably different from the dominant configuration become very improbable as N, the total number of molecules, increases.  The emergence of the dominant configuration as overwhelmingly more probable than any other provides a great simplification for the study of a system at equilibrium. At equilibrium, we can ignore configurations other than the dominant configuration as they make negligible contributions to thermodynamic state functions.   We have now entered the regime where the laws of macroscopic thermodynamics apply.

Figure 2. Case of 100 molecules.

We have drawn important conclusions about the nature of chemical equilibrium from a careful examination of a simple model.  The critical reader may wonder if the model is too simple.  The extension of the coin-flip model to more realistic chemical systems requires considerable mathematical effort but the striking result of the model, the emergence of a dominant configuration as N increases, still holds.  However, two extensions of our model are required.  First, many energy states are possible for each molecular species.  Hence, the problem of calculating the number of microstates for a particular configuration can be a challenging problem.  The mathematics used is the field of combinatorics.  Second, different molecular species usually do not have the same energy.  The fact of life can be handled in a system possessing a fixed number of particles N and total energy E with the requirement that we only accept microstates that satisfy the constraints on energy (i.e. the sum of the energies of the individual molecules must add up to E) and N.  The more realistic chemical system is analogous to flipping an unfair coin or rolling loaded dice where by design Heads and Tails are not equally probable.  Consequently, the position of equilibrium is not simply a consequence of blind chance as the constraints on the system from the First Law of Thermodynamics and the Law of Conservation of Mass must be satisfied.  As it evolves, the system will stumble upon the dominant configuration simply because it is more probable, but the actual position of the dominant configuration will be a consequence of the design of the playing pieces as well as the number of ways of arranging the playing pieces.

Our general model which views chemistry as a game of rolling loaded dice shows that the position of equilibrium is a consequence of two issues: weighting and combinatorics.  Ludwig Boltzmann, the founder of statistical mechanics,  showed for a system with a fixed number of particles that the weighting factor depends on temperature and energy.  If one compares the relative probability of a single molecule in two states, A and B, the famous Boltzmann equation states:

pB/pA = exp(-E/RT)  (1)

where pB is the probability that a molecule will be in state B and E = EB – EA is the energy difference in the two states in Joule/mole.

If the energy difference in the Boltzmann equation is expressed in Joule per molecule, the factor of R in the argument of the exponential function should be replaced with the Boltzmann constant, kB = R/NA = 1.38066 x 10-23 J/K-molecule.

The Boltzmann equation shows that energy differences can be very important at low temperatures but are unimportant in the limit of very high temperatures where the ratio of probabilities is one.

Although the energetic constraints expressed by the Boltzmann equation are important, the number of microstates, W, is more important.  The elementary examples illustrated in this handout show that W quickly becomes astronomically large.  For example, the number of ways of arranging 1000 coins in the configuration 50% Heads far exceeds the total number of particles in the entire universe!  Consequently, instead of dealing with W which is unwieldy (try calculating 1000! with your calculator), chemists instead deal with its natural logarithm which has several virtues.  It increase monotonically with an increase in N and the logarithm of W leads to important mathematical simplifications.  For this reason, Boltzmann proposed a statistical mechanical definition of entropy which can take the place of W:

S = kBln(W)   (2)

Equation (2) might create the impression that absolute entropies are achievable.  This is not the case.  Entropy can only be determined to within a constant.  In the case of equation (2), this constant has been arbitrarily set to zero.

In summary, several important conclusions can be drawn from our short introduction to statistical mechanics.

1) The laws of thermodynamics only apply to a system of a large number of particles where the dominant configuration is overwhelmingly more probable than configurations that are measurably different

2) Thermodynamic state functions are not defined for a system of a small number of particles.  In this case, all configurations have comparable probability and large fluctuations are observable.  You will observe this in the experiment on radioactive decay.  A single molecule therefore does not have a temperature!

3) For a large number of particles (for many purposes N = 1000 may be large enough), the position of equilibrium is a consequence of blind chance and corresponds to the configuration with the maximum number of microstates subject to the constraints posed by energetics and conservation of mass.

4) Entropy, defined from the molecular perspective in equation (2), is the most important thermodynamic state function.

We have started with a molecular view of entropy rather than the traditional approach because chemists study molecules, not heat engines.  We have an intuitive feeling for molecules and our understanding of thermodynamics benefits greatly from the molecular approach.  Before moving on to the macroscopic state of the Second Law of Thermodynamics which can be shown to follow from equation (2), it is worth while to discuss entropy data in light of the molecular perspective.  We shall see that our simple model permits us to understand a wide range of experimental data.  Strictly speaking, entropy is defined by equation (2); the entropy of a configuration is the logarithm of the number of ways of arranging the system consistent with the constraints on the system.  However, it is often convenient to tie our full understanding of a concept to a single word.  In the case of entropy, that word is disorder.  Compare then a perfect crystal with an ideal gas.  In the former case with long-range order, the molecules are locked into a single microstate and an entropy of zero follows from equation (2).  In the latter disorderly case, the molecules are moving randomly and a large number of microstates ispossible.  Furthermore, these microstates are populated as the system which is dynamic evolves in time.  One cannot define the entropy of a single deck of cards.  Entropy, the most important thermodynamic state function, is only defined for large N.  Those who misunderstand the molecular perspective often claim that a well shuffled deck is more disorderly than a deck with the cards all lined up and the suits separated.  However, the designation of such an arrangement as more orderly than that in the well shuffled deck is a matter of taste and inappropriate in a scientific analysis.  Contrast this to the case of the ideal gas which is dynamic, contains many molecules, and samples in time many microstates.

Given these caveats, we shall consider the dependence of entropy on state functions such as pressure and the physical state of the system.  We shall start with the dependence of entropy on pressure, a unit of concentration.  For ideal gases and solutions, this dependence is the sole source of the concentration term in the equilibrium constant expression.  The entropy of a gas or constituent in a solution decreases with concentration.  To see this, consider for a moment the contents of your dorm room as a dynamic system.  I suspect that in most cases the room exhibits many arrangements of your worldly goods as time progresses.  When the room is small, the number of possible arrangements of your possessions is small.  Suppose that you are assigned to a larger room but do not purchase additional possessions.  The concentration of these possessions has decreased but the number of possible arrangements with the increase in space will surely increased.

Starting from equation (2), one can derive, although with considerable mathematical effort, a quantitative relationship between entropy and the partial pressure of a constituent A in an ideal gas.  We shall spare the reader the details and simply present the result.

SA  =  SA  -  Rln[pA(atm)]   (3)

The logarithmic dependence on entropy on pressure should come as no surprise given the form of equation (2).  The important negative sign follows from the simple argument provided above.  Note also that the partial pressure is given in atmosphere.  We use the traditional standard state here.  Equation (3) allows one to separate the contribution of concentration to the total entropy, SA, from that due to other factors such as temperature, SA.

The standard entropy change for a reaction reflects the increase or decrease of complexity and the number of species in the stoichiometric equation.  Entropy increase monotonically with molecular weight.  Symmetry makes very important contributions to entropy.  Fewer microstates or arrangements are possible with symmetric molecules than asymmetric species.  Hence, in the case of a set of isomers, the more asymmetric species will have the higher entropy.  The value of the standard entropy change for decomposition reactions  such as

N2H2(g)    N2(g)  +  H2(g)

has a positive contribution from the net increase in the number of molecules.  If you are unconvinced, consider the carnival game in which the operator hides marbles under cups.  The con artist can produce more arrangements with  5 red and 5 blue marbles than with only 5 white marbles.

The discussion above introduced the dependence on entropy on the state of the substance.  In order to focus on this issue, we have to factor out the contributions from concentration.  We therefore deal with S, the entropy per mole under standard conditions.  At the molecular level, a solid exhibits long-range order; a liquid, only short-range order defined by small loosely bound clusters of molecules; and a gas, random motion.  A liquid has many more accessible microstates than a gas and hence a greater standard entropy.  However, there is considerable residual order in a liquid.  A much larger increase in disorder occurs upon conversion to the gas phase.  Our molecular model provides a satisfying interpretation of the following data for water at 298.15 K.

S(J/K-mole):      44.65    69.91   188.825
phase:               solid     liquid   gas

Consider now the dependence of entropy on temperature.  As the temperature increases, the amplitude of the molecule motions increases.  Even in a crystalline solid a molecule at a lattice site will sample a much wider region of space at higher temperature than low.  X-ray crystallography is performed whenever possible at lower temperatures because of the increased disorder at higher temperature.  Molecular sample a wider range of geometry space and hence microstates as T increases so entropy increases with temperature.  Similarly softer substances with a “floppy” molecular structure possess higher entropies than harder substances.


The discussion in the previous section should have convinced you of the primacy of entropy in thermodynamics.  Entropy must be invoked in an examination of equilibrium.  Although statistical mechanics provides an invaluable view of entropy and in the case of small molecules a means of calculating entropy from spectroscopic data, it is not sufficient, at least not at our present level of understanding.  We must complement our molecular picture with the classic macroscopic picture.  Indeed, the great American thermodynamicists such Josiah Willard Gibbs and Henry Frank (former Adjunct Professor of Chemistry at Pomona College) were equally adept at both statistical mechanics and classical thermodynamics.  Classical thermodynamics is required to obtain conditions for equilibrium and experimental data in those cases in which our understanding of molecular structure is incomplete.  Solutions, especially ionic solutions, are cases where reliable thermodynamic data must be measured experimentally, rather than calculated ab initio, i.e. from first principles.

As in the case of the First Law, a  full statement of the Second Law of Thermodynamics has two parts.  The first makes a statement on how entropy changes can be measured and the second makes a statement about the properties of entropy.

The Second Law
a) Sq/T
b) S is path independent

We note from the second part of the Second Law that entropy like energy is a thermodynamic state function.  The first part involves an inequality and requires comment.  First the denominator on the right side is the absolute temperature.  One can show using both the First and Second Laws that this “thermodynamic” temperature is identical with the ideal gas temperature.   This result is non-trivial and provides the basis for  the absolute calibration of the temperature scale from careful measurements on gases.  Part (a) provides the hope that thermal measurements can provide values of entropy changes.  This is the case, i.e. S = q/T, in the case of a reversible process.  A reversible process is one in which an infinitesimal change occurs over a long time.  A large change that occurs quickly is irreversible and S>q/T. Heating a substance is a controllable process that can be performed in a manner than approaches reversibility and hence thermal measurements are an excellent source of data on the dependence of entropy on temperature.  The leaking of helium from a balloon illustrates the difference between reversibility and irreversibility.  After one fills the balloon, the small helium molecules will slowly leak via effusion from the small pores in the rubber.  In this reversible process, a calorimetric measurement of the heat will yield an accurate value of the entropy change for the loss of the helium.  On the other hand, one could prick the balloon with a pin.  The balloon will pop and the gas will quickly escape.  This second process is definitely irreversible.

We note with some regret that the terms reversible and irreversible which have a strict definition in thermodynamics are used quite differently in kinetics.  In kinetics, a reversible process is one in which there is a facile and hence rapid pathway for the forward and reverse steps.  Time is invoked in the kineticist’s use of the words.  Time is not a factor in thermodynamics.

The equilibrium state is an important special case of reversibility.  Consider a system at equilibrium.  The application of a slight perturbation will lead to a slight shift in the state of the system.  If the perturbation is removed, the system will return to its original state.  If this applies along a particular path, the process is reversible along that path.  If this is the case along all possible paths originating with the supposed equilibrium state, then the system is actually at equilibrium.  Analogies always have to be used with caution but the following mechanical analogy illustrates the point.  Imagine a boulder at the bottom of a bowl-like depression.  If one nudges the rock in any direction, it will move slightly.  After removal of the force, the rock returns to its original position.  The rock is in a state of mechanical equilibrium.  Suppose instead that the rock is on the side of a steep hill.  It is held in a metastable state by friction and perhaps a small pebble.  Now give the rock a slight push.  (Apply the force from the uphill side of the rock unless your wish your heirs to collect your life insurance.)  The rock will move faster and faster and faster down the hill.  It will not roll back after you stopped giving it a gentle nudge.  The rock is not in a state of equilibrium.  There is clearly one path over which a small perturbation initiates a large, irreversible change.

The Second Law then provides an excellent route for converting measurements of heat into values of entropy changes but only if the measurements are performed over reversible paths.  Suppose that one wishes to calculate the standard entropy of condensation of water at 25C.  The equation for this process is

H2O(g)    H2O(l)

At 25C, liquid water and steam are not in equilibrium under standard conditions.  The process of condensation occurs spontaneously and quickly and is most definitely irreversible.  Hence, q/T does not equal the standard entropy change.

The heat for this process can be calculated readily from the difference of the standard enthalpies of formation of liquid and gaseous water.  The two conditions for equating heat and enthalpy changes (constant pressure and only pressure-volume work) apply.  Reversibility is NOT a condition for equating heat and enthalpy changes.

To obtain a correct value of the standard entropy change from calorimetric data, one has to employ the following set of reversible paths:

1) slowly heat the steam up to the normal boiling point,
2) condense the steam at the normal boiling point where the two phases are in equilibrium,
3) slowly cool the liquid water back to 25C.


The molecular approach to equilibrium yields a clear take-home message.  Chemistry is a game of rolling loaded dice.  Both energy and entropy are important and their contributions must in some way be combined.  This combination is not straightforward as energy and entropy reflect radically different properties of molecules.  The synthesis was one of the great achievements of Josiah Willard Gibbs, the Newton of thermodynamics and one of the greatest American scientists.   The road to Gibbs’ master stroke is the inequality in the Second Law.  Consider our total system to be the entire universe which we divide into two subsystems, our chemical system of interest (a small portion of the universe) and the thermal bath in which the system is immersed.  There is nothing external to the universe so the heat of the universe is necessarily zero.  It follows then from the Second Law and our subdivision that

Suniverse   0           (5)
 S  +  S   0

We are interested in a condition for equilibrium.  If the equality applies for all shifts from our starting point, we are at equilibrium.  If the entropy of the universe increases for a path, then we are not at equilibrium and the system can proceed along that path until it reaches equilibrium.

We make no claims on how long it would take a system in a non-equilibrium state to reach equilibrium.  Spontaneous means in thermodynamic that the system will eventually move to the equilibrium state but not when or how fast.  Equation (5) rules out a decrease in the entropy of the universe.  This is an unnatural process which corresponds to conversion to a configuration with a small number of microstates.

Equation (5) is complete and elegant but hardly useful.  We need to express the equation in terms of local variables which can be calculated.  This step begins with the realization that the thermal bath is immense with respect to the chemical subsystem.  Hence, any change in our chemical subsystem, however sudden and irreversible, will always been seen as an infinitesimal and hence reversal change by the thermal bath.

S  =  q /T  =  -q/T  (6)
T  =  T  =  T

Equation (6) can be expressed in terms of thermodynamic state functions if we apply the normal benchtop conditions: only pressure-volume work and constant temperature and temperature.  In this case, the system heat equals the enthalpy change with the result:

S  =  -H/T  (7)

Combining equations (5) and (7) one obtains equation (8) which can be rearranged to yield the desired result, equation (9).

-H/T + S   0  (8)
H - TS    0    (9)

Equation (90 can be expressed in a more compact form by defining a new state function, the Gibbs free energy function G.

G  =  H  -  TS  =  E  +  pV  -  TS  (10)

If one uses the definition of G and constraint of constant temperature, one can quickly show

G  =  H  -  TS  (11)

Hence, under the experimental conditions assumed above, equation (9) can be restated as

G     0  (12)

Equations (9) and (12) has been highlighted because of their importance and deserve further comment before we proceed.

The constraints of constant temperature and pressure may appear to unduly restrictive.  Fortunately, they can be relaxed using the concept of chemical potential, another contribution of Gibbs whose discussion is better deferred until a course in physical chemistry.

The Gibbs free energy is that function which combines in one formula the contributions of entropy and enthalpy.  That is, it allows us to compare apples and oranges.  Note that a process proceeding naturally and spontaneously will take a path that leads to a decrease in G.  A path leading to an increase in G is unnatural.  If G is constant along all infinitesimal paths leading from the starting point, the system is at equilibrium.  For example, under standard conditions, the Gibbs free energy of liquid and gaseous water are equal at 100C.  At 25C, the two phases are not in equilibrium.  Liquid water is the more stable phase and has the lower value of G.  Consequently, gaseous water will spontaneously convert to liquid water.

We can also conclude from equation (11) that at very high temperature, the sign of the entropy change dominates.  Consequently, spontaneous processes must lead to an increase of entropy and therefore life, a process that leads to the increase of order, is thermodynamically infeasible on a very hot planet such as Venus.  In contrast, at low temperatures, G is dominated by the enthalpy term and spontaneous processes must be exothermic.  At intermediate temperatures, both terms are important and small changes in temperature and concentration can have large effects on the position of equilibrium.  That is, temperature can function as a control variable; this is the case with many biochemical processes when control is important to optimize efficiency.

When new tools are developed, it is instructive to apply to a familiar example before proceeding.  We shall follow this advice and examine the vaporization of water as informed by thermodynamics.  The first step is the chemical equation for the process:

H2O(l)    H2O(g)

As a first example, consider whether liquid and gaseous water are in equilibrium under standard conditions at 25C.  We already know the answer.  Under liquid water is the stabler phase under these conditions but the calculations were confirm that our approach is correct and serve to calibrate our intuition.  We assumed standard conditions and can make immediate use of the results in tables tabulated at 298.15 K

H  =  Hf[H2O(g)]  -  Hf[H2O(l)]  =  -241.83 kJ  -  -285.84 kJ  =  44.01 kJ/mole
S  =  S[H2O(g)]  -  S[H2O(l)]  =  188.72 J/K  -  69.94 J/K  =  118.78 J/K-mole

Note that the reaction is endothermic as hydrogen bonds must be broken and therefore is enthalpically unfavorable.  The entropy change is positive as the water is converted into a more disorderly phase.   The standard change in G is readily calculated.

G  =  H  -  T S   =  44010 J  -  (273.15 + 25.00 K)(118.78 J/K)  =  8.6 x 103 J

Nota bene!!  Tables usually tabulate entropies and enthalpies in Joule/K and kiloJoule, respectively.  Don’t make a factor of 1000 error.  Convert the enthalpy change from kiloJoule to Joule.  Also note that the temperature must be Kelvin.                                                                        

The result of the calculation confirms our expectations.  The change in the standard Gibbs free energy for the reaction is positive at 25C.  That is, water does not boil at room temperature.

As a next step, we can estimate the normal boiling point.  At the normal boiling point, the two phases are in equilibrium and hence  G is zero.

H  -  TbS   =  0  or  Tb  =  H/S

At first glance, the result is useless as we require the enthalpy and entropy changes at a temperature that we do not know.  Enthalpy and entropy also depend on temperature via the heat capacity.  Fortunately, their temperature dependence is weak and can often be ignored over small changes in the temperature..  Hence in applying the 25C values of  S and H (but not G) to 100C, one quickly solves for T and obtains 370.5 K.  The experimental value as we know is 373.15 K.  In assuming temperature independence of H and S, we have introduced a very small error.

As a final example, we shall calculate the equilibrium vapor pressure of water at 25C.  We showed earlier that the liquid and gaseous phases are not in equilibrium under standard conditions.  That is, the vapor pressure is not one atmosphere.  The two phases are in equilibrium when the partial pressure equals the equilibrium vapor pressure.  Under these conditions

G = 0  (13)

Nota bene!  We do not conclude from the equilibrium conditions that G = 0.  We are not operating under standard conditions in this problem.  In order to solve the problem, we require the dependence of G on pressure.  The answer for the liquid phase is straightforward.  Liquids are fairly incompressible and hence their entropy is independent of pressure.  Consequently, G[H2O(l)]  =  G[H2O(l)].  In contrast, the entropy of a gas depends strongly on partial pressure.  The enthalpy of an ideal gas is independent of pressure.  These important results and equation (3) yield the dependence of G on pressure.

Gi  =  Gi +  RTln(pi)  (14)

The calculation is now straightforward.

0 =  G  =  {G[H2O(g)] +RTln(pH2O)}  - G[H2O(l)] =  G + RTln(pH2O)
 ln(pH2O) = -G/RT = (-8600 J)/[(8.3141 J/K)(298.15 K)] = -3.47
 pH2O =  0.031 atm = 24 torr.

The calculation is quite satisfying as it replicates the experimental result.  We  have done more than illustrate the use of the Gibbs free energy.  We have also set the state for a rigorous derivation of the equilibrium constant.
thermo_2.doc, WES, 16 Sep. 2002

Source: http://pages.pomona.edu/~wes04747/handout/THERMO_2.doc