The classical thermodynamic definition of entropy
arose from Carnot's construction of a new state function. Its use to predict thermodynamic spontaneity (dS ≥0) is serendipitous. Perhaps more insight can be obtained by Boltzmann's definition of entropy
S = Nk ln W
In this equation N is the number of particles and k is Boltmann's constant (1.38066 x 10-23 J/K), which is simply the ideal gas constant divided by Avogadro's number. W is the number of configurations or ways that a particular system can be generated and still have the same energy. Understanding how to come up with W requires more explanation.
Let's say we have three distinguishable particles, a red one (X), a green one (X) and a blue one (X), and we want to distribute them between two boxes, one on the left and one on the right. There are eight possibilities:
XX X__ or XX_ X__ or XX_ X__
X__ XX or X__ XX_ or X__ XX_
If we were randomly placing the molecules into the box, there would be two ways we could have all our molecules in one box and six ways we could have them distributed between the two boxes. It would be three times more likely (6/2 = 3) that we would have them distributed between the two boxes since there are six ways (W = 6 ) to do this and only two ways (W = 2) to have them in a single box.
Boltzmann's definition of entropy assumes that the most probable distribution is the one that we will actually observe and the field that studies physical phenomena in this manner is called statistical mechanics. Because it is statistical, we cannot use it on only three particles but must have a larger, statistically valid number. While you really need a larger population sample to remove statistical fluctuations from your observations, a simple lab illustrating the concept of number of configurations and probability can be performed with a fairly small number of pennies.
As an example, what is the probability that all the oxygen molecules spread evenly throughout your room will all diffuse into a single cubic millimeter up an the left-hand corner? Not very likely because there are so many more ways to distribute them over the entire room than there are to pack them into the single cubic millimeter. If you were to release some perfume in one small corner of a room, the perfume molecules will diffuse throughout the room until you could detect their odor in the opposite corner. Once diffused throughout the room, the perfume will not spontaneously collect in a single corner. The more random, disordered distribution is the one that you actually observe.
Remember that spontaneous changes must obey the Clausius inequality, dS ≥0. The equality dS = 0 is true only for reversible changes, which must be carried out in equilibrium, infinitely slowly every step of the way. Since real processes occur at a finite rate and thus include irreversibilities, Boltzmann's definition of entropy tells us that since entropy has to decrease as time goes on, the universe will become increasingly disordered and random. This does not mean that entropy cannot decrease locally. You can clean your room! But in order to decrease (order) the entropy of one part of the universe you have to come up with an equal of greater increase (disorder) in another part to balance it out.
Here's another example that will let us calculate an entropy of mixing. Let's start out with two containers of gas of equal volume (V) that are adiabatically insulated from the surroundings and separated by a removable partition. The container on the left has a mole of argon atoms and the one on the right has a mole of nitrogen molecules. We remove the partition separating the two pure gases to begin our experiment. What will happen? You should expect that some of the argon will diffuse into the nitrogen side and some of the nitrogen will diffuse into the argon side until a homogenous gas solution is formed, and that is exactly what entropy predicts will happen.
If we assume that W scales with volume, when we remove the partition between the two containers both the volume available to each gas and W for each gas will double. For one of the gases, say the argon:
Vinitial = V ; Winitial = W
Vfinal = 2V ; Wfinal = 2W
and
ΔmixingSargon = Sfinal -Sinitial
=Nk ln (2V) - Nk (ln V)
= Nk ln(2V/V) = Nk ln (2)
This amount is for one of the two gases (argon). To get the total entropy of mixing, we need to add the change in entropy of the other chemical (nitrogen)
ΔmixingSargon = ΔmixingSargon +ΔmixingSnitrogen
= Nargonk ln (2) + Nnitrogenk ln (2)
= nargonR ln (2) + nnitrogenR ln (2)
= (nargon + nnitrogen)R ln (2)
We can generalize to any number of molecules mixing and to any amount of each molecular substance with the following equation
where we have summed over all i of the components and Xi is the mole fraction of the ith component, defined by Xi = ni/ ∑nj.
This equation allows you to calculate the entropy of mixing or an ideal solution, one where the interactions among the various components of the mixed chemicals is the same as in the pure (unmixed) substances. This is the only way that the distribution of components will be truly random throughout the solution volume. The equation works well for gases under all but very high pressure conditions, for solutions of similar liquids and even for solid solutions such as metal alloys (e.g. platinum and gold) provided you wait long enough for the components to become thoroughly mixed.
For condensed phase solutions with dissimilar components, e.g. ethanol and acetone or sodium chloride and water, the statistical entropy of mixing will only be part of the total entropy of mixing and we will have to add an "excess" quantity excessS to get the entire entropy change
ΔmixingS = ΔidealS ΔexcessS
= -R ∑ Xiln Xi ΔexcessS
The need for "fudging" with the excess entropy of mixing does not result from a breakdown in the Boltzmann concept of entropy, but rather in our assuming that the volumes of the individual components are additive and thus are proportional to W. While the ideal solution entropy of mixing is always positive, the excess entropy of mixing can be either positive or negative. Providing that the components are miscible in one another, the overall sign of ΔmixingS will be positive.
Mixing is a good example of how entropy increases spontaneously to create increased randomness and disorder in the physical world. Remember that to use ΔS ≥ 0 as a condition of spontaneity, you have to include contributions both from the system and from the surroundings. We can determine changes in entropy quantitatively by using tables of thermodynamic values, by measuring heat transfer (calorimetry) and by other experimental methods. However, it is useful to be able to predict the sign of an entropy change qualitatively by evaluating whether the final state is more ordered (ΔS = negative) or less ordered (ΔS = positive) than the initial state.
Here are some guidelines for calculating and estimating entropy changes
1) Changes with temperature
Qualitative: Provided nothing else happens, an increase in temperature always results in an increase in entropy. This is because increasing the temperature causes the molecules to increase their thermal motion increasing the randomness of their location.
Quantitative: At constant pressure and assuming the heat capacity is approximately independent of temperature
ΔS = Cpln[Tf/Ti]
At constant volume, this equation becomes
ΔS = CVln[Tf/Ti]
2) Phase changes
Qualitative: Gases are usually more disordered than liquids and liquids are usually more disordered than solids.
Quantitative: We can easily calculate the entropy change for first order phase transitions by measuring or looking up the enthalpy of the phase change
ΔtransS = ΔtransH/Ttrans.
3) Chemical reactions
Qualitative: Because gases generally have higher entropies than condensed phases, we can look at the number of gases produced (the products) v.s. the number of gases consumed (reactant). If there are more moles of gas products than gas reactants ΔS = ; if there are fewer moles of gas products than gas reactants ΔS = -. If the number of product and reactant gases is the same, then the sign of the entropy change is difficult to predict without additional information, but the entropy change will probably be small.
Quantitative: We can calculate the entropy change of a chemical reaction by subtracting the reactant entropies from the product entropies
where Si and Sj indicate molar entropies and the ni, nj are the number of moles appearing in the stoichiometric equation describing the reaction.
4) Volume changes at constant pressure
Qualitative: An increase in volume increases W and thus results in an increase in entropy.
Quantitative: This will depend upon the substance. For an ideal gas
ΔS = nR ln [Vf/Vi]
5) Pressure changes at constant temperature
Qualitative: Pressure increases generally cause volume decreases at constant temperature provided no phase change occurs. Therefore, pressure increases generally result in a decrease in entropy. This is not always the case for condensed phases.
ΔS = - nR ln [Pf/Pi]
Why does entropy have to increase? Why does time have to progress linearly? Sir Arthur Eddington coined the phrase "time's arrow" in reference to entropy in response to Hubble's description of an ever-expanding universe (1929). From a thermodynamic point of view, it doesn't matter. The second law states that irreversible spontaneous processes are associated with an increase in entropy, and that is the nice thing about laws. You state them, and within the discipline they are by definition true. You cannot prove them; you just have to accept them to work with the theory. You can (and ought to) notice whether they predict things correctly and that they never seem to be wrong.
On a philosophical note or two, the reason that we seem live time only in the forward direction, the one in which entropy always increases, is not necessarily obvious. Time reversal is certainly allowed by Newtonian mechanics and many other fields of physics Time travel is fun to speculate upon and makes for interesting science fiction. Some people actually take it quite seriously. The laws of thermodynamics give us no help except to limit us to cases where ΔS ≥0 for an isolated system.
