Question

Use the second law of thermodynamics to explain briefly what happens, in terms of the entropy of the stable phase, when the temperature of ice is raised from -10 °C to 10 °C.

Answer 1

what does all that "energy spreading out" have to do with entropy?

Entropy change is a measure of the molecular motional energy (plus any phase change energy) that has been dispersed in a system at a specific temperature.

          Always, motional energy flows in the direction of hotter to cooler because that direction of dispersal results in a greater amount of spreading out of energy than the reverse. Entropy change, q(rev)/T, quantitatively measures energy dispersing/spreading out. Its profound importance is that it always increases in a "hotter to cooler" process so long as you consider the ‘universe’ of objects, systems, and surroundings.

          Exactly how entropy measures energy dispersal is mathematically simple in phase change. (It's just the ΔH of the process/T.) Further, from Tables of standard state entropy, the S^o values for substances at 298 K, we can get a general idea of how the amount of energy that has been dispersed in substances differs in types of elements and compounds. Finally, we can measure exactly how much entropy increases when substances are heated, i.e., when energy is dispersed from the surroundings to them. All other areas treated in general chem courses are easily describable to beginning students — and I will do so — but the details of their calculations can be left to your text.

Entropy (change) shown in standard state tables

          A standard state entropy of S⁰ for an element or a compound is the actual change in entropy when a substance has been heated from 0 K to 298 K. [However, determining S⁰ at low temperatures is not a simple calculation or experiment. It is found either from the sum of actual measurements of q(rev)/T at many increments of temperature, or from calculations based on spectroscopy.] The final value of S⁰ in joules/K is related to (but not an exact figure for) how much energy has been dispersed to the material by heating it from absolute zero where perfect crystals have an entropy of 0 to 298 K.

          Thus, in our view of entropy as a measure of the amount of energy dispersed/T, S⁰ is a useful rough relative number or index to compare substances in terms of the amount of energy that has been dispersed in them from 0 K. (For ice and liquid water, the S⁰ at both 273 K and 298 K is listed in Tables.) Let's consider ice at its S⁰₂₇₃ of 41 J/K. Remember, that 41 joules is only an indicator or index of the total thermal energy that was dispersed in the mole of ice as it was warmed from 0 K. (Actually, several thousand joules of energy were added as q(rev) in many small reversible steps.)

          Is that complicated? Abstract and hard to understand? An entropy value of a substance is very approximately related to how much energy that had to be dispersed in it so that it can exist and be stable at a given temperature!

         From standard state tables we can see that liquids need more energy than solids of the same substance at the same temperature. (Of course! Liquids at 298 K have required additional enthalpy of fusion, i.e., phase change energy, to break intermolecular attractions or bonds present in solids so that their molecules could more freely move in the liquid phase. And substances that are gases at 298 K similarly had to have the enthalpy of vaporization supplied to them at some temperature between 0 and 298 K so their intermolecular attractions in the liquid could be broken to allow their molecules to move and rotate as freely as they do in gases.) Heavy elements that are solids at 298 K (and in the same column of the periodic table as lighter elements) need more energy to vibrate rapidly back and forth in one place in their solid state than lighter elements. More complex molecules need more energy for their more complicated motions than do simpler molecules, as do similar ionic solids: those that are doubly charged need more energy than do singly charged to exist at 298 K.

          The causes of all entropy relationships are not obvious from looking at tables of standard entropy values, but many make a lot more sense on the basis of S⁰ being related to the motional energy dispersed in them (plus any phase change energy) that is necessary for a substance's existence at T. (Organic molecules are especially good examples, but are beyond what I have time to talk about here. See "Disorder in Rubber Banks?")

The entropy change of a substance in a phase change

          We all know the Clausius definition of entropy change as dS ≥ dq_rev /T and ΔS = q/T in a reversible process. Let's examine such a process as exemplified by the fusion of ice to form liquid water at 273 K. (Vaporization is parallel, of course.) A very large quantity of energy from warmer surroundings, the ΔH of fusion (6 kJ), must be dispersed within the cooler solid, but the solid's temperature is unchanged until the last crystal of ice melts. Doesn’t something seem wrong here? All that heat input and no increase in temperature?. And the reverse behavior of liquid water might seem odd to a young student: When water is placed in cooler surroundings than 273 K, the same large amount of 6 kJ of energy is transferred to the surroundings before the water all becomes ice.

          This can be rationalized from a strictly macro viewpoint by seeing the process of fusion as a change of motional kinetic energy in the surroundings to potential energy in the water that has nothing to do with temperature change in water. Then, the reverse that occurs when liquid water is placed in surroundings that are 272.9 K , can be understood as merely changing the potential energy of the water system to kinetic energy in the surroundings. (A weak analogy would be the kinetic energy of a pendulum swinging up toward changing totally into potential energy at the end of its arc, and then that potential energy changing back to kinetic energy as the pendulum swings to its low point.)

           Of course, the description of the process in molecular thermodynamics is more detailed and far more enlightening , but our goal here is primarily to see the macro view of thermodynamic changes.     

          As we said at the start, because fusion is an equilibrium process and therefore reversible, ΔS_{ice ->water} = q_rev/T. That q_rev is simply the 6kJ of the enthalpy of fusion of ice and thus, ΔS_{ice ->water} = 6000 J/273 K, or 22 J/K. The entropy change for ice to become water, results from the amount of energy, q_rev that has been spread out in the ice so that it could change it to water — divided by T. Is that mysterious? Hard to comprehend?

          Admittedly, what gives entropy its great power of predicting the direction of energy flow — why spontaneous energy dispersal always occurs only from a hotter to a cooler system — is hidden in the apparently simple process of dividing by T. That tremendously important and relatively invisible predictive property can easily be proved to students as I'll show in a minute.

The entropy change of a substance when it is heated

[The standard procedure for determining the entropy change when a substance is heated involves calculus that may be beyond the background of many AP students. Qualitatively, of course, the process could be described as measuring the amount of energy dispersed from the hot surroundings to the cooler substance, divided by the temperature, at a very large number of small temperature intervals from T1 to T2 and adding all of those entropy results.]

           Qualitatively, from a macro viewpoint, it is obvious that entropy must increase in any system that is heated because entropy measures the increase (or decrease) of energy that is dispersed to a system!

          Quantitatively, determining the entropy change of a substance (such as the iron in a frying pan) as it is heated isn't as easy as in phase change. To keep the process of heating at least theoretically reversible, the substance should be heated in many small increments of energy (dq_rev). That way the temperature remains ‘approximately unchanged’ in each increment and the process almost reversible. This is achieved in calculus. Using the usual symbols, and integrated over the temperature range: ΔS_{T1-> T2} = dq_rev/T. (1) But, how do we find out the value of dq_rev?

          Fortunately, the heat capacity of a substance, C_p, is really an "entropy per degree" because it is the energy that must be dispersed in the substance per one degree Kelvin, i.e., q_rev/"1 K"! Therefore, if we just multiply C_p by "the number of degrees" (in more sophisticated terms: the temperature increment, dT), that will give us dq_rev — i.e., dq_rev = C_pdT.

          Substituting this result in (1) above gives ΔS_{T1-> T2} = C_pdT/T = C_p ln T₂/T_1..

          Then, C_p ΔT is the energy dispersed within a system when it is heated from T₁ to T₂.

The decrease in entropy when a system is cooled
Energy always flows from hotter to cooler
In any spontaneous process, entropy increases

           If anything cools, it clearly has dispersed some of its energy to its cooler surroundings. Lesser energy in it means that its entropy decreases. The cooling of a frying pan is an example that will most easily demonstrate to students how important is entropy. With merely that (overused!) example of a hot iron frying pan we can show them why "heat cannot spontaneously pass from a colder to a warmer body" Clausius' original statement for one version of the second law of thermodynamics. That leads directly to why the universe is always increasing in entropy, another version of the second law.

           Recapping our basic understanding of energy and entropy: Energy spontaneously disperses from being localized to becoming spread out, if it is not hindered from doing so. Entropy change measures that process — how widely spread out energy becomes in a system or in the surroundings — by the relationship, q(rev)/T. (Let's symbolize a high temperature by a bold T, and a lower temperature by an ordinary T.)

          If the pan (system) is hotter than the cool room (surroundings), and q (an amount of motional energy, "heat") might flow from the pan to the room or vice versa, the entropy changes would be: pan, q/T_sys and q/T _surroundings . Then, when q is divided by a large number, i.e., by the bold T of the pan, the result is a smaller entropy change than if q is divided by a small number, i.e., the ordinary T of the surroundings, q/T, so there is a larger entropy change in the surroundings.) (For simplicity in quickly first bringing this conceptual point to a class, perhaps avoid numbers. If you feel that numbers are better, at least avoid dimensions by identification of q = 1, T = 100, and T = 1 so that 1/100 in the hot pan is obvious smaller than 1/1 in the cool surroundings!)

           Now, a larger entropy change — wherever it occus — means that energy would be more widely spread out there. Thus, because energy spontaneously becomes more spread out, if it is not hindered, the q will move from the hotter to the cooler (from a smaller entropy state to a larger entropy state , from our hot pan to our cool room.) This is universally true, as stated by Clausius, as is our common human experience, and as quantified by the relative entropy changes in hotter and cooler parts of this "frying pan - cool room universe". (It should be emphasized that it is true even under conditions in which the process of transferring energy is essentially reversible, i.e., when the difference between q/T_sys and q/T _surroundings is very small.)

          Finally, the spontaneous increase in entropy in the ‘cooler room’ part of this universe is greater than the entropy decrease in the ‘hot pan’ part of this ‘room-pan’ universe. The net result is an increase in entropy in the whole universe, the predicted result for any spontaneous process.

Entropy as "unavailable energy"

          This is a note to clarify an often quoted but confusing sentence about entropy, "Entropy is unavailable energy". The sentence is ambiguous, either untrue or true depending on exactly what is meant by the words. As any of us would predict, the energy q within even a faintly warm iron pan at 298 K, measured by its entropy per mole S^o, will spontaneously cause a 273 K ice cube placed in it to begin to melt. In this sense the pan's entropy represents instantly available energy and the sentence appears untrue.

          However, if any amount of energy is transferred from the 298 K pan, the pan no longer has enough energy for its q/T value to equal the entropy needed for that amount of iron to exist at 298 K. Thus, from this viewpoint, the sentence is true, but tricky: We can easily transfer energy from the pan. It's not "unavailable" at all — except that when we actually transfer the slightest amount of energy, the pan no longer is in its original energy and entropy states! For the pan to remain in its original state, the energy is unavailable..…

          "Entropy is unavailable energy" or "waste heat" is also ambiguous in regard to motional energy that is transferred to the surroundings as a result of a chemical reaction. That energy/T is considered an entropy increase in the surroundings (because it is energy that is spread out in the surroundings and no longer available in the system). However, it is completely available for work in the surroundings or transfer to anything there at a lower temperature; it just is no longer available for the process that occurred in the system at the original temperature.