Much confusion exists about the nature of thermodynamics. This is a brief description of the second law and what it can tell us about possible changes to a system.

 

Thermodynamics & the Second Law

This is a short, but accurate treatment of the changes that can and cannot occur spontaneously in a gaseous system. There are whole textbooks on the subject, but these few pages are sufficient for a working understanding.

The total internal energy of a constant volume system is symbolized by U and any change in the internal energy of the system is symbolized by ΔU such that:

ΔU = Ufinal - Uinitial

ΔU depends upon the initial and final values of the internal energy of the system and does not depend upon the path by which the change occurs.

If ΔU is negative the system has undergone an exothermic process and if ΔU is positive the system has undergone an endothermic process.

If an amount of energy (usually as heat) enters the system ΔU = QV where QV represents the energy entering the system at constant volume.

If energy is transferred from the system to its surroundings ΔU = - QV

For a change carried out at constant pressure the volume is variable and if an amount of energy QP enters the system this will cause a change in internal energy ΔU and as the volume will change by ΔV and the work done by the system as it expands against the external pressure, P, is given by P ΔV.

QP = ΔU + P ΔV

This may be expanded to include functions dependent upon initial and final states of the system:

QP = Ufinal - Uinitial + P (Vfinal – Vinitial)

This may be rearranged to give:

QP = (Ufinal + PVfinal) – (Uinitial + PVinitial)

The quantity U + PV is known as a state function as is U; they depend upon the state of the system and not on its history.

To give U + PV some status it is allotted the symbol H which is the enthalpy of the system:

H = U + PV

As with U, changes in H are important and, for macroscopic changes the equation becomes:

ΔH = ΔU + P ΔV + V ΔP

For a system at constant pressure V ΔP = 0 so:

ΔH = ΔU + P ΔV

As with ΔU, if ΔH is negative the system has undergone an exothermic change and if ΔH is positive the change is endothermic.

ENTROPY

The entropy of a system is a measure of its disorder, randomness or the extent of its molecular chaos. One way of stating the second law of thermodynamics is that if a spontaneous change occurs to a system the entropy of the system and its surroundings must increase. This is very important. The entropy of the system may decrease but providing that the entropy of the surroundings is greater and offsets that decrease the change will occur. An example is the formation of crystals from a saturated solution. The formation of crystals is the production of an ordered array of ions or molecules from the chaos of the solution phase. So there is a decrease of entropy of the system from that part of the change. Normally, when crystals form there is a release of energy and the system warms up and some of that energy is transferred to the surroundings so that the surroundings warm up. This warming of the surroundings constitutes an increase of entropy of the surroundings and so an apparent decrease of the entropy of the system is accompanied by an increase in the entropy of the surroundings which is sufficient to obey the law. But more of that later.

On Ludwig Boltzmann’s gravestone in Vienna is carved one of the most important equations of science:

S = k ln W

S is entropy, k is Boltzmann’s constant (who else’s could it be?) and W is the number of ways that the system can be realized.

An example of the use of the equation is to take two 4 x 4 grids that are coincident with each other.

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The left hand grid [hot] has cells that could be thought of as each containing a dollop [quantum] of energy, the right hand grid [cold] has empty cells. Since the quanta are identical there is only one way of achieving the distribution in the left hand grid and its entropy is zero [ln 1 = 0]. Now take one quantum of energy from one of the cells in the left hand grid, there are 16 ways of doing that, and place the quantum in any of the empty cells in the right hand grid, there are 16 ways of doing that. So, there are 256 ways of achieving the distribution shown next:

 

 

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The other 255 are omitted!

The entropy of this new distribution is positive and in going from the completely separated starting point to the system in which one quantum of energy has been displaced is in line with the 2nd law. This procedure can be modified to be random. A six by six grid with 18 quanta would allow the use of two dice to identify the quantum to be moved and a second shake of the dice would identify the position in which it should be placed. This could continue until an equilibrium position was achieved in which further random changes had no effect on the distribution of quanta. Computerization of this procedure allows the derivation of the Maxwell-Boltzmann distribution which is that corresponding to maximum entropy and the second law which is concerned with the even temperature distribution in an isolated column of gas – in the 4 x 4 case there would be an equal number of quanta in each grid, but not necessarily evenly distributed; that outcome would be very unusual.

The Boltzmann equation is the basis of statistical mechanics and from which the major equations of thermodynamics may be derived.

The 2nd Law

The simplest statement of the 2nd law is that in a spontaneous process the change in entropy is equal to or greater than zero and the particular change refers to the system and its surroundings:

ΔS ≥ 0

The equals sign may be dropped for all real changes and only refers to changes carried out reversibly.

Another statement concerns temperature changes: heat never spontaneously flows from a colder body to a hotter one.

Some statistical mechanics can help now.

Suppose the volume of a gaseous system of N molecules increases from V1 to V2 and the number of ways of achieving the system varies from W1 to W2. The probability of any one molecule occupying the initial volume is V1/V2 and the probability of all N molecules occupying the original volume is (V1/V2)N. The ratio of the number of ways of achieving the system in its two volumes is:

W1/W2 = (V1/V2)N

The Boltzmann equation allows this to be re-written as:

ΔS = k ln W2 - k ln W1 = kN ln V2 - kN ln V1

This leads to the generalized differential form:

dS = kN ln V

If we take N as the number of molecules in one mole of gas (Avogadro) this becomes:

dS = R dV/V

If the system expands by dV and takes place against an external pressure P, an amount of heat dQ must enter the system to allow the work done as P dV.

dQ = P dV

Since, for one mole, P = RT/V

dQ = RT dV/V

As shown above R dV/V = dS

So dS = dQ/T

That applies to the ideal reversible change, but if the expansion of the gas is into a vacuum where no work is done by the gas, dQ = 0

So, we may write:

dS > dQ/T

This inequality is vital to what follows.

In terms of measurable quantities the equation becomes:

ΔS > ΔQ/T

Now consider a spontaneous change in which there is a change of enthalpyof the system, ΔHsystem. The first law indicates that the enthalpy change to the surroundings is - ΔHsystem.

The second law is:

ΔSsystem + ΔSsurroundings > 0

The entropy change of the surroundings is: ΔSsurroundings = - ΔHsystem/T

This gives: ΔSsystem - ΔHsystem/T > 0

Or: TΔSsystem - ΔHsystem > 0

Or: ΔHsystem - TΔSsystem < 0

The quantity on the left of the inequality sign is a state function and is symbolized by ΔG in honour of Josiah Willard Gibbs, a famous American thermodynamicist who worked all this out many moons ago. G is now known as the Gibbs energy [a previous description was Free Energy or sometimes Gibbs Free Energy] and ΔG [= ΔHsystem - TΔSsystem] is the change in Gibbs energy.

The importance of the value of ΔG is that it allows the criterion of equilibrium to be enunciated in terms that are usually known or measurable. This follows from the inequality:

ΔG < 0

This is the condition for a spontaneous change to occur, the position of equilibrium is reached when ΔG = 0, its minimum value without contravening the 2nd law. That is, if ΔG = 0 is positive the change won’t happen.

As ever with thermodynamics, the change it refers to might be shown to be spontaneous, but it tells us nothing about the rate of the change; that is the province of kinetics. For example, the thermal decomposition reaction of ozone:

2O3 → 3O2

ΔH = - 2 x 142 kJ and ΔS = + (3 × 29.4) – (2 × 38.2) = 11.8 J K-1

ΔG(298 K) = - 242000 - 3516 = - 245.5 kJ

The reaction is spontaneous, yet in the absence of suitable photons a sample of ozone may be kept for weeks, the thermal reaction is very slow. The feasibility of the reaction would seem to be the very exothermic enthalpy change, but that is strictly not the case. The main driving ‘force’ for the reaction is the increase in entropy of the surroundings when all that energy is released. The energy may be dissipated, but it is the positive entropy change of the surroundings that the reaction feasible.

The dissociation of hydrochloric acid in water is almost instantaneous;

HCl(g) → H+(aq) + Cl-(aq)

ΔH = - 74.6 kJ mol-1 and ΔS = - 130.4 J K-1 mol-1

ΔG(298 K) = - 74600 + 38859 = - 35.7 kJ mol-1

In this reaction the entropy change of the system is negative, but this is overcome by the exothermicity of the reaction. The real criterion of feasibility is the effect that the exothermicity has on the entropy of the surroundings. ΔG takes care of such things.