From WolfWikis

LECTURE 9

One item that is left on our wish list

Above we have answered the question: what is entropy really, but we still do not have a general criterium for spontaneity, just one that works in an isolated system. Let's fix that now. It leads to two new state functions that prove to be most useful ones of thermodynamics.

Helmholtz energies

As we saw above, the expression

dU = TdS -PdV

was only valid for reversible changes.

Let us consider a spontaneous change. If we assume constant volume, the work term drops out. From the Clausius inequality dS>δq/T we get:

dU≤ TdS (V constant)

dU-TdS≤0 (V constant)

Consider a new state function

A ≡ U -TS

dA = dU -TdS - SdT

If we also set T constant we see that the above expression becomes:

dA=dU-TdS≤0 (V,T constant)

This means that the Helmholtz energy A is a decreasing quantity for spontaneous processes (regardless of isolation!) and A becomes constant once a reversible equilibrium is reached.

What A stands for

A good example is the case of the mixing of two gases. Let's assume isothermal conditions and keep the total volume constant.

For this process the ΔU is zero (isothermal, ideal) but the ΔS_molar = -y₁Rlny₁-y₂Rlny₂.

This means that ΔA_molar = RT [y₁lny₁+y₂lny₂].

This is a negative quantity because the mole ratios are smaller than unity. So yes this spontaneous process has a negative ΔA.

If we look at ΔA = ΔU - TΔS we should see that the latter term is the same thing as -q_rev So we have :

ΔA = ΔU - q_rev = w_rev

This is however the maximal work that a system is able to produce and so the Helmholtz energy is a direct measure of how much work one can get out of a system. A is therefore often called the Helmholtz free energy.

Interestingly this work can not be volume work as volume is constant. so it stands for the maximal other work (e.g. electrical work) that can be obtained under the unlikely condition that volume is constant.

Natural variables of A

Because A≡ U-TS we can write

dA = dU -TdS -SdT

dA = TdS -PdV -TdS -SdT = -PdV - SdT

The natural variables of A are volume V and temperature T.

Gibbs energy

The Helmholtz energy A is developed for isochoric changes and as we have often said before it is much easier to deal with isobaric ones where P= constant. We can therefore repeat the above treatment for the enthalpy and introduce another state function the Gibbs (free) energy

G≡ H -TS = U + PV - TS = A + PV

If we take both T and P constant (pleasantly achievable conditions!) we get

dU-TdS + PdV≤0

dG≤0

G either decreases (spontaneously) or is constant (at equilibrium).

Calculating the state function between two end points we get:

ΔG = ΔH - TΔS ≤ 0 (T,P constant)

This quantity is key to the question of spontaneity under the conditions we usually work under. If for a process ΔG is positive it does not occur spontaneous and can only be made to occur if it is 'pumped', i.e. coupled with a process that has a negative ΔG. The latter is spontaneous.

Phase transitions

A phase transition like melting is often done under equilibrium conditions. We have seen that both the H and the S curves undergo a dicontinuity at such a temperature, because there is an enthalpy of fusion to overcome. For a general phase transition at equilibrium at constant T and P, we can say that:

Δ_trsG = Δ_trsH - T_trsΔ_trsS = 0

Δ_trsH = T_trsΔ_trsS

Δ_trsH / T_trs=Δ_trsS

For melting of a crystalline solid, we now see why there is a sudden jump in enthalpy. The reason is that the solid has a much more ordered structure than the melt. The decrease in order implies a finite Δ_trsS.

We should stress at this point that we are talking about first order transitions here. The reason for this terminology is that the discontinuity is in a function like S, that is a first order derivative of G (or A). Second order derivatives (e.g. the heat capacity) will display a singularity (+∞) at the transition point.

Direction of the spontaneous change

Because the ΔS term contains the temperature T as coefficient the spontaneous direction of a process, e.g. a chemical reaction can change with temperature depending on the values of the enthalpy and the entropy change ΔH and ΔS.

This is true for the melting process, e.g. for water below 0^oC we get water=>ice, above this temperature ice melts to water, but it also goes for chemical reactions.

Take: NH_3(g) + HCl_(g) => NH₄Cl_(s)

Δ_rH at 298K / 1 bar is -176.2 kJ. The change in entropy is -0.285 kJ/K so that at 298K ΔG is -91.21 kJ. Clearly this is a reaction that will proceed to the depletion of whatever is the limiting reagent on the left.

However at 618K this is a different story. Above this temperature ΔG is positive! (assuming enthalpy and entropy have remained the same, which is almost but not completely true) The reaction will not proceed. Instead the reverse reaction would proceed spontaneously. The salt on the right would decompose in the two gases -base and acid- on the left.

Meaning of the ΔG term

As we have seen, ΔA can be related to the maximal amount of work that a system can perform at constant V and T. We can hold an analogous argument for ΔG except that V is not constant so that we have to consider volume work (zero at constant volume) .

dG = d(U+PV-TS) = dU -TdS - SdT - PdV +VdP

As dU = TdS + δw_rev

dG = δw_rev -SdT + VdP + PdV

As the later term is -δw_volume

dG = δw_rev -SdT + VdP - δw_volume

At constant T and P the two middle terms drop out

dG = δw_rev - δw_volume = δw_{other useful work}

Conjugate variables

As discussed before there are many other forms of work possible, such as electrical work, magnetic work or elastic work. Although Simon and McQuarri do not discuss this they are commonly incorporated in the formulism of thermodynamics by adding other terms, e.g:

dG = -SdT + VdP + ℰde + MdH + FdL + γdA

1. ℰde stands for the electromotoric force ℰ and de the amount of charge transported against it.
2. MdH stand for magnetization and (change in) magnetic field.
3. F stands for the elastic force of e.g. a rubber band dL for the length it is stretched
4. γ stands for the surface tension (e.g. of a soap bubble), A for its surafce area.

The terms always appear in a pair of what is known as conjugate variables. That is even clearer if we write out the state function rather than its differential form:

G = U + PV -TS + ℰe + MH + FL + γA + etc.

The PV term can also be generalized -and needs to be so- for a viscous fluid to a stress-strain conjugate pair. It then involves a stress tensor.

We will soon encounter another conjugate pair: μdn that deals with changes in composition (n) and the thermodynamic potential μ

Natural variables of G

Because G ≡ H-TS we can write

dG = dH -TdS -SdT

dG = TdS +VdP -TdS -SdT = VdP - SdT

The natural variables of G are pressure P and temperature T. This is what makes this function the most useful of the four U,H,A and G: these are the natural variables of most of your experiments!

We are now ready to begin applying thermodynamics to a number of very diverse situations, but we will first develop some useful partial differential machinery.