The Helmholtz Energy

Maximum work

Starting with the first law:

because dS ≥ δq/T, we can rewrite the equation above as

If dT = 0 (that is form an isothermal change), this can be written as

If we define the Helmholtz energy as

Because the quantity inside the parenthesis is the definition of A, we can substitute:

which integrated yields to

This says that the isothermal change in A is less than or, for reversible changes, equal to the work done by the system on the surrounding. Because work done by the system has a negative value, equation 4.1 means that the ΔA of an isothermal process is the maximum amount of work, a system can do on the surroundings. The conenction between work and the Helmholtz energy is the reason that A is represented by A from Arbeit, meaning "work".

The "TdS" term in the definition of dA and dG represents the amount of internal energy or enthalpy that is tied up in random molecular energy states, and so is not avalable to perform work of any sort, whether it be pressure-volume work or non-PV work. It is also why A and G are sometimes referred to as Helmholtz free energy and Gibbs free energy, repsectevely. It is the amount of energy that is "free" to do work.

If the change occurs with an increase of entropy of the system (in which case T∆S > 0), the right-hand side of the equation is more negative than ∆U. In this case, the maximum work that can be obtained from the system is greater than ∆U. The explanation of this apparent paradox is that the system is not isolated and energy may flow in as heat as work is done. Because the entropy of the system increases, we can afford a reduction of the entropy of the surroundings yet still have, overall, a spontaneous process. Therefore, some energy (no more than the value of T∆S) may leave the surroundings as heat and contribute to the work the change is generating.

Criteria for spontaneity

The Clausius inequality implies a number of criteria for spostaneuous changer under a variety of conditions that may be expressed in terms of the properties of the system; They are summarized by introducting the Helmholtz and Gibbs energies.

When a change in the system occurs and there is a transfer of energy as heat between the system and the surroundings, the Clausius inequality, reads

If we consider a closed system with only P-V work, dq = dU + PdV so that at constant volume dq_V = dU. Consequently

At either constant internal energy (dU = 0) or constant entropy (dS = 0), this expression becomes, respectively,

Recall that ‘additional work’ is work other than expansion work. The first is the basis for saying that in a spontaneuous process in an isolated system the entropy increases to a maximum.

The second says that if an infinitesimal change takes place in a system of constant volume and entropy, dU is negative if the change is spontaneous and zero if the change is reversible. Do not interpret this criterion as a tendency of the system to collapse to lower energy. Is is a disguised statement abount entropy and it implies that if the entropy of the system is unchanged, then there must be an increase in entropy of the surroundings, which can be achieved only if the energy of the system decreases as energy flows out as heat.

When energy is transferred as heat at constant pressure, and there is no work other than expansion work, we can write dq_P = dH and obtain

At either constant enthalpy or constant entropy this inequality becomes, respectively,

The entropy of the system at constant pressure must increase if its enthalpy remains constant, for there can then be no change in entropy of the surroundings). Alternatively, the enthalpy must decrease if the entropy of the system is constant, for then it is essential to have an increase in entropy of the surroundings.

None of these three criteria for spontaneuos change is immediately useful in the chemistry laboratory where processes are carried out at constant temperature and volume or, more frequentely, at constant temperature and pressure.

From

we know that at constant volume TdS ≥ dU, which implies that

Thus for any irreversible process at constant T and V, the Helmholtz energy A decreases, and for a reversible process it is consntant.

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