Chem 351 Help — \( \Delta A, \Delta G \), and Maximum Work

Why Are \(A\) and \(G\) Called "Free Energies"?

In the previous module, we introduced the Helmholtz function,

\[ A = U - TS \]

and the Gibbs function,

\[ G = H - TS \]

as criteria for spontaneity.

At constant temperature and volume,

\[ \Delta A < 0 \]

indicates a spontaneous process, while at constant temperature and pressure,

\[ \Delta G < 0 \]

indicates a spontaneous process.

But why are these functions called free energies?

The answer is that they measure the amount of energy that is available to perform useful work.

The Helmholtz Function and Maximum Work

Begin with the definition of the Helmholtz function:

\[ A = U - TS \]

Differentiating gives

\[ dA = dU - TdS - SdT \]

For a reversible process,

\[ dU = dq + dw \]

and

\[ dq = TdS \]

so

\[ dU = TdS + dw \]

Substituting into the differential for \(A\):

\[ dA = (TdS+dw) - TdS - SdT \]

\[ dA = dw - SdT \]

At constant temperature,

\[ dT = 0 \]

and therefore

\[ \boxed{ dA = dw_{\mathrm{rev}} } \]

Because a reversible process produces the maximum possible amount of work, the change in the Helmholtz function is equal to the maximum work obtainable from a process at constant temperature and volume.

The Gibbs Function and Maximum Non-\(pV\) Work

Most chemical reactions occur at constant temperature and pressure rather than constant volume. In this case, expansion work is unavoidable.

We therefore split the work into two contributions:

\(dw_{pV}\): expansion work
\(dw_e\): all other forms of work (electrical, mechanical, etc.)

so that

\[ dw = dw_{pV} + dw_e \]

Starting with

\[ G = H - TS \]

and

\[ H = U + pV \]

it can be shown that

\[ dG = dw_e + Vdp - SdT \]

At constant temperature and pressure,

\[ dT = 0 \qquad dp = 0 \]

leaving

\[ \boxed{ dG = dw_{e,\mathrm{rev}} } \]

Thus, the change in the Gibbs function is equal to the maximum non-\(pV\) work obtainable from a process at constant temperature and pressure.

Why This Matters

The term free energy refers to energy that is free to perform useful work on the surroundings.

The Helmholtz function measures the maximum work available from a process when temperature and volume are held constant.

The Gibbs function measures the maximum useful work available from a process at constant temperature and pressure after accounting for any expansion work that must occur.

For example, a battery operating at constant temperature and pressure can perform electrical work. The maximum electrical energy that can be obtained from the battery is determined by

\[ \Delta G \]

which is why Gibbs free energy is so important in chemistry and electrochemistry.

Big picture: The Helmholtz and Gibbs functions are called free energies because they quantify the amount of energy available to perform work. At constant temperature and volume, \(\Delta A\) gives the maximum work obtainable. At constant temperature and pressure, \(\Delta G\) gives the maximum non-\(pV\) work obtainable.

Worked examples

Worked example: Maximum electrical work from a battery

A battery reaction has a Gibbs free-energy change of

\[ \Delta G = -250\ \mathrm{kJ} \]

under the operating conditions of the battery. If the battery operates at \(6.0\ \mathrm{V}\), calculate the maximum amount of charge that can be transferred.

At constant temperature and pressure, the change in Gibbs free energy gives the maximum non-\(pV\) work:

\[ w_{e,\max} = \Delta G \]

The negative sign means that the system is doing work on the surroundings. Therefore, the maximum electrical work delivered by the battery is

\[ |w_{e,\max}| = 250\ \mathrm{kJ} = 2.50\times10^5\ \mathrm{J} \]

Electrical work is related to charge and voltage by

\[ |w_e| = q_{\mathrm{charge}}E \]

where \(q_{\mathrm{charge}}\) is the charge transferred and \(E\) is the voltage.

Solving for charge:

\[ q_{\mathrm{charge}} = \frac{|w_e|}{E} \]

Substitute the values:

\[ q_{\mathrm{charge}} = \frac{2.50\times10^5\ \mathrm{J}} {6.0\ \mathrm{V}} \]

\[ q_{\mathrm{charge}} = 4.2\times10^4\ \mathrm{C} \]

\[ \boxed{ q_{\mathrm{charge}} = 4.2\times10^4\ \mathrm{C} } \]

Interpretation: The battery can transfer at most \(4.2\times10^4\ \mathrm{C}\) of charge at \(6.0\ \mathrm{V}\) if the process occurs reversibly. A real battery would transfer less useful charge or deliver less useful work because real processes are irreversible and produce entropy.

Key points (one glance)

The Helmholtz and Gibbs functions are called free energies because they measure the amount of energy available to perform useful work.
The Helmholtz function is defined by \[ \boxed{ A = U - TS } \]
At constant temperature and volume, \[ \Delta A < 0 \] indicates a spontaneous process.
The Gibbs function is defined by \[ \boxed{ G = H - TS } \]
At constant temperature and pressure, \[ \Delta G < 0 \] indicates a spontaneous process.
For a reversible process, \[ dq_{\mathrm{rev}} = TdS \]
Using the First and Second Laws, the differential form of the Helmholtz function can be shown to be \[ dA = dw_{\mathrm{rev}} \] at constant temperature.
Because reversible processes produce the maximum possible work, \[ \boxed{ \Delta A = w_{\max} } \] at constant temperature and volume.
The Gibbs function can be used when pressure-volume work is separated from other forms of work.
At constant temperature and pressure, \[ \boxed{ \Delta G = w_{e,\max} } \] where \(w_{e,\max}\) is the maximum non-\(pV\) work obtainable from the process.
Examples of non-\(pV\) work include:
- electrical work performed by a battery,
- mechanical work performed by a motor,
- work required to stretch a material.
The maximum useful work is obtained only for a reversible process.
Real processes are irreversible and therefore always produce less useful work than the theoretical maximum predicted by \(\Delta A\) or \(\Delta G\).
For an electrochemical cell operating reversibly, \[ |w_e| = qE \] where \(q\) is the charge transferred and \(E\) is the cell voltage.
The Gibbs free-energy change therefore provides a direct connection between thermodynamics and the performance of batteries and other electrochemical devices.

Big picture: The Helmholtz and Gibbs functions do more than predict spontaneity. They also quantify the maximum useful work that can be extracted from a process. This interpretation is the origin of the term free energy: it is the energy that is free to perform work on the surroundings.