# 8.3: ΔA, ΔG, and Maximum Work

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- Contributed by Patrick Fleming
- Assistant Professor (Chemistry and Biochemistry) at California State University East Bay

The functions \(A\) and \(G\) are oftentimes referred to as **free energy** functions. The reason for this is that they are a measure of the maximum work (in the case of \(\Delta A\)) or non p-V work (in the case of \(\Delta G\)) that is available from a process. To show this, consider the total differentials.

First, consider the differential of \(A\).

\[dA = dU -TdS - SdT\]

Substituting the combined first and second laws for \(dU\), but expressing the work term as \(dw\), yields

\[dA = TdS -dw -TdS - SdT\]

And cancelling the \(TdS\) terms gives

\[ dA = dw - SdT\]

or at constant temperature (\(dT = 0\))

\[dA = dw\]

Since the only assumption made here was that the change is reversible (allowing for the substitution of \(TdS\) for \(dq\)), and \(dw\) for a reversible change is the maximum amount of work, it follows that \(dA\) gives the maximum work that can be produced from a process at constant temperature.

Similarly, a simple expression can be derived for \(dG\). Starting from the total differential of \(G\).

\[dG = dU + pdV – pdV + Vdp – TdS – SdT\]

Using an expression for \(dU = dq + dw\), where \(dq = TdS\) and \(dw \) is split into two terms, one (\(dw_{pV}\)) describing the work of expansion and the other (\(dw_e\)) describing any other type of work (electrical, stretching, etc.)

\[ dU - TdS + dW_{pV} + dW_e\]

\(dG\) can be expressed as

\[dG = \cancel{TdS} - \cancel{pdV} +dw_e + \cancel{pdV} + Vdp – \cancel{TdS} – SdT\]

Cancelling the \(TdS\) and \(pdV\) terms leaves

\[dG = +dw_e + Vdp – SdT\]

So at constant temperature (\(dT = 0\)) and pressure (\(dp = 0\)),

\[dG = dw_e \]

This implies that \(dG\) gives the maximum amount of non p-V work that can be extracted from a process.

This concept of \(dA\) and \(dG\) giving the maximum work (under the specified conditions) is where the term “free energy” comes from, as it is the energy that is *free* to do work in the surroundings. If a system is to be optimized to do work in the soundings (for example a steam engine that may do work by moving a locomotive) the functions A and \(G\) will be important to understand. It will, therefore, be useful to understand how these functions change with changing conditions, such as volume, temperature, and pressure.

Contributors

Patrick E. Fleming (Department of Chemistry and Biochemistry; California State University, East Bay)