# Thermodynamic potentials

In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials:

 Internal energy [itex]U[itex] The energy needed to create a system. Also represented by E Helmholtz free energy [itex]F=U-TS[itex] Also represented by A Enthalpy [itex]H=U+PV[itex] Gibbs free energy [itex]G=U+PV-TS[itex]

where T = temperature, S = entropy, P = pressure, V = volume

 Contents

## Differential definitions

Any differential change in the internal energy U of a system can be written as the sum of heat flowing into the system and work done on the system by the environment:

[itex]dU = dq + dw[itex]

where dq is the infinitesimal heat flow into the system, and dw is the infinitesimal work done on the system.

Assuming only reversible mechanical work (PV work), we can express the internal energy change in terms of state functions and their differentials:

[itex]dq = TdS[itex]
[itex]dw = -PdV[itex]

where T is temperature, S is entropy, P is pressure, and V is volume.

This leads to the standard differential form of the internal energy:

[itex]dU = TdS - pdV[itex]

Applying Legendre transforms repeatedly, the following differential relations hold for the four potentials:

 dU = TdS - PdV dF = - SdT - PdV dH = TdS + VdP dG = - SdT + VdP

If we write the above four equations generally as

Then it is seen that

[itex]A=\left(\frac{\partial \Phi}{\partial x}\right)_y[itex]
[itex]B=\left(\frac{\partial \Phi}{\partial y}\right)_x[itex]

yielding expressions for T, P, S, and V in terms of derivatives of the potentials

[itex]

+T=\left(\frac{\partial U}{\partial S}\right)_V

 =\left(\frac{\partial H}{\partial S}\right)_P


[itex]

[itex]

-P=\left(\frac{\partial U}{\partial V}\right)_S

 =\left(\frac{\partial F}{\partial V}\right)_T


[itex]

[itex]

+V=\left(\frac{\partial H}{\partial P}\right)_S

 =\left(\frac{\partial G}{\partial P}\right)_T


[itex]

[itex]

-S=\left(\frac{\partial G}{\partial T}\right)_P

 =\left(\frac{\partial F}{\partial T}\right)_V


[itex]

Furthermore, mathematically we have

[itex]

\left(\frac{\partial}{\partial y} \left(\frac{\partial \Phi}{\partial x}\right)_y \right)_x = \left(\frac{\partial}{\partial x} \left(\frac{\partial \Phi}{\partial y}\right)_x \right)_y [itex]

which gives:

[itex]

\left(\frac{\partial A}{\partial y}\right)_x = \left(\frac{\partial B}{\partial x}\right)_y [itex]

which are known as Maxwell's relations

## Chemical reactions

Changes in these quantities are useful for assessing the degree to which a chemical reaction will proceed. The relevant quantity depends on the reaction conditions, as shown in the following table. Δ denotes the change in the potential and at equilibrium the change will be zero.

Constant V Constant P ΔU ΔH ΔF ΔG

Most commonly one considers reactions at constant P and T, so the Gibbs free energy is the most useful potential in studies of chemical reactions.

## References

• Lewis, Gilbert Newton; Randall, Merle; Revised by Pitzer, Kenneth S. & Brewer, Leo "Thermodynamics" 2nd Editon, New York, NY USA: McGraw-Hill Book Co. 1961.

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