# 9: Helmholtz and Gibbs Energies

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• 9.1: Helmholtz Energy
We have answered the question: what is entropy, but we still do not have a general criterion for spontaneity, just one that works in an isolated system. Let's fix that now by showing that Helmholtz energy is the indicator for spontaneity when temperature and volume are held constant.
• 9.2: Gibbs Energy Determines the Direction of Spontaneity at Constant Pressure and Temperature
Gibbs energy is the maximum amount of non-$$PV$$ work that can be extracted from a thermodynamically closed system. At constant temperature and pressure, Gibbs energy determines the direction of spontaneous processes, such as chemical reactions.
• 9.3: The Maxwell Relations
To fully exploit the power of the state functions we need to develop some mathematical machinery by considering a number of partial derivatives.
• 9.4: Thermodynamic Functions have Natural Variables
The fundamental thermodynamic equations follow from five primary thermodynamic definitions and describe internal energy, enthalpy, Helmholtz energy, and Gibbs energy in terms of their natural variables. Here they will be presented in their differential forms.
• 9.5: The Enthalpy of an Ideal Gas is Independent of Pressure
Ideal gases do not interact with each other (no intermolecular forces), so the enthalpy of an ideal gas is independent of pressure.
• 9.6: The Gibbs-Helmholtz Equation
The first order partial on G versus P is the volume V; this allows us to find the dependence of G on P by simply integrating over the volume V from one pressure to the other.

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