Thermodynamic Potential - The Equations of State

The Equations of State

We can use the above equations to derive some differential definitions of some thermodynamic parameters. If we define Φ to stand for any of the thermodynamic potentials, then the above equations are of the form:

where and are conjugate pairs, and the are the natural variables of the potential . From the chain rule it follows that:

Where is the set of all natural variables of except . This yields expressions for various thermodynamic parameters in terms of the derivatives of the potentials with respect to their natural variables. These equations are known as equations of state since they specify parameters of the thermodynamic state. If we restrict ourselves to the potentials U,F,H and G, then we have:

$+T=\left(\frac{\partial U}{\partial S}\right)_{V,\{N_i\}} =\left(\frac{\partial H}{\partial S}\right)_{p,\{N_i\}}$

$-p=\left(\frac{\partial U}{\partial V}\right)_{S,\{N_i\}} =\left(\frac{\partial F}{\partial V}\right)_{T,\{N_i\}}$

$+V=\left(\frac{\partial H}{\partial p}\right)_{S,\{N_i\}} =\left(\frac{\partial G}{\partial p}\right)_{T,\{N_i\}}$

$-S=\left(\frac{\partial G}{\partial T}\right)_{p,\{N_i\}} =\left(\frac{\partial F}{\partial T}\right)_{V,\{N_i\}}$

$~\mu_j= \left(\frac{\partial \phi}{\partial N_j}\right)_{X,Y,\{N_{i\ne j}\}}$

where, in the last equation, is any of the thermodynamic potentials U, F, H, G and are the set of natural variables for that potential, excluding . If we use all potentials, then we will have more equations of state such as

$-N_j=\left(\frac{\partial U}{\partial \mu_j}\right)_{S,V,\{N_{i\ne j}\}}$

and so on. In all, there will be D equations for each potential, resulting in a total of D 2D equations of state. If the D equations of state for a particular potential are known, then the fundamental equation for that potential can be determined. This means that all thermodynamic information about the system will be known, and that the fundamental equations for any other potential can be found, along with the corresponding equations of state.

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