Thermodynamic Equations - The Fundamental Equation

The Fundamental Equation

The first and second law of thermodynamics are the most fundamental equations of thermodynamics. They may be combined into what is known as fundamental thermodynamic relation which describes all of the thermodynamic properties of a system. As a simple example, consider a system composed of a number of k different types of particles and has the volume as its only external variable. The fundamental thermodynamic relation may then be expressed in terms of the internal energy as:

Some important aspects of this equation should be noted: (Alberty 2001), (Balian 2003), (Callen 1985)

  • The thermodynamic space has k+2 dimensions
  • The differential quantities (U, S, V, Ni) are all extensive quantities. The coefficients of the differential quantities are intensive quantities (temperature, pressure, chemical potential). Each pair in the equation are known as a conjugate pair with respect to the internal energy. The intensive variables may be viewed as a generalized "force". An imbalance in the intensive variable will cause a "flow" of the extensive variable in a direction to counter the imbalance.
  • The equation may be seen as a particular case of the chain rule. In other words:
dU= \left(\frac{\partial U}{\partial S}\right)_{V,\{N_i\}}dS+ \left(\frac{\partial U}{\partial V}\right)_{S,\{N_i\}}dV+ \sum_i\left(\frac{\partial U}{\partial N_i}\right)_{S,V,\{N_{j \ne i}\}}dN_i
from which the following identifications can be made:
These equations are known as "equations of state" with respect to the internal energy. (Note - the relation between pressure, volume, temperature, and particle number which is commonly called "the equation of state" is just one of many possible equations of state.) If we know all k+2 of the above equations of state, we may reconstitute the fundamental equation and recover all thermodynamic properties of the system.
  • The fundamental equation can be solved for any other differential and similar expressions can be found. For example, we may solve for and find that

Read more about this topic:  Thermodynamic Equations

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