Fundamental thermodynamic relation

In thermodynamics, the combined law of thermodynamics is simply a mathemtical summation of the first law of thermodynamics and the second law of thermodynamics subsumed into a single concise mathematical statement as shown below:

$dU-TdS+PdV\leq 0$

Here, U is internal energy, T is temperature, S is entropy, P is pressure, and V is volume.

Derivation

Starting from the first law, and neglecting differential details:

$dU=dQ-dW\,$

From the second law we have:

$dS=dQ/T\,$

Hence:

$dQ=TdS\,$

By substituting this into the first law, we have:

$dU=TdS-dW\,$

Rearranging we have:

$dU+dW-TdS=0\,$

Letting dW be pressure-volume work, we have:

$dU+PdV-TdS=0\,$

By assigning the quantity to the left of the equals sign the symbol G, as Willard Gibbs did in 1876, this reduces to the following at thermodynamic equilibrium:

$dG=0\,$

Or for a spontaneous process:

$dG\leq 0\,$

Thus, this expression is referred to by many as the combined law of thermodynamics; Gibbs showed that deviations of this quantity could be used to predict the direction of various natural chemical processes.

External links

• Combined Law of Thermodynamics - Wolfram's World of Science