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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:

${\displaystyle dU-TdS+PdV\leq 0}$

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:

${\displaystyle dU=dQ-dW\,}$

From the second law we have:

${\displaystyle dS=dQ/T\,}$

Hence:

${\displaystyle dQ=TdS\,}$

By substituting this into the first law, we have:

${\displaystyle dU=TdS-dW\,}$

Rearranging we have:

${\displaystyle dU+dW-TdS=0\,}$

Letting dW be pressure-volume work, we have:

${\displaystyle dU+PdV-TdS=0\,}$

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:

${\displaystyle dG=0\,}$

Or for a spontaneous process:

${\displaystyle dG\leq 0\,}$

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

• Thermodynamics