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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:

<math>dU - TdS + PdV \le 0</math>

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

[edit] Derivation

Starting from the first law, and neglecting differential details:

<math>dU = dQ - dW\,</math>

From the second law we have:

<math>dS = dQ/T\,</math>

Hence:

<math>dQ = TdS\,</math>

By substituting this into the first law, we have:

<math>dU = TdS - dW\,</math>

Rearranging we have:

<math>dU + dW - TdS = 0\,</math>

Letting dW be pressure-volume work, we have:

<math>dU + PdV - TdS = 0\,</math>

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:

<math>dG = 0\,</math>

Or for a spontaneous process:

<math>dG \le 0\,</math>

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.

[edit] External links

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