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In ], the '''combined law of thermodynamics''' is simply a mathemtical summation of the ] and the ] subsumed into a single concise mathematical statement as shown below:
⚫ #REDIRECT ]

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

Here, ''U'' is ], ''T'' is ], ''S'' is ], ''P'' is ], and ''V'' is ].

==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 ] did in 1876, this reduces to the following at ]:

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

Or for a ]:

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

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

== External links ==

* - Wolfram's World of Science

⚫ ]

Revision as of 18:52, 26 July 2006

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:

d U − T d S + P d V ≤ 0 {\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:

d U = d Q − d W {\displaystyle dU=dQ-dW\,}

From the second law we have:

d S = d Q / T {\displaystyle dS=dQ/T\,}

Hence:

d Q = T d S {\displaystyle dQ=TdS\,}

By substituting this into the first law, we have:

d U = T d S − d W {\displaystyle dU=TdS-dW\,}

Rearranging we have:

d U + d W − T d S = 0 {\displaystyle dU+dW-TdS=0\,}

Letting dW be pressure-volume work, we have:

d U + P d V − T d S = 0 {\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:

d G = 0 {\displaystyle dG=0\,}

Or for a spontaneous process:

d G ≤ 0 {\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

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