Mathematical Statement of Second Law

\[\Delta S_{\text{total}} \ge 0\] The word “total” is meant to imply that we take into account of all changes in both the system and the surroundings. i.e., \[\Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \ge 0\] Second law applies to the system and its surroundings taken together and does not impose any general restraint on the system alone.

\(\Delta S_{\text{total}} = 0\) for a reversible process.

  • Isolated System: For the isolated system, which is completely cut off from its surroundings, neither mass nor energy exchange with its surroundings is possible. Thus changes which occur in such a system can not cause changes in the surroundings, and we need to consider the system alone. In this case the equation becomes, \[\Delta S_{\text{system}} \ge 0\] and \[\Delta E_{\text{system}} = 0\] For such a system the total energy must remain constant, and the total entropy can increase or stay constant.

  • Combination of First and Second Laws

    From first law of thermodynamics, we have \(dU = dQ + dW\). From second law, we have \(dQ=TdS\). And for a reversible process, \(dW=-PdV\). Using these in the first law, we have, \[dU = TdS - PdV \tag*{(20)}\] Eqn.(20) is called the “Fundamental Equation”, as it combines laws (the first and second laws) of thermodynamics.

Third Law of Thermodynamics

Absolute entropy is zero for all perfectly crystalline substances at absolute zero temperature. (i.e., \(S = 0\) at \(T=0\) K)

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