In this PDF by MIT, the condition that mechanical system evolves by the law that the internal change is negative is derived. In the derivation, the assumptions used as the systems are of constant volume and entropy, how can we justify these assumptions?
Asked
Active
Viewed 24 times
1 Answers
1
Those are not assumptions but possible conditions: In the special case of constant volume $V$ and entropy $S$—if we happen to encounter it—then the Second Law statement that total entropy always tends to increase is equivalent to the statement that the internal energy $U$ always tends to decrease.
This is shown mathematically at the question Why are thermodynamic potentials minimised?. But we can also articulate the argument in words: Assume a constant-volume, constant-entropy system putatively at equilibrium (i.e., with maximum entropy). If the energy isn't minimized, then we could extract some as work—which carries no entropy—and return that energy to the system irreversibly (e.g., by using an electrical current to heat up a resistor), which generates entropy. But then the entropy wasn't maximized after all, and so the system wasn't originally at equilibrium. This is a contradiction. Therefore, entropy maximization implies internal energy minimization at constant entropy and volume.
Various other thermodynamic potentials can be defined and used if the volume and/or entropy aren't constant, emphasizing that we don't need to apply those conditions (and therefore that no justification is necessary). For example, under the familiar conditions of constant temperature $T$ and pressure $P$, the energy that is minimized is the Gibbs free energy $G\equiv U+PV-TS$. This is discussed in the document you linked. Does this make sense?
Chemomechanics
- 25,546