# Thermodynamics Second Law thought experiment - where am I wrong?

I thought of the following situation:

Suppose there are two separate perfectly rigid vessels enclosing fixed and euqal volumes of the same gas at some state A. An amount E of heat is supplied in an internally reversible manner to the volume of gas in the first vessel, causing the gas to reach a state B. The second vessel, however, contains a wire filament with a certain resistance that crosses its boundary twice and is hooked to an external voltage source. The mass of the filament is negligible. Electrical work in the same amount $$E$$ is supplied to the gas in the second vessel in a likewise internally reversible manner -- that is, the wire continuously raises the temperature of the gas in a reversible process.

Since the same net amount $$E$$ of energy was transferred to both vessels -- the control volumes -- the internal energy of each volume of gas must increase by the same amount. The volumes remain the same throughout each process and the initial state is the same. Therefore, the end states of each gas must coincide.

The heat transfer to the first vessel is accompanied by an entropy transfer in the amount $$\int \frac{\delta Q}{T}$$, by definition. Since the energy transfer to the second vessel was carried out entirely through work and in an internally reversible manner, the entropy transfer has to be zero. Also, given that the process was internally reversible, no entropy was generated within the system boundaries.

This scenario gives rise to an apparent paradox. The first law ensures that the final states are the same: equal increases in internal energy and same constant volume. Through this reasoning, the entropy of both systems must be the same. However, the process relative to the first vessel involved no internal entropy generation but net entropy transfer to the enclosed gas. No transfer occured to the second vessel and no entropy was generated within its boundaries.

So the first law assures that both volumes of gas are at the same state while the second law calls for a difference in entropies, which would prescribe different end states.

Where is the fault in my thought process?

• The heating of a wire through electrical resistance is not reversible. Resistance can be thought of as analogous to friction for electrons. Dec 20, 2017 at 23:11