What is the second law of thermodynamics

[Greg Bernhardt]

Definition/Summary

The second law has various forms. I shall give these here then show how they are logically connected.

Entropic Statement of the Second Law:

There exists an additive function of thermodynamic state called entropy which never decreases for a thermally isolated system.

Clausius' Statement of the Second Law:

No process exists in which heat is transferred from a cold body to a less cold body in such a way that the constraints on the bodies remain unaltered and the thermodynamic state of the rest of the universe does not change.

Equations

$$\Delta S_{tot} = \Delta S_{sys} + \Delta S_{surr} \geq 0$$

Extended explanation

The entropic statement of the second law requires that systems are thermally isolated. This is rarely the case and so the second law becomes:

$$\Delta S_{tot} = \Delta S_{sys} + \Delta S_{surr} \geq 0$$

Thermodynamic processes can be reversible and so $\Delta S_{sys}$ can be < 0.

Lets consider the transfer of heat from body B to body A in such a way that the constraints on those two bodies never change.

From the second law:

$$dS_{tot} = dS_A+dS_B = \left(\frac{\partial S_A}{\partial E_A}\right)_{PV}dE_A + \left(\frac{\partial S_B}{\partial E_B}\right)_{PV} dE_B$$

therefore;

$$dS_{tot} = \left[\left(\frac{\partial S_A}{\partial E_A}\right)_{PV} -\left(\frac{\partial S_B}{\partial E_B}\right)_{PV}\right]dq_A \geq 0$$

where, $dq_A = - dq_B$, from the first law. Also the work is zero since $P$ and $V$ don't change.

We now define the coldness of a body to be:

$$\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)_{PV}$$

and thus:

$$\left(\frac{1}{T_A} - \frac{1}{T_B} \right) dq_A \geq 0$$

So for $dq_A > 0$ we must have:

$$\frac{1}{T_B} < \frac{1}{T_A}$$

Body B must be less cold than body A and we conclude Clausius' statement of the second law.

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[APhysicist]

This is a great explanation of the Second Law, both in its entropic and Clausius' statements. It's very clearly laid out and makes the logical connection between the two statements easy to understand. Thank you for sharing this!