# What is the second law of thermodynamics

1. Jul 24, 2014

### 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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