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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.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
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.
* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!