# Clausius theorem

The Clausius theorem (1855) states that in a cyclic process

$\oint \frac{\delta Q}{T} \leq 0.$

The equality holds in the reversible case[1] and the '<' is in the irreversible case. The reversible case is used to introduce the state function entropy. This is because in cyclic process the variation of a state function is zero.

## Proof

Proving Clausius Inequality

Suppose a system absorbs heat δQ at temperature T. Since the value of $\frac{\delta Q}{T}$ does not depend on the details of how the heat is transferred, we can assume it is from a Carnot engine, which in turn absorbs heat δQ0 from a heat reservoir with constant temperature T0.

According to the nature of Carnot cycle,

$\frac{\delta Q}{T}=\frac{\delta Q_0}{T_0}$
$\Rightarrow\delta Q_0=T_0\frac{\delta Q}{T}$

Therefore in one cycle, the total heat absorbed from the reservoir is

$Q_0=T_0\oint\frac{\delta Q}{T}$

Since after a cycle, the system and the Carnot engine as a whole return to its initial status, the difference of the internal energy is zero. Thus according to First Law of Thermodynamics,

Q0 = ΔU + W + W0 = W + W0 = Wtotal

According to the Kelvin statement of Second Law of thermodynamics, we cannot drain heat from one reservoir and convert them entirely into work without making any other changes, so

$W_\text{total}\leq 0$

Combine all the above and we get Clausius inequality

$\oint\frac{\delta Q}{T}\leq 0$

If the system is reversible, then reverse its path and do the experiment again we can get

$-\oint\frac{\delta Q}{T}\leq 0$

Thus

$\oint\frac{\delta Q}{T}=0$

## References

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