Clausius theorem

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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 = W_total

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

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

See also

• Introduction to entropy

References

1. ^ Clausius theorem at Wolfram Research

External links

• Judith McGovern (2004-03-17). "Proof of Clausius's theorem". http://theory.ph.man.ac.uk/~judith/stat_therm/node30.html. Retrieved Octorber 4,2010. 
• "The Clausius Inequality And The Mathematical Statement Of The Second Law". http://ronispc.chem.mcgill.ca/ronis/chem213/hnd10.pdf. Retrieved October 5, 2010.