<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/9826592e-96d9-4c6d-bb75-e9b1040eb0bc/Clausius_inequality.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/9826592e-96d9-4c6d-bb75-e9b1040eb0bc/Clausius_inequality.png" width="40px" /> Clausius inequality states that over a cycle:

$$ \oint\frac{\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q}{T}\le0 $$

</aside>

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/c88543a4-dc47-4576-bbde-0c7cff90933c/Entropy.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/c88543a4-dc47-4576-bbde-0c7cff90933c/Entropy.png" width="40px" /> Entropy is defined as:

$$ \text dS=\frac{\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q}{T} $$

</aside>

• $\text dS=0$ over a full cycle

• $\text dS$ is path independent

• For a reversible process entropy can be defined as:

$$ S(A)=\int^A_O\frac{\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q_\text{rev}}{T} $$

Where $O$ is an arbitrary point

• We can define a change in entropy for a reversible process as:

$$ \Delta S=S(B)-S(A)=\int^B_A\frac{\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q_\text{rev}}{T}\ge 0 $$

Reversible vs Irreversible

Consider free expansion against reversible isothermal expansion with the same initial and final equilibrium states

$$ \text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q_\text{irrev}+\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} W_\text{irrev}=\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q_\text{rev}+\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} W_\text{rev} $$

We get the following relations:

$$ \text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} W_\text{irrev} > \text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} W_\text{rev} \quad ; \quad \text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q_\text{irrev}<\text{d}\hspace*{-0.16em}\bar{}\hspace*{0.2em} Q_\text{rev} $$

Isolated systems

The entropy of an isolated system can only increase or stay the same

• Increase when the isolated system is not at equilibrium
• Stay the same when in equilibrium ( equilibrium = maximum entropy )

An isolated system at equilibrium must be in the state of maximum entropy

Entropy and temperature

• If $⛵$ volume or $🌋$ pressure is constant then:

$$ ⛵\; C_V=T\left(\frac{\partial S}{\partial T}\right)_V \qquad ; \qquad 🌋 \; C_p=T\left(\frac{\partial S}{\partial T}\right)_p $$

Definitions

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/498198de-9755-4079-9e56-ecc7ae2493fe/Isentropic.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/498198de-9755-4079-9e56-ecc7ae2493fe/Isentropic.png" width="40px" /> Isentropic refers to a process in which there is no change in entropy $\text dS=0$

</aside>