📂Thermal Physics

In Thermodynamics, What is Entropy?

Definition

The quantity $S$ is defined as entropy if it satisfies the following equation.

$$dS = {{ \delta Q_{\text{rev} } } \over { T }}$$

Explanation

Entropy is a physical quantity representing ‘disorder,’ and it’s challenging to understand why it indicates disorder just by looking at its mathematical definition. Explanations for non-specialists like ‘messing up a room’ or ‘dropping ink into a glass of water’ can only explain ‘disorder,’ not $dS = \dfrac{\delta Q_{\text{rev}} }{ T }$.

To grasp the concept, it’s essential to think of entropy not as something derived but as a ‘definition.’ If you’ve ever wondered, ‘Why define it this way?’ a brief visual explanation might help. Consider the following two situations:

• Case 1. When the system’s temperature is low

  

  Imagine inputting thermal energy $Q$ into a cooled space. This is giving a change in thermal energy $\delta Q$ at a low temperature $T_{1}$. Let’s say the change in entropy is $d S_{1}$.

• Case 2. When the system’s temperature is high

  

  Now, imagine adding the same thermal energy $Q$ to an already hot space as in Case 1. This is giving a change in thermal energy $\delta Q$ at a high temperature $T_{2}$. Let’s say the change in entropy is $d S_{2}$ in this case.

The inputted energy will diffuse to a colder area according to the Second Law of Thermodynamics. The space in Case 1. turns into a murky color that is hard to describe, compared to its original blue color, while the space in Case 2. barely becomes redder than before. This difference in color change signifies how much the space has changed due to the inputted energy, meaning $d S_{1} > d S_{2}$. Mathematically, since $T_{1} < T_{2}$, it follows that:

$${{1} \over {T_{1} }} > {{1} \over {T_{2}}}$$

Thus, according to the definition of entropy, the following equation is a natural conclusion.

$$d S_{1} = {{\delta Q_{\text{rev} }} \over {T_{1} }} > {{\delta Q_{\text{rev} }} \over {T_{2}}} = d S_{2}$$

In thermodynamic terms, this means the increase in entropy in Case 1. is greater than in Case 2.. If explained by messing up a room, it means ’the same mischief causes a tidy room to become messier easier than a messy room.’ If explained by dropping ink, ’the same amount makes clear water dirtier easier than murky water.’