Increase of Entropy Principle

According the increase of entropy principle, “the entropy of an isolated system during a process always increases, or in the limiting case of a reversible process, remains constant.” What this means is that entropy can never decrease. In addition, entropy change is only due to irreversibilities .

Increase of Entropy Principle Proof

To prove out the increase of entropy principle we will consider a cycle that consists of two processes. The first process, process 1-2, is reversible or irreversible. Next, the second process, process 2-1 is internally reversible. Taking in consideration of the Clausius inequality the following equation is derived.

(Eq 1) $\oint \frac{δ Q}{T} \leq 0$

$\int_{1}^{2} \frac{δ Q}{T} + \int_{2}^{1} {(\frac{δ Q}{T})}_{i n t r e v} \leq 0$

In equation 1 the second integral represents the entropy change $S_{1} - S_{2}$ . As a result,

(Eq 2) $\int_{1}^{2} \frac{δ Q}{T} + S_{1} - S_{2} \leq 0$

$S_{2} - S_{1} \geq \int_{1}^{2} \frac{δ Q}{T}$

In addition, equation 2 can also be expressed in differential form.

(Eq 3) $d S \geq \frac{δ Q}{T}$

$T$ = thermodynamic temperature at the boundary

$δ Q$ = differential heat transferred between the system and the surroundings

For the above, the equality occurs for internally reversible process. On the other hand, the inequality occurs for an irreversible process. As a result, the entropy change for a closed system will always be greater than $δ Q / T$ for an irreversible process. In contrast, for the limited case where the process is reversible than the two quantities will be equal. Finally, entropy that is generated during a process is called entropy generation. Entropy generation is represented by the variable $S_{g e n}$ . In turn, equation 2 can be rewritten as the following.

(Eq 4) $Δ S_{s y s} = S_{2} - S_{1} = \int_{1}^{2} \frac{δ Q}{T} + S_{g e n}$

$S_{g e n}$ is always a positive quantity or zero. Hence, it is not a property of the system. In addition, if there is an absence of any entropy transfer, than the entropy change of the system will equal the entropy generation.

When the heat transfer is zero, such as for adiabatic closed system, than equation 2 is reduced to the following.

(Eq 5) $Δ S_{i s o l a t e d} \geq 0$

Equation 5 represents the entropy of an isolate system which during its process always increases, or in limited cases of a reversible process remains constant. It however will never decrease. In turn, this is known as the increase of entropy principle.

Entropy and its Effects on the Universe

Entropy is consider an extensive property. As a result, the total entropy of a system is determined from the sum of the entropies of the parts of the system. For example a system and its surroundings can be two subsystems. However, for this to be the case a sufficiently large arbitrary boundary that has no heat, work, or mass transfer must enclose the system and its boundary. Refer to the equation below.

(Eq 6) $S_{g e n} = Δ S_{t o t a l} = Δ S_{s y s} + Δ S_{s u r r} \geq 0$

$Δ S_{s u r r}$ = change in entropy of the surroundings

$Δ S_{s y s}$ = change in entropy of the system

Due to the fact that no process is truly reversible, it can be concluded that some entropy is always generated during a process. As a result, the entropy of the universe, which can be considered an isolated system, is always increasing. Entropy is a measurement of disorder. Hence, as entropy increases in the universe’s disorder increases. Eventually, it is theorized, that entropy will cause the universe to die since energy will no longer be in usable state. This is known as heat death.

Additional Remarks on Entropy

1. Entropy is directional. This means that it can only occur one direction not any direction. In turn, entropy direction must comply with the increase in entropy principle, $S_{g e n} \geq 0$ . If a process violates the increase in entropy principle than it is impossible.
2. Entropy is a nonconserved property. In other words, there isn’t a conservation of entropy principle.
3. An engineering systems performance is degraded by the presence of irreversibilities. In addition, entropy generation is a measurement of the magnitudes of irreversibilities that are present in the process. As a result, the greater the irreversibilities the greater the entropy generation. This means that entropy is a quantitative measurement of the irreversibilities of a process.