Entropy

Entropy is a property

Equations:

Reversible:

d S = \frac{δ Q_{r e v}}{T}

Irreversible:

d S \geq \frac{δ Q_{r e v}}{T}

Formed from the Clausius Statement:
Clausius Statement
and Kelvin-Planck Statement:
Kelvin Planck Statement

Alternate Expressions:

Intensive form:

d s = \frac{d u}{T} + \frac{P}{T} d v

Extensive form:

d S = \frac{d U}{T} + \frac{P}{T} d V

Reversible Systems:

Cyclic reversible transformation of a system

Cycle 1:
- $1 \to 2$ via $A$
- $2 \to 1$ via $B$

\int_{1}^{2} \frac{δ Q}{T} |_{A} + \int_{2}^{1} \frac{δ Q}{T} |_{B} = 0

Cycle 2:
- $1 \to 2$ via $C$
- $2 \to 1$ via $B$
$\int_{1}^{2} \frac{δ Q}{T} |_{C} + \int_{2}^{1} \frac{δ Q}{T} |_{B} = 0$ Thus resulting in: $\int_{1}^{2} \frac{δ Q}{T} |_{C} = \int_{1}^{2} \frac{δ Q}{T} |_{A}$

Result:

Independent of path
Depends only on initial and final states
Entropy is a property $d S = \frac{δ Q_{r e v}}{T}$ where $d S$ is path-independent and $δ Q_{r e v}$ is path-dependent

Irreversible and Reversible Systems:

Cyclic general transformation of a system

Cycle 1 (Irreversible):
- $1 \to 2$ via $A$
- $2 \to 1$ via $B$

\int_{1}^{2} \frac{δ Q}{T} |_{A} + \int_{2}^{1} \frac{δ Q}{T} |_{B} < 0

Cycle 2 (Reversible):
- $1 \to 2$ via $C$
- $2 \to 1$ via $B$
$\int_{1}^{2} \frac{δ Q}{T} |_{C} + \int_{2}^{1} \frac{δ Q}{T} |_{B} = 0$ Thus resulting in: $\int_{1}^{2} \frac{δ Q}{T} |_{C} > \int_{1}^{2} \frac{δ Q}{T} |_{A}$

Result:

$A$ is reversible and path-independent
Depends only on initial and final states
Entropy is a property $d S \geq \frac{δ Q_{r e v}}{T}$ where $d S$ is path-independent and $δ Q_{r e v}$ is path-dependent

Impacts systems Isolated Systems and Non-Isolated Systems differently.

Entropy Generation:

Second Law of Thermodynamics:

Reversible:

d S_{c m} = \frac{δ Q_{r e v}}{T}

Irreversible:

d S_{c m} > \frac{δ Q_{i r r}}{T}

Definition:

Entropy generated due to irreversibility:
$d S_{c m} = \frac{δ Q}{T} + δ S_{g e n}$

Properties:

Path dependent!
Always positive $δ S_{g e n} \geq 0$
Reversible $δ S_{g e n} = 0$

Control Mass/System:

Entropy Difference:

Δ S_{c m} = \int \frac{d Q}{T} + S_{g e n}

Rate Equation:

\frac{d S_{c m}}{d t} = \sum \frac{\dot{Q}}{T} + {\dot{S}}_{g e n}

Control Volume:

Differential:

d S_{c v} = \frac{δ Q}{T} + s_{i n} δ m_{i n} - s_{o u t} δ m_{o u t} + δ S_{g e n}

Rate Equation:

\frac{d S_{c v}}{d t} = \frac{\dot{Q}}{T} + s_{i n} {\dot{m}}_{i n} - s_{o u t} {\dot{m}}_{o u t} + {\dot{S}}_{g e n}

Steady State:

Control Mass/System:

0 = \sum \frac{\dot{Q}}{T} + {\dot{S}}_{g e n}

Control Volume:

s_{o u t} = s_{i n} + \frac{q}{T} + s_{g e n}