Entropy

Entropy Intro

Ideal processes are reusable but real processes contain irreversibilities, entropy is a thermodynamic property of a system used to quantify the irreversibilities of a process or cycle.

This is a measure of the number of possible ways energy can be distributed in a system of molecules.

$S=K_B\ln \Omega$
Where $\Omega$ is the number of ways energy can be distributed, higher entropy means the microscopic state of the system is more uncertain.

If you have a higher temperature, you usually have more micro states.

Once two things come in contact, thermal energy transfers from the hotter state to the colder state where the 1st Law of Thermo states that the energy quanta is conserved.

Change in Entropy

Total entropy ($\Delta S_{tot}$) must increase when heat transfers

The change in entropy can also be less than zero or greater than zero for a given system but the change in total entropy must be greater than 0.

For an isolated system where $S_{gen}=\Delta S_{tot}$

$$\{\begin{array}{l}>0\text{ irreversible process}\\ =0\text{ reversible process}\\ <0\text{ impossible process}\end{array}$$

Entropy Balanced for a Closed System
$\Delta S_{system}=S_{in}-S_{out}+S_{gen}$
$\Delta S_{sys}=\Sigma \frac{Q_k}{T_k}+S_{gen}$ Where $S_{gen}\ge 0$ and the $\frac{Q_k}{T_k}$ term is the sum of all differential amounts of heat transfer divided by $T_k$ at location k on a boundary.

Adding or subtracting W does not change the enthalpy

The second law lets us

• Predict the direction of the processes
• Determine the best theoretical performance of cycles
• Evaluate quantitatively the factors that preclude maximum performance

Isentropic

Note that isentropic means $S_{gen}=0$ or “constant entropy” which is used to describe a reversible adiabatic process

For an internally reversible process
$S_{gen}=0$ and $\Delta S_{sys}={\frac{Q_k}{T_k}}_{}$ and
$Q_{rev}={\int }_1^{2}TdS$
The area under a TS graph is heat input and the amount of heat in/out of a system depends on the process path.

For an internally reversible heat process
$\delta Q=TdS$
$TdS=dU+PdV$
The TdS Equations
$Tds=du+pdv$
$Tds=dh-vdP$
The TdS equations relate state properties. To use them, the equation of state must first be defined.

The TdS equations relate state properties therefore, we actually don’t directly rely on the process path making it valid for open and closed, and reversible or irreversible processes.

To use these Tds equations, T, P during the process must be known and entropy may still be generated.

Recalling how the specific heat is related to u, h, and T, we can apply the TdS equations in convenient forms depending on the phase

For incompressible solids or liquids
$s_2-s_1=\overline{c}\ln \frac{T_1}{T_2}$
For in or near the vapour dome
Use the tables

For Ideal Gases
$s_2-s_1=\overline{c_v}\ln {\frac{T_2}{T_1}}_{}+R\ln {\frac{v_2}{v_1}}_{}$
$s_2-s_1=\overline{c_p}\ln \frac{T_2}{T_1}-R\ln {\frac{P_2}{P_1}}_{}$