Thermodynamics Concept Page - 13

Definition

Entropy

Entropy is a thermodynamic property that measures the degree of randomization or disorder at the microscopic level. The natural state of affairs is for entropy to be produced by all processes.
If a system has a temperature T$T$ (in absolute scale) and a small quantity of heat ΔQ$\mathrm{\Delta }Q$ is given to it, we define the change in the entropy of the system as: 
Δs=ΔQT$\mathrm{\Delta }s={\displaystyle \frac{\mathrm{\Delta }Q}{T}}$

Definition

Entropy of the universe always increases

Entropy is a measure of the disorder of a system. That disorder can be represented in terms of energy that is not available to be used. Natural processes will always proceed in the direction that increases the disorder of a system. When two objects are at different temperatures, the combined systems represent a higher sense of order than when they are in equilibrium with each other. The sense of order is associated with the atoms of system A and the atoms of system B being separated by average energy per atom - those of A being the higher energy atoms if system A is at a higher temperature. When they are put in thermal contact, energy flows from the higher average energy system to the lower average energy system to make the energy of the combined system more uniformly distributed - ie, less ordered. So the disorder of the system has increased - and we say the entropy has increased. But the process of increasing the disorder has removed the possibility that the energy that was transferred from A to B can be used for any other purpose - for example, work cannot be extracted from the energy by operating a heat engine between the two reservoirs of different temperatures. So although energy was conserved in the transfer (the first law), the entropy of the universe has increased in becoming more disordered (the second law) and consequently the availability of energy for doing work has decreased.

Definition

Entropy as a degree of randomness

Entropy is related to the disorder in the system. Thus, if all the molecules in a given sample of a gas are made to move in the same direction with the same velocity, the entropy will be smaller than that in the actual situation in which the molecules move randomly in all directions. Moreover, entropy is not a conserved quantity. Once some entropy is created in a process,it is there forever. 

Example

Entropy for a given process

100 g$100g$ of water is slowly heated from 270C${27}^{0}C$ to 870C${87}^{0}C$. Calculate the change in the entropy of water (special heat capacity of water =$=$ 4200 J/kg-k).
Solution:Change in entropy = m×Cplog(T2T1)$m\times C_{p}log({\displaystyle \frac{T_2}{T_1}})$
m$m$ - mass of water (Kg$Kg$)
Cp$C_p$ - Specific heat capacity of Water (liquid) = 4200 J/Kg.K
T2$T_2$ =360 K$360K$
T1$T_1$ = 300 K$300K$
So Change in entropy =4200×log(360300)$4200\times log({\displaystyle \frac{360}{300}})$= 76.575053855 J/K$76.575053855J/K$

Definition

Entropy change in a Carnot cycle

Process	Δs$\mathrm{\Delta }s$
Isothermal expansion	−nRln(V2V1)$-nRln({\displaystyle \frac{V_2}{V_1}})$
Adiabatic expansion	0
Isothermal compression	−nRln(V4V3)$-nRln({\displaystyle \frac{V_4}{V_3}})$
Adiabatic compression	0

Formula

Entropy change in a reversible process for an ideal gas

Δs=s2−s1=∫21dQT=∫21(CvdTT+PTdv)=∫21(CvdTT+Rdvv)$\mathrm{\Delta }s=s_2-s_1={\int }_1^2{\displaystyle \frac{dQ}{T}}={\int }_1^2(C_v{\displaystyle \frac{dT}{T}}+{\displaystyle \frac{P}{T}}dv)={\int }_1^2(C_v{\displaystyle \frac{dT}{T}}+R{\displaystyle \frac{dv}{v}})$
where s$s$ is the entropy, subscripts 2 and 1 represents final and initial states respectively, Cv$C_v$ is the specific heat capacity at constant volume, T$T$ is the temperature in the absolute scale.

Definition

Reversible process for an ideal gas

Consider a system in contact with a heat reservoir during a reversible process. If there is heat Q$Q$ absorbed by the reservoir at temperature T$T$ , the change in entropy of the reservoir is ΔS=QT$\mathrm{\Delta }S={\displaystyle \frac{Q}{T}}$.During any infinitesimal portion, heat dQrev$d{Q}_{rev}$ will be transferred between the system and one of the reservoirs which is at T$T$ . If dQrev$d{Q}_{rev}$ is absorbed by the system, the entropy change of the system isdSsystem=dQT$d{S}_{system}=\frac{dQ}{T}$
The entropy change of the reservoir is dSreservoir=−dQT$d{S}_{reservoir}=-\frac{dQ}{T}$
The total entropy change of system plus surroundings isdStotal=dSsystem+dSreservoir=0$d{S}_{total}=d{S}_{system}+d{S}_{reservoir}=0$
This is also true if there is a quantity of heat rejected by the system.

The conclusion is that for a reversible process, no change occurs in the total entropy produced, i.e., the entropy of the system plus the entropy of the surroundings: dStotal=0$d{S}_{total}=0$

Definition

Change in entropy in an irreversible process

For an irreversible process, dQactual−dWactual=TdS−dWrev$d{Q}_{actual}-{dW}_{actual}=TdS-d{W}_{rev}$
The subscript ``actual'' refers to the actual process (which is irreversible). The entropy change associated with the state change is
dS=dQactualT+1T[dWrev−dWactual]$dS={\displaystyle \frac{d{Q}_{actual}}{T}}+{\displaystyle \frac{1}{T}}[d{W}_{rev}-{dW}_{actual}]$

Law

Second Law in terms of entropy

 In any cyclic process the entropy will either increase or remain the same.
ΔS≥ΔQT$\mathrm{\Delta }S\ge {\displaystyle \frac{\mathrm{\Delta }Q}{T}}$

Definition

Heat engine

A heat engine is a device in which a system undergoes a cyclic process resulting in conversion of heat into work.

BookMarks

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