Entropy

Two reversible adiabatic can not intersect each otherCA

B

C

isotherm

RA

AB and CB are two reversible adiabatic and they intersect at B.AC is a reversible isotherm. Through AC we can transfer heat to the cycle ABC which isA close cyclic process. From the cycle ABC there is no heat loss but work out put. SoThis violates 2nd law of thermodynamics. Hence two RA can not intersect each other.

pv

i

fa b

RA

Rev isothermAny rev path

Through i, f there is aRev process. We constructtwo rev adiabatic throughi and f, then a rev isotherm ab such that:

Area under iabf = area under if,

First law for if : ,

Process iabf:

𝑤𝑖𝑓=𝑤𝑖𝑎𝑏𝑓 ,𝑠𝑜𝑞𝑖𝑓=𝑞𝑖𝑎𝑏𝑓

p

v

=

Thus any rev path can be substituted by a reversible Zig-zag path, between the same end states, consisting of a rev adiabatic + rev isotherm+ another rev adiabatic.

Clausius theorem

p

v

Any rev cyclic process

RA

Rev isotherm

T1

T2

Lets us consider a smooth close curve which represents a rev cycle. Then we take two Elemental parts from this cycle as shown in the picture (upper part and lower part). The upper part cab represented by an isotherm+ two RA and similarly the lower part of the curve. The upper isotherm is at a temp say T1 and the lower isotherm is at T2. Now this elementary process can be thought of as a Carnot engine which receives heat dq1 at T1 and rejects heat dq2 at T2. So for this elementary cycle Signs of heat transfer then we get . If we add all the elementary cycles such that we cover the entire cyclic process then we get :

dq1

dq2

, hence is called entropy , ds.

R1

R2

a

b,=

Hence entropy is a property, independent of pathSo, or,

Clausius theorem

; If then the process is called rev adiabatic and then ds=0Hence s=Const;

T

S

A B

CD

𝑑𝑞

𝑑𝑞2

RA

AB is a general processEither rev or irreversible,All other processes are rev.

Consider an elemental cycle, as shown in the picture. = heat supplied at,

1−𝑑𝑞2𝑑𝑞𝑎𝑛𝑦

≤1−𝑑𝑞2𝑑𝑞𝑟𝑒𝑣

𝑑𝑞2𝑑𝑞𝑎𝑛𝑦

≥𝑑𝑞2𝑑𝑞𝑟𝑒𝑣

𝑑𝑞𝑎𝑛𝑦

𝑑𝑞2≤𝑑𝑞𝑟𝑒𝑣

𝑑𝑞2

=

Or,

Or,

Or,

Or,

For a reversible process: For any other process :

For any cyclic process:

, cycle is reversible,, cycle is irreversible and feasible, cycle is impossible

Clausius Inequality

Entropy principle

For any process undergone by the system, , if the process is reversible then

For isolated system, dq=0, so,

For irreversible process,

Causes of entropy increase: for a closed system entropy increases due to(a) External interaction, (b) internal irreversibility

𝑑𝑠=𝑑𝑠𝑒𝑥𝑡+𝑑𝑠𝑖𝑛𝑡 so,

𝑑𝑠𝑑𝑡

=˙𝑞𝑐𝑣

𝑇+ ˙𝑠𝑔𝑒𝑛

Entropy change Entropy transfer Entropy generation

dqT𝑠𝑔𝑒𝑛

𝑠2−𝑠1

For an open system �̇�𝑖

�̇�˙𝑠𝑔𝑒𝑛

�̇�𝑒

, for steady state

𝑚𝑖 𝑠𝑖+𝑞𝑐𝑣

𝑇+𝑠𝑔𝑒𝑛=𝑚𝑒𝑠𝑒+𝑚2𝑠2−𝑚1 𝑠1 Integral form of the equation

Change of entropy within the CV

system

relation

First law and second law combined:

𝑑𝑠=𝑑𝑢𝑇

+𝑝𝑇𝑑𝑣 ∫𝑑𝑠=∫ 𝑑𝑢

𝑇+∫ 𝑅

𝑣𝑑𝑣

𝑑𝑞𝑟𝑒𝑣=𝑇𝑑𝑠

𝑠2−𝑠1=∫𝑐𝑣𝑑𝑇𝑇

+∫ 𝑅𝑣𝑑𝑣

𝑠2−𝑠1=𝑐𝑣 ln (𝑇2

𝑇1)+𝑅 ln( 𝑣2𝑣1 )

1st Tds relation

Only for ideal gases

h=𝑢+𝑝𝑣h𝑑 =𝑑𝑢+𝑝𝑑𝑣+𝑣𝑑𝑝h𝑑 =𝑇𝑑𝑠+𝑣𝑑𝑝

T 2nd Tds relation

𝑑𝑠= h𝑑𝑇−𝑣𝑇𝑑𝑝𝑠2−𝑠1=∫

𝑐𝑝𝑑𝑇𝑇

−∫ 𝑅𝑝𝑑𝑝𝑐𝑝 ln (𝑇2

𝑇1)−𝑅 ln(𝑝2𝑝1 )Only for ideal gases