Corollaries Second law of thermodynamics

1 corollaries

1.1 perpetual motion of second kind
1.2 carnot theorem
1.3 clausius inequality
1.4 thermodynamic temperature
1.5 entropy
1.6 energy, available useful work

corollaries
perpetual motion of second kind

before establishment of second law, many people interested in inventing perpetual motion machine had tried circumvent restrictions of first law of thermodynamics extracting massive internal energy of environment power of machine. such machine called perpetual motion machine of second kind . second law declared impossibility of such machines.

carnot theorem

carnot s theorem (1824) principle limits maximum efficiency possible engine. efficiency solely depends on temperature difference between hot , cold thermal reservoirs. carnot s theorem states:

all irreversible heat engines between 2 heat reservoirs less efficient carnot engine operating between same reservoirs.
all reversible heat engines between 2 heat reservoirs equally efficient carnot engine operating between same reservoirs.

in ideal model, heat of caloric converted work reinstated reversing motion of cycle, concept subsequently known thermodynamic reversibility. carnot, however, further postulated caloric lost, not being converted mechanical work. hence, no real heat engine realise carnot cycle s reversibility , condemned less efficient.

though formulated in terms of caloric (see obsolete caloric theory), rather entropy, insight second law.

clausius inequality

the clausius theorem (1854) states in cyclic process

∮



δ
q

t


≤
0.


{\displaystyle \oint {\frac {\delta q}{t}}\leq 0.}

the equality holds in reversible case , < in irreversible case. reversible case used introduce state function entropy. because in cyclic processes variation of state function 0 state functionality.

thermodynamic temperature

for arbitrary heat engine, efficiency is:

η
=



w

n



q

h




=




q

h


−

q

c




q

h




=
1
−



q

c



q

h





(
1
)


{\displaystyle \eta ={\frac {w_{n}}{q_{h}}}={\frac {q_{h}-q_{c}}{q_{h}}}=1-{\frac {q_{c}}{q_{h}}}\qquad (1)}

where wn net work done per cycle. efficiency depends on qc/qh.

carnot s theorem states reversible engines operating between same heat reservoirs equally efficient. thus, reversible heat engine operating between temperatures t1 , t2 must have same efficiency, say, efficiency function of temperatures only:






q

c



q

h




=
f
(

t

h


,

t

c


)

(
2
)
.


{\displaystyle {\frac {q_{c}}{q_{h}}}=f(t_{h},t_{c})\qquad (2).}

in addition, reversible heat engine operating between temperatures t1 , t3 must have same efficiency 1 consisting of 2 cycles, 1 between t1 , (intermediate) temperature t2, , second between t2 andt3. can case if

f
(

t

1


,

t

3


)
=



q

3



q

1




=




q

2



q

3





q

1



q

2





=
f
(

t

1


,

t

2


)
f
(

t

2


,

t

3


)
.


{\displaystyle f(t_{1},t_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(t_{1},t_{2})f(t_{2},t_{3}).}

now consider case




t

1




{\displaystyle t_{1}}

fixed reference temperature: temperature of triple point of water. t2 , t3,

f
(

t

2


,

t

3


)
=



f
(

t

1


,

t

3


)


f
(

t

1


,

t

2


)



=



273.16
⋅
f
(

t

1


,

t

3


)


273.16
⋅
f
(

t

1


,

t

2


)



.


{\displaystyle f(t_{2},t_{3})={\frac {f(t_{1},t_{3})}{f(t_{1},t_{2})}}={\frac {273.16\cdot f(t_{1},t_{3})}{273.16\cdot f(t_{1},t_{2})}}.}

therefore, if thermodynamic temperature defined by

t
=
273.16
⋅
f
(

t

1


,
t
)



{\displaystyle t=273.16\cdot f(t_{1},t)\,}

then function f, viewed function of thermodynamic temperature, simply

f
(

t

2


,

t

3


)
=



t

3



t

2




,


{\displaystyle f(t_{2},t_{3})={\frac {t_{3}}{t_{2}}},}

and reference temperature t1 have value 273.16. (of course reference temperature , positive numerical value used—the choice here corresponds kelvin scale.)

entropy

according clausius equality, reversible process

∮



δ
q

t


=
0


{\displaystyle \oint {\frac {\delta q}{t}}=0}

that means line integral




∫

l





δ
q

t




{\displaystyle \int _{l}{\frac {\delta q}{t}}}

path independent.

so can define state function s called entropy, satisfies

d
s
=



δ
q

t





{\displaystyle ds={\frac {\delta q}{t}}\!}

with can obtain difference of entropy integrating above formula. obtain absolute value, need third law of thermodynamics, states s=0 @ absolute 0 perfect crystals.

for irreversible process, since entropy state function, can connect initial , terminal states imaginary reversible process , integrating on path calculate difference in entropy.

now reverse reversible process , combine said irreversible process. applying clausius inequality on loop,

−
Δ
s
+
∫



δ
q

t


=
∮



δ
q

t


<
0


{\displaystyle -\delta s+\int {\frac {\delta q}{t}}=\oint {\frac {\delta q}{t}}<0}

thus,

Δ
s
≥
∫



δ
q

t






{\displaystyle \delta s\geq \int {\frac {\delta q}{t}}\,\!}

where equality holds if transformation reversible.

notice if process adiabatic process,



δ
q
=
0


{\displaystyle \delta q=0}

,



Δ
s
≥
0


{\displaystyle \delta s\geq 0}

.

energy, available useful work

an important , revealing idealized special case consider applying second law scenario of isolated system (called total system or universe), made of 2 parts: sub-system of interest, , sub-system s surroundings. these surroundings imagined large can considered unlimited heat reservoir @ temperature tr , pressure pr — no matter how heat transferred (or from) sub-system, temperature of surroundings remain tr; , no matter how volume of sub-system expands (or contracts), pressure of surroundings remain pr.

whatever changes ds , dsr occur in entropies of sub-system , surroundings individually, according second law entropy stot of isolated total system must not decrease:

d

s


t
o
t



=
d
s
+
d

s

r


≥
0


{\displaystyle ds_{\mathrm {tot} }=ds+ds_{r}\geq 0}

according first law of thermodynamics, change du in internal energy of sub-system sum of heat δq added sub-system, less work δw done sub-system, plus net chemical energy entering sub-system d ∑μirni, that:

d
u
=
δ
q
−
δ
w
+
d
(
∑

μ

i
r



n

i


)



{\displaystyle du=\delta q-\delta w+d(\sum \mu _{ir}n_{i})\,}

where μir chemical potentials of chemical species in external surroundings.

now heat leaving reservoir , entering sub-system is

δ
q
=

t

r


(
−
d

s

r


)
≤

t

r


d
s


{\displaystyle \delta q=t_{r}(-ds_{r})\leq t_{r}ds}

where have first used definition of entropy in classical thermodynamics (alternatively, in statistical thermodynamics, relation between entropy change, temperature , absorbed heat can derived); , second law inequality above.

it therefore follows net work δw done sub-system must obey

δ
w
≤
−
d
u
+

t

r


d
s
+
∑

μ

i
r


d

n

i





{\displaystyle \delta w\leq -du+t_{r}ds+\sum \mu _{ir}dn_{i}\,}

it useful separate work δw done subsystem useful work δwu can done sub-system, on , beyond work pr dv done merely sub-system expanding against surrounding external pressure, giving following relation useful work (exergy) can done:

δ

w

u


≤
−
d
(
u
−

t

r


s
+

p

r


v
−
∑

μ

i
r



n

i


)



{\displaystyle \delta w_{u}\leq -d(u-t_{r}s+p_{r}v-\sum \mu _{ir}n_{i})\,}

it convenient define right-hand-side exact derivative of thermodynamic potential, called availability or exergy e of subsystem,

e
=
u
−

t

r


s
+

p

r


v
−
∑

μ

i
r



n

i




{\displaystyle e=u-t_{r}s+p_{r}v-\sum \mu _{ir}n_{i}}

the second law therefore implies process can considered divided subsystem, , unlimited temperature , pressure reservoir in contact,

d
e
+
δ

w

u


≤
0



{\displaystyle de+\delta w_{u}\leq 0\,}

i.e. change in subsystem s exergy plus useful work done subsystem (or, change in subsystem s exergy less work, additional done pressure reservoir, done on system) must less or equal zero.

in sum, if proper infinite-reservoir-like reference state chosen system surroundings in real world, second law predicts decrease in e irreversible process , no change reversible process.

d

s

t
o
t


≥
0


{\displaystyle ds_{tot}\geq 0}

equivalent



d
e
+
δ

w

u


≤
0


{\displaystyle de+\delta w_{u}\leq 0}

this expression associated reference state permits design engineer working @ macroscopic scale (above thermodynamic limit) utilize second law without directly measuring or considering entropy change in total isolated system. (also, see process engineer). changes have been considered assumption system under consideration can reach equilibrium reference state without altering reference state. efficiency process or collection of processes compares reversible ideal may found (see second law efficiency.)

this approach second law utilized in engineering practice, environmental accounting, systems ecology, , other disciplines.